Projective objects and the modified trace in factorisable finite tensor categories
Azat M. Gainutdinov, Ingo Runkel
TL;DR
This paper explores the structure of projective objects in factorisable finite tensor categories, establishing their properties, actions, and traces, with applications to symplectic fermion categories and a Verlinde-like formula.
Contribution
It introduces new results on the existence and properties of projective objects, their internal characters, and the modified trace in such categories, including a Verlinde-like formula.
Findings
Existence of a simple projective object in C
Internal characters of projectives span a submodule for SL(2,Z) action
Modified trace of open Hopf links expressed via S-matrix elements
Abstract
For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple. Results 1-3 remain true in positive characteristic under an extra assumption. Result 1 implies that the tensor ideal of projective objects in C carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of S-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
ZMP-HH/17-11
Hamburger Beiträge zur Mathematik 651
Projective objects and the modified trace
in factorisable finite tensor categories
Azat M. Gainutdinov a,b and Ingo Runkel b ***Emails: [email protected], [email protected]
a Institut Denis Poisson, CNRS, Université de Tours, Université d’Orléans,
Parc de Grandmont, 37200 Tours, France
b Fachbereich Mathematik, Universität Hamburg
Bundesstraße 55, 20146 Hamburg, Germany
Abstract
For a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:
always contains a simple projective object; 2. 2.
if is in addition ribbon, the internal characters of projective modules span a submodule for the projective -action; 3. 3.
the action of the Grothendieck ring of on the span of internal characters of projective objects can be diagonalised; 4. 4.
the linearised Grothendieck ring of is semisimple iff is semisimple.
Results 1–3 remain true in positive characteristic under an extra assumption. Result 1 implies that the tensor ideal of projective objects in carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of -matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular -transformation restricted to internal characters of projective objects.
We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.
Keywords: Braided and pivotal categories, projective objects, modified traces,
Grothendieck rings, Vertex-Operator Algebras, Verlinde-like formula.
MSC codes: 18D10, 18G05, 18F30, 17B69.
Contents
- 1 Introduction
- 2 Factorisable finite tensor categories
- 3 Existence of a simple projective object
- 4 Non-degenerate trace on the projective ideal
- 5 Symmetric algebras and ideals in
- 6 Characters and the modified trace
- 7 Properties of the Reynolds and Higman ideals
- 8 -invariance of projective characters
- 9 Example: symplectic fermions
- A Proofs of Lemma 2.1 and Corollary 2.3
- B Proofs of Lemma 5.6 and Proposition 5.8
1 Introduction
Let be a finite-dimensional quasi-triangular Hopf algebra over a field and denote by its universal -matrix. is called factorisable if the map (the Drinfeld map [Dr]) is bijective. If is factorisable and ribbon, the centre of carries a projective representation of , the mapping class group of the torus [LM]. In this case, the Drinfeld map furthermore allows one to give a faithful representation of the -linear Grothendieck ring of on [Dr].
A factorisable Hopf algebra can be endowed with a central form such that it becomes a symmetric Frobenius algebra [CW, Rem. 3.1]. The central form induces an isomorphism between and the space of all central forms on :
[TABLE]
Characters of finite-dimensional -modules are examples of central forms. We are particularly interested in the preimage in of the characters of projective modules. This subspace is called the Higman ideal (see e.g. [Br]).
Denote by the category of finite-dimensional representations of . In [CW], Cohen and Westreich prove, amongst other things, the following remarkable results.
Theorem 1.1** ([CW]).**
Let be a factorisable ribbon Hopf algebra over an algebraically closed field of characteristic zero.
There is at least one simple and projective -module. 2. 2.
* is an -submodule of .* 3. 3.
* is a submodule for the action of the Grothendieck ring of on , and this action can be diagonalised on (“the fusion rules can be diagonalised on ”).*
Part 2 generalises [La1, Thm. 5.5], [La2, Cor. 4.1] established for small quantum groups associated to simple Lie algebras of ADE type.
The motivation of the present paper is to give a generalisation of these results to factorisable finite tensor categories. Let us describe our setting in more detail.
Let be a field. An abelian -linear category is called finite if it is equivalent to the category of finite-dimensional representations of a finite-dimensional -algebra. A finite tensor category is a finite abelian category which is rigid monoidal with bilinear tensor product and simple tensor unit. If the category is equipped with a natural monoidal isomorphism from the identity functor to the endofunctor given by taking double duals, it is called pivotal. An important class of examples of finite tensor categories are the categories for finite-dimensional Hopf algebras . A Hopf algebra is called pivotal if the squared antipode of is given by conjugating with a group-like element [AAGTV]. In this case, is pivotal. A ribbon Hopf algebra is automatically pivotal.
For a braided finite tensor category one can write down at least four natural non-degeneracy conditions for the braiding. These have only recently been shown to all be equivalent [Sh3], and we briefly review this result in Section 2. We refer to braided finite tensor categories satisfying these equivalent conditions as factorisable. Indeed, for a quasi-triangular finite-dimensional Hopf algebra , is factorisable iff is factorisable [Ly1].
Our first main result generalises part 1 of Theorem 1.1 (see Theorem 3.4).
Theorem 1.2**.**
A factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero contains a simple projective object.
Remark 1.3**.**
One can formulate the above theorem for positive characteristic if one adds an extra assumption, which we call “Condition P” (Section 3). Namely, we say that a finite braided tensor category satisfies Condition P if there exists a projective object such that is not nilpotent in the linearised Grothendieck ring (that is, in the Grothendieck ring tensored with the field). In characteristic zero, Condition P is always satisfied (Lemma 3.2). Theorems 1.2, 1.4 and part 1 of Theorem 1.5 remain true if instead of requiring the field to be of characteristic zero, one demands that the category satisfies Condition P (but for part 2 of Theorem 1.5 we still need to require characteristic zero). The details are given in the body of the paper.
One noteworthy consequence is that the category in the above theorem allows for a unique-up-to-scalars non-zero modified trace function on the tensor ideal , that is, on the full subcategory of all projective objects in [GKPM1, GPMV, GKPM2], see Section 4.
A modified trace on is a family of -linear maps , , subject to a cyclicity and partial trace condition. Recall that the categorical trace defined in terms of the pivotal structure on vanishes identically on unless is semisimple (Remark 4.6). In contrast to this, for the modified trace the pairings , are non-degenerate for all (see [CGPM] and Proposition 4.2 below). In particular, the turn into a Calabi-Yau category (Definition 5.4).
One can use modified traces to define new link invariants in finite ribbon categories which are not accessible via the usual quantum traces [GPMT]. However, we will not pursue this point in the present paper.
Let be a finite abelian category over a field and assume that is Calabi-Yau. In Section 5 we give a categorical definition of the Higman ideal as an ideal
[TABLE]
in the algebra of natural endomorphisms of the identity functor on . Recall that for algebras, , i.e. the centre of is isomorphic as a -algebra to the algebra of natural endomorphisms of the identity functor on the category of finite-dimensional (right, say) -modules. The definition of is such that via this isomorphism, gets mapped to .
As in Theorem 1.2 above, let be a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero. Let be a projective generator of and . Then is equivalent as a -linear category to . As discussed above, we obtain a modified trace which turns into a symmetric Frobenius algebra, and as in (1.1) we get isomorphisms
[TABLE]
Our second main result is Theorem 6.1, which links characters of -modules to the modified trace over certain elements in obtained from the internal character of the corresponding object in . The details of this are too lengthy for this introduction, and we refer to Section 6.
Theorem 6.1 is important for what follows as it implies that is spanned by the images of internal characters of projective objects (Proposition 7.1).
For a factorisable finite ribbon category one obtains a projective -action on , and in fact actions of all surface mapping class groups on appropriate Hom-spaces [Ly2].
Our third main result generalises part 2 of Theorem 1.1 (see Corollary 8.5)
Theorem 1.4**.**
Let be a factorisable finite ribbon category over an algebraically closed field of characteristic zero. Then is an -submodule of .
We note that under the conditions of this theorem, iff is semisimple (Proposition 5.10).
Still under the conditions of the above theorem, one can define an injective ring homomorphism from the Grothendieck ring to [Sh2] (see Section 2). This, in particular, turns into a faithful -module by defining the representation map to be left multiplication. We show the following statement, the first part of which generalises part 3 of Theorem 1.1 (see Proposition 8.6).
Theorem 1.5**.**
Let be a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero.
The restriction of to the submodule is diagonalisable. 2. 2.
The action can be diagonalised on if and only if is semisimple.
See Proposition 8.6 for an explicit choice of basis diagonalising the -action. In particular, the -linear Grothendieck ring is semisimple iff is semisimple (Corollary 8.7).
We also investigate in Section 8 how the restriction of the projective -action to can be used to gain information about the decomposition of tensor products of projective objects, see Proposition 8.8 for details. This can be thought of as a non-semisimple variant of the Verlinde formula [Ve]. See e.g. [FHST, Fu, GR2, CG] for discussions of the Verlinde formula in the finitely non-semisimple setting, and [CW] for a related result in the context of factorisable ribbon Hopf algebras. As a corollary of the projective -action on , we also give a purely categorical counterpart of the conjectural relation between modular properties of pseudo-trace functions and modified trace of Hopf link operators [CG] — we prove a special case of this (namely when all labels are projective) in Proposition 8.10.
Remark 1.6**.**
A semisimple factorisable finite ribbon category is a modular tensor category [Tu, BK]. Since the qualifier “modular” refers to the projective action of the modular group , it would also be reasonable to refer to modular tensor categories in the sense of [Tu, BK] as modular fusion categories, and to call factorisable finite ribbon categories modular finite tensor categories. An important application of modular fusion categories is that they are precisely the data needed to construct certain three-dimensional topological field theories [RT, Tu, BDSPV]. For modular finite tensor categories in the above sense, a corresponding result is not known. However, a version of three-dimensional topological field theory build form such categories is studied in [KL]. 2. 2.
So far, we have motivated our results from the theory of Hopf algebras. Another reason to look at factorisable finite tensor categories is provided by vertex operator algebras. A particularly benign class of vertex operator algebras are the so-called -cofinite ones, for which we in addition require that they are simple, non-negatively graded and isomorphic to their contragredient module. Let be such a vertex operator algebra.
If , the category of (quasi-finite dimensional, generalised) -modules, is in addition semisimple, it is in fact a modular fusion category [Hu2]. What is more, it is proved in [Zh] that the characters of -modules form a projective representation of the modular group, which by [Hu1] agrees with the representation obtained from the modular fusion category .
If is not semisimple, it is known from [HLZ, Hu3] that satisfies all properties of a finite braided tensor category except for rigidity (which is, however, conjectured in [Hu4, GR2] and proven in a different setting for models in [TW]). It is furthermore conjectured that also in the non-semisimple case, is factorisable [GR2, CG]. For the modular group action one now has to pass to pseudo-trace functions [Mi, AN], and it is tempting to conjecture that this modular group action agrees with the one on discussed above, see [GR2, Conj. 5.10] for a precise formulation. For so-called symplectic fermion conformal field theories [Ka, GK, Ab], the modular group actions obtained from pseudo-trace functions and from have been compared in [GR1, FGR2] and they indeed agree.
If these conjectures were true, Theorems 1.2 and 1.4 would imply that such a always has at least one simple and projective module, and that the characters (not the pseudo-trace functions) of projective modules transform into each other under the modular group action. Both properties hold in the very few non-semisimple examples where they can be verified [Ka, FHST, FGST, NT, Ab].
The study of properties of factorisable tensor categories and in particular of is also important input for the construction of bulk conformal field theory correlators in the non-semisimple setting as in [FS1, FSS, FS2].
We conclude the paper in Section 9 with the computation of the modified trace in a class of examples of factorisable finite ribbon categories [DR1, Ru] arising in symplectic fermion conformal field theories. We also illustrate in this example how the projective -action on the Higman ideal can be used to obtain information about the tensor product of projective objects.
Acknowledgements: We thank Anna Beliakova, Christian Blanchet, Alexei Davydov, Jürgen Fuchs, Nathan Geer, Jan Hesse, Dimitri Nikshych, Ehud Meir, Victor Ostrik and Christoph Schweigert for helpful discussions, and we are grateful to Miriam Cohen, Alexei Davydov, Ehud Meir and Sara Westreich for comments on a draft of this paper. The work of AMG was supported by DESY and CNRS.
Convention: Through this paper, denotes an algebraically closed field, possibly of positive characteristic.111 Some definitions and results of this paper do not actually require algebraic closedness of . For example, algebraic closedness is not required in the first half of Section 2, up to and excluding Theorem 2.6, or in the definition of a modified right trace in Section 4, etc. But to avoid confusion we prefer to require algebraic closedness for the entire paper.
2 Factorisable finite tensor categories
In this section we introduce notation and recall some definitions and results that will be used later, such as the different characterisations of a factorisable finite tensor category. The reader is invited to just skim through this section and return to it when necessary.
For an essentially small abelian category we write (see e.g. [EGNO] for more details)
- •
: a choice of representatives for the isomorphism classes of simple objects of , i.e. every simple object is isomorphic to one and only one element of ,
- •
: the full subcategory of consisting of all projective objects,
- •
: the Grothendieck group of . We write for the class of in the Grothendieck group,
- •
: if is -linear we set . Note that the canonical ring homomorphism is injective iff has characteristic zero.
A -linear abelian category is called locally finite if all Hom-spaces are finite dimensional over and every object of has finite length. A locally finite abelian category is called finite if is finite and every simple object has a projective cover. For a finite abelian category we set
- •
: a choice of projective cover for each simple .
One can show that a -linear abelian category is finite iff it is equivalent as a -linear category to the category of finite-dimensional modules over a finite-dimensional -algebra.
For a finite abelian category we have . (On the other hand, if infinite direct sums are allowed in we have because of the short exact sequence .)
A monoidal category is called rigid if for each there is (the left dual) and (the right dual), together with morphisms
[TABLE]
where is the tensor unit. These morphisms have to satisfy the zig-zag identities, see e.g. [EGNO] for more details. We will later use string diagram notation. Our diagrams will be read bottom to top and the above duality morphisms will be written as
[TABLE]
A monoidal category with left duals is pivotal if it is equipped with a natural monoidal isomorphism
[TABLE]
It is then automatically rigid via and (we omit the tensor product between objects for better readability)
[TABLE]
Let be a monoidal category with right duals. Below we will make use of the functor , , and of dinatural transformations between this functor and the constant functor with value for some . We refer to [ML] for dinatural transformations in general and to [FS1, Sec. 4] or [FGR1, Sec. 3] for these specific functors.
The following lemma is a slight generalisation of [GKPM1, Lem. 2.5.1] which in turn follows [De]. It is proven in the same way as in [GKPM1] and we relegate the proof to Appendix A.
Lemma 2.1**.**
Let be an abelian monoidal category with right duals and biexact tensor product functor. Suppose that there is a projective object and a surjection . Let and let be a dinatural transformation from to . Consider the following commuting diagram in with exact rows:
[TABLE]
In this case, the following equality of morphisms holds for any choice of endomorphisms , , and that make (2.5) commute:
[TABLE]
Remark 2.2**.**
A corresponding statement as that in Lemma 2.1 holds for a dinatural transformation from to . In this case, for as in (2.5) we have
[TABLE] 2. 2.
As a consequence of Lemma 2.1, the morphism only depends on the class . Indeed, if one chooses in (2.5) to be identities, then (2.6) states that given a short exact sequence we have the identity
[TABLE]
of morphisms in . For example, if is in addition pivotal, one can take and the dinatural transformation . One obtains the statement that the quantum dimension only depends on the class in .
A finite tensor category is a category which (see [EO])
- •
is a -linear finite abelian category,
- •
is a rigid monoidal category with -bilinear tensor product functor,
- •
has a simple tensor unit.
The tensor product functor of an abelian rigid monoidal category is automatically biadditive and biexact [EGNO, Prop. 4.2.1]. A finite tensor category is called unimodular if , see [ENO, EGNO] for more details.
The following technical corollary to Lemma 2.1 generalises [GKPM1, Cor. 2.5.2] and will be needed later. The proof is given in Appendix A.
Corollary 2.3**.**
Let be a finite tensor category over . Let and let be a dinatural transformation from to and from to . Let and be such that for all simple and all we have . Then:
* and .* 2. 2.
If is in addition braided and pivotal, then for any ,
[TABLE]
For the rest of this section, we fix
- •
: a braided and pivotal finite tensor category over .
Our notation for the coherence and braiding isomorphisms in is, for ,
[TABLE]
We write for the natural endomorphisms of the identity functor on .
Given , consider the natural transformation : for all ,
[TABLE]
where in (*) we used properties of the braiding to deform the string diagram, as well as pivotality to exchange left and right duality morphisms. This natural transformation appears in [Tu, Sec. I.1.5] and also in [CGPM, CG] where it is called “general Hopf link” and “open Hopf link operator”, respectively (see [GR2, Rem. 3.10 (2)] for the conventions used here).
Lemma 2.4**.**
* only depends on the class of in , and the resulting map is a ring homomorphism.*
Proof.
Let be arbitrary and consider the dinatural transformation from to given by (see also [FGR1, Sec. 4.4])
[TABLE]
Note that . By Lemma 2.1 (or rather Remark 2.2 (2)), the right hand side factors through , hence so does . That the resulting map is a ring map follows from the identity , which one can verify by a straightforward calculation [Tu, Sec. 4.5] (see also [CG, Sec. 3.1]). ∎
By abuse of notation, we also use the symbol for the resulting ring homomorphism
[TABLE]
Let be the coend (see e.g. [KL] or the review in [FS1, Sec. 4] or in [FGR1, Sec. 3])
[TABLE]
and denote by , , the corresponding dinatural transformation. The coend exists since is a finite tensor category (see e.g. [KL, Cor. 5.1.8]). It carries the structure of a Hopf algebra in and is equipped with a Hopf pairing [Ly1]. We denote the product, coproduct, unit, counit and antipode of the Hopf algebra by , , , , , respectively (we use the conventions in [FGR1, Sec. 3]).
If is unimodular, the coend admits a two-sided integral [Sh1, Thm. 6.8].222 For factorisable (see Definition 2.9 below), the existence of two-sided integrals has been shown in [Ly1, Thm. 6.11], see also [KL, Cor. 5.2.11]. The integral is unique up to a scalar factor. Dually, admits a two-sided cointegral which can be normalised such that (see e.g. [KL, Sec. 4.2.3]).
We will need the linear maps
[TABLE]
which are defined as follows.
- •
For , the value is defined uniquely by the universal property of the coend via, for all ,
[TABLE]
- •
is defined via the Hopf pairing of as, for ,
[TABLE]
- •
The definition of requires to be unimodular and the choice of a non-zero integral . In this case we set, for ,
[TABLE]
The bialgebra structure on allows one to endow and with the structure of a -algebra. is equally a -algebra. We have:
Lemma 2.5**.**
* is an isomorphism of -algebras.* 2. 2.
* is a -algebra homomorphism (but not necessarily an isomorphism).* 3. 3.
For unimodular, is an isomorphism (but not necessarily a -algebra homomorphism).
For the proof of this lemma, see e.g. [Ly1], [KL, Prop. 5.2.5], [KL, Cor. 4.2.13] and [Sh3, Sec. 3.1] (and also [GR2, Sec. 2]). One verifies that the inverses of and are, for , , ,
[TABLE]
The internal character of is the element given by [FS1, Sh2] (we use the convention in [GR2])
[TABLE]
By Lemma 2.1, this map factors through the Grothendieck ring, i.e. we obtain a map
[TABLE]
By abuse of notation, we will denote the induced map , by as well.
Theorem 2.6** ([FS1, Sec. 4.5] and [Sh2, Thm. 3.11, Prop. 3.14, Cor. 4.2, Thm. 5.12]).**
The linear map is an injective -algebra homomorphism. 2. 2.
Suppose in addition that is unimodular. Then is surjective iff is semisimple.
This theorem remains true for not braided (but still a pivotal finite tensor category over an algebraically closed field).
Define, for unimodular and ,
[TABLE]
After substituting the definitions, one can check that, for all ,
[TABLE]
As only depends on , so does . Since and are isomorphisms, by Theorem 2.6 the set is linearly independent in . We note that since is not necessarily an algebra map, neither is the linear map , .
We define the linear map , the modular -transformation, as
[TABLE]
We have seen that and are algebra maps, while is in general not. Thus in general is not an algebra map, either. Recall the definition of in (2.13). A straightforward calculation shows that . Combining this with (2.22) and (2.24) gives, for all ,
[TABLE]
cf. [GR2, Rem. 3.10 (2)]. In particular, by Lemma 2.4, the combination is an algebra homomorphism from to .
Remark 2.7**.**
Since the coend is unique up to unique isomorphism, and the cointegral is unique up to sign, it is straightforward to verify that the , and depend on the choice of only up to an overall sign, see [FGR1, Prop. 5.3]. In particular, is independent of the choice of , as we already saw explicitly in (2.25).
Next we introduce four equivalent non-degeneracy requirement for the braiding. (This does not require a pivotal structure, hence we use the letter and reserve for the pivotal case as declared above.)
Theorem 2.8** ([Sh3, Thm. 1.1]).**
Let be a braided finite tensor category over an algebraically closed field. The following conditions are equivalent:
Every transparent object in is isomorphic to a direct sum of tensor units. (* is transparent if for all , .)* 2. 2.
The canonical braided monoidal functor is an equivalence. (Here, is the Deligne product, is the same tensor category as , but has inverse braiding, and is the Drinfeld centre of .) 3. 3.
The pairing is non-degenerate (in the sense that there exists a copairing ). 4. 4.
* is an isomorphism.*
Definition 2.9**.**
as in Theorem 2.8 is called factorisable if it satisfies the equivalent conditions 1–4 there.
In [ENO, Prop. 4.5] the following result is proved (using formulation 2 of factorisability in Theorem 2.8):
Theorem 2.10**.**
Let be factorisable. Then is unimodular.
In particular, for factorisable the Hopf algebra given by the coend admits integrals and cointegrals. Moreover, in this case we can normalise the integral such that
[TABLE]
see e.g. [KL, Sec. 5.2.3]. Since the space of integrals is one-dimensional, this determines up to a sign.
Remark 2.11**.**
Suppose is factorisable. Then by the above theorem, is unimodular and we have the isomorphism from (2.18) at our disposal (Lemma 2.5). Hence the linear endomorphism of in (2.24) is defined, and by condition 4 in Theorem 2.8, is invertible.
Using Lemma 2.4 and (2.22), (2.25), we get the following corollary to Theorem 2.6.
Corollary 2.12**.**
Let be factorisable. Then the algebra map is injective. If in addition , the ring homomorphism is injective, too.
The following important theorem is proved in [Ly1], see also [FGR1, Sec. 5.1] for a summary in the notation used here.
Theorem 2.13**.**
Let be factorisable and ribbon with ribbon twist . Then there is a projective representation of on for which the and generators act as
[TABLE]
Remark 2.14**.**
For for a finite-dimensional ribbon Hopf algebra over , we have , and the coend is given by with the coadjoint action [Ly2, Ke2]. Let be the antipode of , the universal R-matrix, the ribbon element, and the Drinfeld element. We will (for the sake of this remark) identify with the centre of . Then we have:
- •
The internal characters are the -characters: the images are linear forms on invariant under the coadjoint action. They are the trace functions introduced and studied in [Dr], see also [Ke1].
- •
The map is then the Drinfeld mapping from the space of -characters to given by for the monodromy matrix , see [Dr, Prop. 1.2]. The central elements (where is the left regular -module) are the images of under the Drinfeld mapping composed with .
- •
The map is the Radford mapping from the space of -characters to given by , for the (co)integral (that can be computed from using the duality maps), see [Ra] for properties of this map. The central elements from (2.23) are the images of under the Radford mapping composed with , see also [FGR1, Rem. 7.10].
- •
The modular -transformation on a quasi-triangular Hopf algebra was introduced in [LM] following the categorical construction of [Ly1]. The images of under the Drinfeld and Radford mappings are related by the modular -transformation [Ke1, Sec. 2], see [FGST, Sec. 5] for the statement in this form. This result is generalised by (2.25).
3 Existence of a simple projective object
Throughout the rest of this paper, the following technical condition will play an important role.
Condition P: A finite tensor category over satisfies Condition P if there exists a projective object such that is not nilpotent in the linearised Grothendieck ring .
Two important classes of categories which satisfy Condition P are described in the next two lemmas.
Lemma 3.1**.**
Every semisimple finite tensor category satisfies Condition P.
Proof.
Since is semisimple, every simple object is projective. Furthermore, for every simple object , is non-zero in . Hence we may take , the tensor unit. Then for all , in . ∎
Lemma 3.2**.**
Let be a locally finite braided tensor category over some field. For all , and we have in . If in addition is finite and if the field is of characteristic zero, then satisfies Condition P.
Proof.
Since by assumption all objects have finite length, we have . Thus for it follows that iff .
The map is non-zero, hence , and so . Iterating this argument shows , i.e. , etc. We used here Drinfeld’s canonical isomorphism given by
[TABLE]
Using commutativity of , after steps we find .
As a finite has enough projectives, it contains a non-zero projective object . Since in characteristic zero, the canonical ring homomorphism is injective, for all also holds in . ∎
Remark 3.3**.**
In positive characteristic, Condition P may or may not be satisfied. For example, suppose that is of characteristic and let be a finite group. If does not divide , the category -mod of finite-dimensional -modules is semisimple and hence satisfies Condition P by Lemma 3.1.
On the other hand, if is a -group the trivial -module is up to isomorphism the unique simple -module (see e.g. the corollary to Proposition 26 in Section 8.3 of [Se]). The projective cover of the trivial -module is (see [Se, Sec. 15.6]). The image of in is , as divides . By the previous observations, every projective -module is isomorphic to a direct sum of ’s, and so -mod does not satisfy Condition P.
We can now state our first main result:
Theorem 3.4**.**
Let be a factorisable and pivotal finite tensor category over . If satisfies Condition P, it contains a simple projective object.
The proof relies on a series of lemmas and will be given at the end of this section. The idea is very simple: we show that in the absence of simple projectives, the injective algebra homomorphism from Corollary 2.12 would map the class of a projective object to a nilpotent natural endomorphism, which is a contradiction to Condition P.
Remark 3.5**.**
Theorem 3.4 is a generalisation of a result by Cohen and Westreich [CW, Cor. 3.6]. There, the authors show that a factorisable ribbon Hopf algebra over an algebraically closed field of characteristic zero has an irreducible projective module. The proof strategy in [CW] is different from ours: in [CW] it is observed that Hopf algebras with the above properties are also symmetric algebras (see Section 4 for the definition) and that fact is used to show existence of a simple projective module. Our approach is in some sense opposite: we first show that there exists a simple projective module and then in Section 4 use the theory of modified traces of [GKPM1] to deduce the existence of certain symmetric algebras. In this sense, applying Theorem 3.4 to gives an alternative proof of the result in [CW] (in addition to having the benefit of not being restricted to characteristic zero). 2. 2.
In characteristic , a category as in Theorem 3.4 may or may not satisfy Condition P. To see this, we use the observations in Remark 3.3. Let be a finite abelian group. The Drinfeld double is a factorisable Hopf algebra, and hence -mod is a factorisable and pivotal finite tensor category over . But as algebras, , and is semisimple.
Suppose that does not divide . Then -mod is semisimple and hence satisfies Condition P (Lemma 3.1).
On the other hand, if is a -group, the indecomposable projective modules in -mod are isomorphic to a simple -module tensored with . As in Remark 3.3, each of these have zero image in . 3. 3.
The example in part 2 (combined with Remark 3.3) also shows that for and a finite abelian -group, -mod does not contain a simple projective object. Thus we cannot drop Condition P from Theorem 3.4.
We now turn to the proof of Theorem 3.4. We first gather the tools to see that is nilpotent for projective unless there is a simple projective object.
Lemma 3.6**.**
Let be an abelian category and let be a natural endomorphism of the identity functor. If is zero on all simple objects, then it is nilpotent on all objects of finite length.
Proof.
We use induction on the length of an object. Given an object of finite length , choose a short exact sequence such that have smaller length. Then
[TABLE]
commutes. By induction hypothesis, and are nilpotent. Hence there is such that
[TABLE]
commutes. Then and , and since the image of is the kernel of we have the inclusions . This shows that . ∎
Corollary 3.7**.**
Let be a finite abelian category over some field and let . If is zero on all simple objects, then is nilpotent, i.e. there is such that .
Proof.
By finiteness, contains a projective generator (of finite length). By Lemma 3.6, for some . Since determines , it follows that . ∎
Lemma 3.8**.**
Let be a pivotal finite braided tensor category over some field and let with projective and non-projective and simple. Then .
Proof.
By (2.11), is a composition of maps . Since is projective, the map factors as , for the projective cover of . Thus can be written as a composition . By assumption , and so this composition is zero. ∎
Combining the above results, the proof of Theorem 3.4 is now a straightforward consequence of Corollary 2.12:
Proof of Theorem 3.4.
333 We thank Victor Ostrik for suggesting a simplification of our first version of the proof.
Suppose that does not contain a simple projective object. By Condition P, we can find a projective object such that in for all .
By Corollary 2.12, is an injective algebra homomorphism. In particular, since we find that for each , . But by Lemma 3.8, the natural transformation is zero on all non-projective simples. By assumption, all simples in are non-projective, and so Corollary 3.7 implies that is nilpotent, which is a contradiction.
Hence must contain a simple projective object. ∎
4 Non-degenerate trace on the projective ideal
In this section we will review some aspects of the theory of modified traces from [GKPM1, GPMV, GKPM2]. In particular, we will see that the existence of a simple projective object established in Theorem 3.4 guarantees the existence of a unique-up-to-scalars non-zero modified trace on the projective ideal, and that this trace induces non-degenerate pairings on Hom-spaces.
Let be a pivotal finite tensor category over . The full subcategory of all projective objects in is a tensor ideal. Following [GPMV], a modified right trace on is a family of -linear functions such that
(partial trace) For all , and we have
[TABLE]
where denotes the partial right trace
[TABLE] 2. 2.
(cyclicity) For all , , we have
[TABLE]
Given a modified right trace, one can define a family of pairings by setting, for all , ,
[TABLE]
Note that the cyclicity property (4.3) only applies when also is projective.
Remark 4.1**.**
There are also natural notions of left and left-right (or two-sided) modified traces for a pivotal category, see [GPMV] for details. For in addition ribbon, each modified right trace on is also a two-sided modified trace [GKPM1, Thm. 3.3.1]. In this case, one can construct an isotopy invariant of ribbon graphs in , where at least one strand is coloured by a projective object of , by cutting the strand and computing of the corresponding endomorphism of , see [GPMT, Thm. 3]. (There, the modified trace was used implicitly as the modified dimension for simple projectives.)
In this paper we will only use modified right traces, and we will refer to these just as “modified traces”.
Proposition 4.2**.**
Let be a unimodular pivotal finite tensor category over . Suppose that contains a simple projective object.
There exists a nonzero modified trace on , and this modified trace is unique up to scalar multiples. 2. 2.
For any choice of non-zero modified trace on and for all , , the pairing (4.4) is non-degenerate.
The first part of the proposition is proved in [GKPM2, Cor. 3.2.1]444 In [GKPM2, Sect. 3.2] it is assumed that every indecomposable object of is absolutely indecomposable (see [GKPM2] for definitions). This condition does typically not hold (e.g. in the example in Section 9). However, the arguments in [GKPM2, Sect. 3.2] actually only require this condition on , where it holds for any finite tensor category over an algebraically closed field.. We will review the argument below as we hope it to be helpful to have the more general construction of [GPMV, GKPM2] specialised to the present setting, and since we will need the explicit construction of the modified trace below anyway. The second part of the proposition is proved in [CGPM, Prop. 6.6] for the Hopf algebra , but the same proof works in general as we also review below.
Before turning to the proof of Proposition 4.2, we need some preparation. For the rest of this section, we fix:
- •
: a unimodular pivotal finite tensor category over ,
- •
: a simple projective object in (in particular we assume that such a exists).
Lemma 4.3** ([GKPM2, Thm. 3.1.3 & Sec. 3.2]).**
For all we have
[TABLE]
Proof.
Write and let and be the corresponding embedding and projection maps. The resulting idempotent will be denoted by . Since is one-dimensional ( being simple and algebraically closed), we have .
Since is indecomposable, there is an and a nilpotent element such that (using again algebraically closedness of )555 Since is indecomposable, by Fitting’s Lemma an element is either nilpotent or invertible. As is algebraically closed, is not invertible for some (as the -linear map on has an eigenvector), hence it is nilpotent, and therefore , as we state.
[TABLE]
For all and we have
[TABLE]
To see , first note that by unimodularity (i.e. since ) is one-dimensional. Suppose and hence forms a basis of . Then there is such that . But is nilpotent, say , and so applying on both sides of the latter equality times we get and therefore . For the argument is analogous.
Next insert the identity into (4.5). Since for , and (using unimodularity), only the term with will contribute on either side of (4.5).
The claim of the lemma now follows from the observation that taking and we have by (4.6) and (4.7)
[TABLE]
where we used that and (because ). Both the left and right hand sides of (4.5) are therefore equal to . ∎
The above lemma will be used in the following form: for all ,
[TABLE]
This is a consequence of (4.5) upon substituting
[TABLE]
Let us fix a non-zero linear function for the given simple projective ,
[TABLE]
Since , such a function is unique up to a constant, and it is uniquely determined by its value .
Let and choose an object such that there is a surjection . Such an always exists, for example . As is projective, there is a morphism such that . That is, realise as a direct summand of . Consider the function
[TABLE]
The next lemma shows in particular that this notation is indeed consistent with (4.11) when setting (in the lemma, take and (inverse) unit isomorphisms).
Lemma 4.4**.**
The function in (4.12) is independent of the choice of and of , .
Proof.
This proof is taken from [GPMV, Sec. 4.5]. The argument is more general than needed for the statement of the lemma, but we can reuse it later to show cyclicity.
Let and pick , , . For and arbitrary, consider the morphism given by
[TABLE]
It is easy to check that inserting into (4.9) results in the identity
[TABLE]
where for RHS we used cyclicity of the categorical trace, the zig-zag axiom and that . For and , , this shows the statement of the lemma. ∎
Proof of Proposition 4.2.
Part 1. (following [GPMV, Sec. 4.5])
Existence: By Lemma 4.4 we have a family of functions \big{(}t_{P}:\operatorname{End}(P)\to k\big{)}_{P\in\mathcal{P}roj(\mathcal{C})}. This family is not identically zero since by the definition in (4.11), is not zero. By (4.14) in the proof of Lemma 4.4, the satisfy the cyclicity requirement (4.3).
For the partial trace property (4.1) choose such that there are , , , in order to compute from the definition in (4.12) with replaced by . Then fix
[TABLE]
to compute from (4.12). A short calculation shows (4.1).
Uniqueness: First note that for the modified trace in (4.12) (or in fact for any modified trace) we have, for ,
[TABLE]
To see this, apply to both sides. Let now \big{(}t_{P}^{\prime}:\operatorname{End}(P)\to k\big{)}_{P\in\mathcal{P}roj(\mathcal{C})} be a modified trace. Let and be given. Pick , , such that and compute
[TABLE]
Part 2. (following the proof of [CGPM, Prop. 6.6])
We will show non-degeneracy in the first argument of the pairing (4.4). Non-degeneracy in the second argument can be seen analogously.
Let , and , , be given. We need to show that there is a such that . Pick , , such that . Then also . Define
[TABLE]
Using the zig-zag identity, we can recover from as
[TABLE]
In particular, since , we must have . Since is simple, is surjective. Since is projective, there is such that . Define
[TABLE]
Then
[TABLE]
where for the first equality we used the partial trace property (4.1) on RHS and then the pivotal structure in (2). ∎
Lemma 4.5**.**
Suppose is in addition braided. Let be a modified trace on and let and , . Then
[TABLE]
Proof.
One computes
[TABLE]
where
[TABLE]
Step (*) follows from cyclicity of the modified trace, as well as naturality of the braiding. In step (**) we used that the partial trace over amounts to closing the double string, see (4.2). In the last equality, to straighten the line coloured by we also used (2) and the property of the pivotal structure that . ∎
Remark 4.6**.**
The result of Proposition 4.2 is remarkable, because the categorical trace defined on all of via the pivotal structure vanishes on unless is semisimple. Indeed, is semisimple iff , which in turn is equivalent to the existence of such that is non-zero. But for any projective object , the categorical trace
[TABLE]
factors through in this way. 2. 2.
For each , turns the -algebra into a symmetric Frobenius algebra, or a symmetric algebra for short. We will review some aspects of symmetric algebras and their categories of modules in Section 5.
Since a factorisable finite tensor category is automatically unimodular [ENO, Prop. 4.5], we state the following consequence of Theorem 3.4 and Proposition 4.2 for later use:
Corollary 4.7**.**
A factorisable and pivotal finite tensor category over which satisfies Condition P admits an up-to-scalar unique modified trace on . For each non-zero modified trace, the pairings (4.4) are non-degenerate.
This result generalises the existence of a modified trace for factorisable Hopf algebras (over an algebraically closed field of characteristic zero) proven in [GKPM2, Thm. 4.3.1].
Remark 4.8**.**
The converse of Corollary 4.7 does not hold, i.e. it is not true that a pivotal finite tensor category over which satisfies Condition P and admits a non-zero modified trace on can be equipped with a braiding that makes it factorisable. The simplest example is the category of super-vector spaces, which only admits symmetric braidings. A non-semisimple example is provided by the restricted quantum group . This is a unimodular Hopf algebra which has a simple projective module but whose category of modules does not admit a braiding [FGST, KS, GR1].
5 Symmetric algebras and ideals in
In this section we will collect some facts about symmetric algebras following [Br, CW] and translate them into categorical statements. In the end we give implications for categories with modified trace as in the previous section.
For a -algebra , denote by
[TABLE]
the space of central forms on . A symmetric algebra over (aka a symmetric Frobenius algebra) is a finite-dimensional -algebra together with a central form such that the induced pairing is non-degenerate. In this case the map
[TABLE]
is an isomorphism, and the invertible elements of are precisely mapped to those elements of that induce a non-degenerate pairing (see e.g. [Br, Lem. 2.5]).
Let be a symmetric algebra and denote by , , the copairing, that is, for all we have . We recall that the copairing is unique and that, since is central, the copairing is symmetric. Define the map
[TABLE]
It is easy to check that the image of does indeed lie in the centre of (see e.g. [Br, Sec. 3.A]). The argument uses that the copairing of a non-degenerate invariant pairing satisfies for all .
We will need the following chains of inclusions in and , compatible with the map from (5.2) [CW]:
[TABLE]
The individual subsets are defined as follows:
- •
is the Reynolds ideal. (Symmetry of is not required for this definition.) Here, is the Jacobson radical of . is an ideal in .
- •
is the Higman ideal or *projective centre666 The name “projective centre” can be motivated from the observation that can be described as all elements in such that the corresponding endomorphism of the --bimodule factors through a projective --bimodule [LZZ, Prop. 2.3 & 2.4]. *. It is an ideal in as for , and by [HHKM, Lem. 4.1]777 In [HHKM, Sec. 4] the underlying field is assumed to be algebraically closed, but this is not used in the proof of Lemma 4.1 in [HHKM]. it is contained in . Note that even though depends on the choice of a non-degenerate central form , does not. Indeed, any two , would be related by an invertible as , and one checks that .
- •
R(A)=\mathrm{span}_{k}\{\,\chi_{M}\,|\,M\text{ right A-module}\}, where , the trace in over the linear map given by acting with . That is shown in [Lo, Thm. 1.6] (for finite-dimensional not necessarily symmetric algebras).
- •
I(A)=\mathrm{span}_{k}\{\,\chi_{P}\,|\,P\text{ projective right A-module}\}. In [CW, Prop. 2.1] it is shown that .
Lemma 5.1**.**
For an algebra over some field, if and only if is semisimple.
Proof.
If is semisimple, and hence . Conversely, if then and since annihilates we must have . ∎
The Cartan matrix of a finite-dimensional algebra over some field is the matrix with entries
[TABLE]
that is, the multiplicity of the simple -module in the composition series of the projective cover of .
Let , be a choice of primitive idempotents such that as right -modules. If the underlying field is given by , which we assumed to be algebraically closed in the outset of the paper, the endomorphism spaces of simple -modules are one-dimensional and we can write
[TABLE]
The last equality follows by noting that every right -module map is given by left multiplication with an element , . Namely, in one direction we set , while in the other direction which is in , and one checks that both the maps are inverses of each other.
Proposition 5.2**.**
Let be a symmetric algebra over . Then:
The Cartan matrix of is symmetric, . 2. 2.
Let and let be the image of under the canonical homomorphism . We have
[TABLE]
Proof.
For part 1 one checks that the non-degenerate pairing on descends to a non-degenerate pairing between and . Indeed, for every such that there exists such that . But then also since . Hence, . The more surprising part 2 is proved in [LZZ, Cor. 2.7]. ∎
Note that if has characteristic [math], then is just the rank of over .
We now give a categorical formulation of the inclusions in (5.4), which will be applicable in particular in the presence of a modified trace as in Section 4. For the rest of this section, let us fix
- •
: a finite abelian category over ,
- •
: a projective generator of ,
- •
the -algebra of endomorphisms of .
Recall the Hom-tensor adjoint equivalence
[TABLE]
of -linear categories between and the category of finite-dimensional right -modules (see e.g. [EGNO, Sect. 1.8]).
It is known that being symmetric is a Morita-invariant property of an algebra, as is the chain of ideals (see e.g. [HHKM, Br, LZZ] for related results). We can therefore use the equivalence in (5.8) and the ideals in (5.4) for the choice and for a given projective generator to define a chain of ideals in as:
[TABLE]
Rather than showing directly that the above definition is independent of the choice of , we will now give a different definition of and which does not require a choice of a projective generator. We show in Propositions 5.3 and 5.8 that this definition agrees with the one above.
For each define the subspace as follows:
[TABLE]
By definition, the Jacobson radical of a ring consists of all which act as zero on all simple -modules. Since is a projective generator, , where runs over all simple , gives all simple right -modules up to isomorphism (by the equivalence (5.8)). From (5.10) we therefore get
[TABLE]
In the case we can take , so that and hence .
Using the subspaces (5.10), we define
[TABLE]
From this definition it is immediate that is an ideal.
Recall that for a projective generator , the map
[TABLE]
is an algebra isomorphism (that is central in follows from the naturality of ; that (5.13) is a bijection is clear in the case , for general use the equivalence (5.8)). We can use this to relate to the previous definition in terms of algebras:
Proposition 5.3**.**
Let be a finite-dimensional -algebra. Then in (5.13) restricts to an isomorphism of algebras .
Proof.
We set and choose the projective generator and thus . We have to show that .
“”: Let . Then . By definition of we have in particular that for all (recall (5.11)), . Thus annihilates , i.e. .
“”: Conversely, let and let . Let , and . We need to show . Pick a surjection and a right-inverse . We have since for all , where is simple, since and . Note then that and , therefore . Since we get
[TABLE]
But is injective and surjective, and so , as required. ∎
The condition replacing that the algebra is symmetric will be that is Calabi-Yau (see e.g. [He] for a recent treatment of the relation between symmetric Frobenius algebras and Calabi-Yau categories in the semisimple case):
Definition 5.4**.**
A -linear additive category with finite-dimensional Hom-spaces is called Calabi-Yau if it is equipped with a family of -linear maps (the trace maps)
[TABLE]
such that for all ,
(cyclicity) for all , we have . 2. 2.
(non-degeneracy) the pairing , is non-degenerate.
We denote by the set of all families of trace maps which turn into a Calabi-Yau category.
A Calabi-Yau category with one object is the same as a symmetric Frobenius algebra. The generalisation of the isomorphism (5.2) to the present setting is:
Lemma 5.5**.**
Let be as in Definition 5.4 and let be the subset of invertible endo-transformations. Then for each the map
[TABLE]
is a bijection.
Proof.
Clearly, defines a family of cyclic and non-degenerate traces (for non-degeneracy, one notes that is non-zero for any non-zero because is invertible). Conversely, given , by non-degeneracy for each there exists a unique such that . Rewriting the cyclicity condition for all and in terms of the family and using its cyclicity property shows \mathsf{t}_{M}\bigl{(}g\circ(\eta_{N}\circ f-f\circ\eta_{M})\bigr{)}=0 for any , and thus by non-degeneracy of we have for all , , and . ∎
Suppose now that the finite abelian category is such that is Calabi-Yau with trace maps . For the pairings of morphisms between the projective objects we write
[TABLE]
Denote by the corresponding (unique) copairing, that is and for all , ,
[TABLE]
We need the following lemma, whose proof is given in Appendix B.
Lemma 5.6**.**
For each and there exists a unique such that for all .
This lemma allows us to define, for each , a -linear map by
[TABLE]
These maps are the analogue of the map from (5.3). Indeed, is now defined as the joint image of all the maps :
[TABLE]
The point of the above definition is to avoid the choice of a projective generator. To actually compute , choosing a projective generator may nonetheless be useful. For example, we show in Corollary B.3 that .
As was the case for the Higman ideal of a symmetric algebra, also is independent of the choice of traces:
Proposition 5.7**.**
Let be a finite abelian category over such that is Calabi-Yau. Then is independent of the choice of the family of traces .
Proof.
Let . By Lemma 5.5 there is such that for all and . Let and be the copairings arising from and as in (5.18). If we write , it is easy to see that
[TABLE]
Write for the map (5.19) computed from . It is immediate from the above equality that for ,
[TABLE]
Thus indeed the joint image of the is independent of the choice of traces . ∎
The next proposition makes the connection between the definition (5.20) of and for a symmetric algebra when taking . The proof has been relegated to Appendix B.
Proposition 5.8**.**
Let be a finite-dimensional -algebra.
* admits a central form turning it into a symmetric algebra if and only if admits a family of traces turning it into a Calabi-Yau category.* 2. 2.
If is symmetric, the map from (5.13) for restricts to an isomorphism of algebras .
Combining this proposition and Proposition 5.3 with the equivalence (5.8) and the inclusions (5.4) we get that we indeed have the inclusions (5.9).
As for algebras, the Cartan matrix of is the -matrix with entries , i.e. in the Grothendieck group we have
[TABLE]
Since is algebraically closed, we can write
[TABLE]
In view of Proposition 5.8 and the equivalence (5.8), we have the following corollary to Proposition 5.2.
Corollary 5.9**.**
Let be a finite abelian category over such that is Calabi-Yau. Then
The Cartan matrix of is symmetric, . 2. 2.
Let and let be the image of under the canonical homomorphism . We have
[TABLE]
In the context of this paper we are particularly interested in the case of factorisable and pivotal finite tensor categories . In this case we will later give a more direct description of and in terms of the defined in (2.22), see Proposition 7.1 below. The next proposition shows in particular that Corollary 5.9 is applicable to such .
Proposition 5.10**.**
Let be a factorisable and pivotal finite tensor category over which satisfies Condition P.
* admits a family of traces turning it into a Calabi-Yau category.* 2. 2.
* is semisimple if and only if any one of the inclusions is an equality.*
Proof.
Part 1 is immediate from Corollary 4.7 by setting for any , where is the modified trace on . For Part 2, the equivalence of semisimplicity to follows from Proposition 5.3 and Lemma 5.1.
To show that is equivalent to semisimplicity of , we pick a projective generator and write . Via the equivalence (5.8) to , and by Propositions 5.3 and 5.8, the equality is equivalent to , and by the isomorphisms in (5.4) in turn to .
Let be the image of in as in Corollary 5.9. We now show:
Claim: iff is non-degenerate.
This claim completes the proof of part 2, since by [EGNO, Thm. 6.6.1], a pivotal finite tensor category over is semisimple if and only if is non-degenerate.
Proof of claim: Recall from (5.4) that the characters of -modules span (over ) and those of projective -modules span . Using again the equivalence (5.8), we have , where is the image of under the equivalence (5.8). In particular, the image of the -linear map described by the matrix in the basis of , is precisely . We conclude that is equivalent to non-degeneracy of , proving the claim. ∎
Part 2 of the proposition also follows from [Sh2, Cor. 4.2 & Thm. 5.12] together with Proposition 7.1 below. 888 In [CW, Cor. 2.3] it is stated (among other things) that for a symmetric algebra over an algebraically closed field of characteristic zero, implies that is semisimple. This would also imply part 2 of Proposition 5.10. However this statement is not true, as illustrated by the following counterexample (due to Ehud Meir): Take and , . Then is a symmetric algebra. Its unique simple module is and the projective cover of is . We have . Thus , but is not semisimple. In [CW], Corollary 2.3 is only used in Theorem 2.9, and there only the remaining parts of Corollary 2.3 are relevant, so that the above counterexample does not affect the other results in [CW] (we thank Miriam Cohen and Sara Westreich for correspondence on this point). On the other hand, it is shown in [Lo, Thm. 2] that if is in addition a Hopf algebra such that the square of the antipode is inner, does indeed imply that is semisimple. This is a special case of [EGNO, Thm. 6.6.1].
6 Characters and the modified trace
In this section we fix
- •
: a factorisable and pivotal finite tensor category over which satisfies Condition P given in the beginning of Section 3,
- •
: a choice of non-zero modified trace on via Corollary 4.7,
- •
: a projective generator for ,
- •
, a symmetric -algebra via .
Recall from (2.22) the natural endomorphism assigned to each . Note that is uniquely determined by , and that in fact .
For a given we can now define two central forms on . Let . The first central form uses the Hom-tensor equivalence (5.8) and is given by , the trace in the right -module over the right action of . The second form is . The aim of this section is to prove our second main result:
Theorem 6.1**.**
The modified trace is non-zero, and for all the following equality of central forms on holds:
[TABLE]
The proof relies on several intermediate results, which we present now.
Lemma 6.2**.**
Let and suppose that for all we have
[TABLE]
Then there is and , such that .
Proof.
Write f=\big{[}A\xrightarrow{a}M\xrightarrow{b}B\big{]}, where is epi and is mono. Rewrite (6.2) as
[TABLE]
By exactness of , also is epi and is mono. We conclude that for all . From characterisation 1 of factorisability in Theorem 2.8 it follows that for some . ∎
Recall the coend with its family of dinatural transformations and its non-zero cointegral from Section 2.
Lemma 6.3**.**
For each there exist , , (possibly zero) such that
[TABLE]
where , are the components of .
Proof.
By definition of the cointegral, we have
[TABLE]
Combining this with the pairing we obtain the following equality:
[TABLE]
where we used the property of the Hopf pairing that . After pre-composing with and substituting the defining equations, a short calculation shows
[TABLE]
Using the isomorphism ,
[TABLE]
and similarly for , the latter equality can be brought to the form with
[TABLE]
Lemma 6.2 now implies the claim. ∎
Corollary 6.4**.**
For simple, , we have . Furthermore, there exists a unique such that
[TABLE]
where is the canonical projection of the projective cover.
Proof.
This follows from Lemma 6.3 and the fact that and are both zero-dimensional for and one-dimensional for (by unimodularity of ). ∎
Recall the natural endomorphism in (2.23). The next proposition computes its values on projective objects and also shows that (and all defined there) are non-zero.
Proposition 6.5**.**
Let . Then and there exists a unique non-zero such that
[TABLE]
where is the canonical projection of the projective cover, and where is the same as in Corollary 6.4. Moreover, there exists a choice of , which realise as a direct summand of (i.e. ), such that
[TABLE]
Proof.
In the expression (2.23) for set and . Write and pick and , , which realise the direct sum decomposition:
[TABLE]
For the dinatural transformation in (2.23) consider the equalities
[TABLE]
From Corollary 6.4, for , and so in the expression for the sum over in (6.14) reduces to . Since for , the multiplicities are given by . In particular,
[TABLE]
So in (6.14) the sum over is zero for and reduces to for . Together with (6.10) we obtain
[TABLE]
Substituting this into (2.23) yields where
[TABLE]
It remains to show that we can achieve , the chosen projection , and that for , agrees with in (6.10).
By Theorem 2.6, in particular . Since and are isomorphisms, also , and thus (or otherwise would vanish on all indecomposable projectives and hence be zero). Since is one-dimensional, by rescaling if necessary (and thus rescaling accordingly by the inverse factor) one can achieve . This finally gives (6.12).
For , one can alternatively compute directly from (2.23) and (6.10). A short calculation confirms that from (6.10) agrees with . ∎
Corollary 6.6**.**
Let , then unless and is projective.
Proof.
By naturality of we have . Applying (6.11) to LHS of the last equality and using for non-projective, we get the statement. ∎
Proof of Theorem 6.1.
We will show and then determine to be . Write and let , , be the corresponding primitive orthogonal idempotents. Pick , such that
[TABLE]
As for any finite-dimensional associative algebra, the quotient is a direct sum over of -matrix algebras, and the form a basis in these matrix algebras. Therefore, the endomorphism algebra decomposes (as a vector space) as
[TABLE]
To show it is enough to consider . In this case, is an irreducible -module of dimension . The trace vanishes on while on semisimple part it gives
[TABLE]
It remains to check that evaluating on leads to the same result, up to a factor of . We have, for all ,
[TABLE]
By (5.11) the map annihilates and therefore . Thus also vanishes on . Furthermore,
[TABLE]
and where we used first cyclicity of , then naturality of and then (6.18) and (6.11). To arrive at , we first observe that
[TABLE]
To see this, first use (6.11) and substitute the explicit expressions for from (6.12). Next use the pivotal structure to insert id_{M}=\big{[}M\xrightarrow{\delta_{M}}M^{**}\xrightarrow{\delta_{M}^{-1}}M\big{]} and combine with the evaluation and coevaluation maps as in (2), thereby replacing by in the right-most strand of the above diagram. We can now compute
[TABLE]
where in (*) we have rewritten (6.23) as a partial trace and used compatibility of the modified trace with partial traces in (4.1).
So far we proved with . It remains to show that . By Theorem 2.6 and (2.22), for . Hence also . By non-degeneracy of the pairings (4.4) (Corollary 4.7), there is an such that . In particular, . ∎
As a corollary to the proof of Theorem 6.1, specifically (6.24) (using also Proposition 6.5), we obtain that, for ,
[TABLE]
7 Properties of the Reynolds and Higman ideals
In this section , , , and have the same meaning as in the beginning of Section 6. We give an alternative description of the Reynolds and Higman ideals defined in Section 5 in terms of the . Then we describe the multiplication in these ideals.
We start with the alternative description of and .
Proposition 7.1**.**
We have
[TABLE]
Proof.
Abbreviate S=\mathrm{span}_{k}\big{\{}\,(\phi_{M})_{G}\,\big{|}\,M\in\mathcal{C}\,\big{\}} and S^{\prime}=\mathrm{span}_{k}\big{\{}\,(\phi_{P})_{G}\,\big{|}\,P\in\mathcal{P}roj(\mathcal{C})\,\big{\}}. By Propositions 5.3 and 5.8 it is enough to show and .
The image of under the map from (5.2) (we take ) is given by \zeta(S)=\mathrm{span}_{k}\big{\{}\,t_{G}\big{(}(\phi_{M})_{G}\circ-\big{)}\,\big{|}\,M\in\mathcal{C}\,\big{\}}. Theorem 6.1 allows us to rewrite this as
[TABLE]
where for the last equality we used the definition of below (5.4) and that every -module is isomorphic to for some (equivalence (5.8)). But by the isomorphisms in (5.4) the equality implies .
An analogous argument shows . ∎
The next proposition shows that the action of on is diagonal in the basis .
Proposition 7.2**.**
For and we have
[TABLE]
where is determined by . If lie in the same block of , then .
Proof.
We pick a projective generator and use the equivalence in (5.8). Let be the primitive central idempotents. The decompose as an algebra into indecomposable direct summands, or, equivalently, via (5.8) they decompose into blocks.
Abbreviate . We can write for some , . Suppose is in the ’th block of , that is, acts on the simple -module by . Then , proving the second statement.
By Proposition 6.5, factors through the semisimple -module for some . Hence , proving the first statement. ∎
Remark 7.3**.**
Together with Proposition 7.1, Proposition 7.2 states that acts diagonalisably on , with eigenbasis given by . Since the value of on only depends on the block of , and since all composition factors of an indecomposable projective belong to the same block, we equally have
[TABLE]
with as in Proposition 7.2.
By Proposition 7.1, is spanned by the , . From this formulation, we see that contains a natural subspace, namely the span of all for is both simple and projective.999 Via the equivalence (5.8) to symmetric algebras, this subspace corresponds to the subspace of the centre denoted by in [CW]. For each simple projective object , by Proposition 6.5 and since , there is a non-zero such that
[TABLE]
This allows for another expression of the normalisation constant in (6.1), namely, for a simple projective object,
[TABLE]
For later use we define
[TABLE]
By Theorem 3.4, this set is not empty. We can use this notation to describe the product in and :
Proposition 7.4**.**
We have, for ,
[TABLE]
Proof.
By Proposition 7.2, . By Corollary 6.6, unless and is projective. The definition of in (7.5) now implies the first equality. For the second equality, expand , and similar for . Since can only contain a composition factor from if already , the second equality follows from the first. ∎
8 -invariance of projective characters
In this section , , , and have the same meaning as in the beginning of Section 6. Here we prove our third main result, namely that the Higman ideal of gets mapped to itself under the action of . As applications, we investigate the action of the Grothendieck ring on and , then give a variant of the Verlinde formula for projective objects and we show how matrix coefficients of on are related to the modified trace.
Recall the definition of from (2.13).
Lemma 8.1**.**
Let be projective and let . Then .
Proof.
Since induces a non-degenerate pairing on (Corollary 4.7), it is enough to show that for all , t_{G}\big{(}\sigma([P])_{G}\circ f\circ x\big{)}=0. Combining the explicit expression of as a partial trace in (2.11) and Lemma 4.5 we see
[TABLE]
As (recall (5.11)), for all simple and all , we have , and hence also . Corollary 2.3 now shows that the right hand side of (8.1) is zero. ∎
Corollary 8.2**.**
Let be projective. Then .
For each choose a primitive idempotent corresponding to the projective cover of the irreducible representation of . That is, as right -modules. We remark here that by (6.22) we have for ,
[TABLE]
where we set as above. From this equality and Proposition 7.1 it follows that the pairing
[TABLE]
is non-degenerate.
Let us abbreviate the image of the Cartan matrix in by , cf. Corollary 5.9. For write . We define
[TABLE]
Lemma 8.3**.**
.
Proof.
We first show that . By Proposition 7.1 we need to show that for all and , t_{G}\big{(}(\phi_{P})_{G}\circ\tilde{x}\big{)}=0. It is enough to consider the projective covers , . By (5.23) we have . Using this, we compute
[TABLE]
By non-degeneracy of the pairing (8.3), we have , the rank of the Cartan matrix, seen as a matrix over . Hence by Corollary 5.9, . ∎
The following theorem generalises results of [La1, CW] as mentioned in the introduction.
Theorem 8.4**.**
.
Proof.
We will show . Let . By Corollary 8.2 and (2.25), . It remains to check that , i.e. that t_{G}\big{(}(\mathscr{S}_{\mathcal{C}}(\phi_{P}))_{G}\circ\tilde{x}\big{)}=0 for all . We compute
[TABLE]
In (*) we inserted and used that , for , realising the direct summand in , and moving around the loop we get . The reason that appears instead of is the extra dual in (2.11), together with (as follows from uni-modularity of ) 101010Applying (2.11) for , then inserting and moving once around the loop, we get a loop coloured by instead of .. This isomorphism, combined with (5.24), shows (and hence ), which explains (**).
The inclusion implies the theorem since is an isomorphism (Remark 2.11) and so together with Lemma 8.3 we have . Thus , once more appealing to Lemma 8.3. ∎
Recall the projective -action on from Theorem 2.13 in case is in addition ribbon.
Corollary 8.5**.**
Let be in addition ribbon. Then is a submodule for the projective -action on .
Proof.
By Theorem 8.4, is stable under the action of the -generator of . The action of is given by , and since is an ideal, it is stable under the -action as well. ∎
We now give applications of Theorem 8.4 to the problem of diagonalisability of the Grothendieck ring of (Proposition 8.6), to the Verlinde formula (Proposition 8.8) and to the modified trace (Proposition 8.10). Namely, we ask how much can be learned about the tensor products of projective objects from knowing the restriction of to , and give a formula for the modified trace of the “open Hopf-link operator” in terms of the -matrix elements. See e.g. [FHST, Fu, GR2, CG] for discussions of the Verlinde formula in the finitely non-semisimple setting and for more references.
We start with the application to the Grothendieck ring. Recall from Corollary 2.12 that is an injective ring homomorphism from to . For denote by
[TABLE]
the left multiplication by . Then and , i.e. defines a faithful representation. Since and are ideals, they are submodules for the representation .
We can now state the following result, part 1 of which is a generalisation of (part of) [CW, Thm. 3.14].
Proposition 8.6**.**
Consider the action .
The restriction of to the submodule , respectively , is diagonalisable with action
[TABLE]
where and \ell^{M}_{V}:=\big{\langle}\sigma([M])_{V}\big{\rangle}\in k (see Proposition 7.2). 2. 2.
If is semisimple, can be diagonalised on . 3. 3.
If and can be diagonalised on , then is semisimple.
Proof.
Part 1 is immediate from Proposition 7.2 and Remark 7.3. Part 2 follows from part 1 together with Proposition 5.10, which states that for semisimple we have .
Part 3 we prove by contradiction. Suppose that is not semisimple, so that not every indecomposable projective object is simple. Let be such a non-simple indecomposable projective object. Since , the map is injective, and hence by Theorem 2.6 (and the definition in (2.22)), . Furthermore, by Proposition 7.4, is nilpotent, in fact .
Since (Proposition 7.1), by Theorem 8.4, we can write and this linear combination is nilpotent because . Then, since is an injective algebra map (Corollary 2.12), contains a non-zero nilpotent element, and so (which factors through ) cannot be diagonalised. ∎
In particular, we have the following consequence for the linearised Grothendieck ring:
Corollary 8.7**.**
If , the linearised Grothendieck ring is semisimple iff is semisimple.
Next we turn to an application of Theorem 8.4 to a Verlinde-type formula.
Write for the isomorphism class of an object (rather than its class in ). Define the ring as the free abelian group spanned by the isomorphism classes of indecomposable projective objects (see e.g. [EGNO, Sec. 1.8]),
[TABLE]
The product of and is defined by the decomposition of into indecomposable projectives. Thus encodes the tensor products of projective objects in up to isomorphism. We denote the structure constants of by , that is,
[TABLE]
or, equivalently,
[TABLE]
The map , is a ring homomorphism. The matrix elements of this map are precisely the entries of the Cartan matrix of .
For the application to the Verlinde formula, we will need two bases of . For the first basis, choose a subset such that
[TABLE]
is a basis of . Recall that denotes the image of in . By Corollary 5.9 we have . By Theorem 8.4, is an invertible endomorphism of . The second basis then is
[TABLE]
Let be the -matrix expressing in the first basis,
[TABLE]
The change of basis map between (8.12) and (8.13) will be denoted by the -matrix , that is
[TABLE]
Finally, let be the -matrix which describes the effect of taking duals,
[TABLE]
By construction, and are invertible, while is invertible if and only if is semisimple (Proposition 5.10).
Recall the definition of the constants from (7.5) and of the subset from (7.7). We remark that in order to get a basis in (8.12) one necessarily has .
Proposition 8.8**.**
For all and we have the following identity in ,
[TABLE]
Proof.
From (8.10) we get the identity
[TABLE]
Substituting (8.14) and (8.15) gives
[TABLE]
Next note that for all (as the square of results in the action of the antipode of , see [Ly1] and e.g. [FGR1, Lem. 5.7]). This results in the matrix identity . Hence and commute and , as is the identity matrix. Multiplying both sides of (8.19) by yields the claim. ∎
Note that (8.17) does not determine uniquely unless is semisimple and is of characteristic zero. Indeed, otherwise does not have a right inverse and is considered as an element of , i.e. modulo .
Remark 8.9**.**
Recall from Remark 1.6 (2) that one of the motivations for this paper comes from vertex operator algebras. Let be a vertex operator algebra as in Remark 1.6 (2). In [GR2, Conj. 5.10], a precise conjecture is made on how the behaviour of so-called pseudo-trace functions of under the modular -transformation is related to for . In this relation, corresponds to the character of the -module (and not to a pseudo-trace function). Characters and their modular properties are much more accessible than those of pseudo-trace functions. That is, from a vertex operator algebra perspective, the following pieces of information are typically more accessible than the structure constants themselves:
- •
, (find the projective modules of and analyse their composition series as well as those of their contragredient modules),
- •
(compute the modular -transformation of characters of projective modules),
- •
(compute the modular -transformation of the vacuum character and take the inverse of the coefficient of the character of ).
Here the text in brackets indicates the computation one needs to do for -modules, provided one assumes the conjectures in [GR2]. For the last point note that by (2.11) and (2.25) we have
[TABLE]
Thus one can expect all the data on the right hand side of (8.17) to be relatively accessible for a vertex operator algebra (as compared to computing fusion of -modules). We will see an example of this in Section 9. 2. 2.
If is semisimple, then Proposition 8.8 is precisely the categorical Verlinde formula: Firstly, , so that are the structure constants of . Next, by [GR2, Rem. 3.10 (3)] we have , where and . Thus . Finally, , and from (8.20) we see . Altogether, (8.17) becomes (using )
[TABLE]
where is considered as an element of . This is the semisimple categorical Verlinde formula, see [Tu, Thm. 4.5.2]. 3. 3.
A variant of the fusion algebra of projective characters was studied [La2, Cor. 4.4] in the context of the small quantum groups for simply laced simple Lie algebras. For factorisable ribbon Hopf algebras, a Verlinde-like formula for the action of the Grothendieck ring on the Higman ideal is given in [CW, Thm. 3.14].
It is an interesting problem to relate the modified trace on “open Hopf links” and properties of the modular -transformation [CG]. The above results allow us to do this in the case the open Hopf link is coloured by projective objects:
Proposition 8.10**.**
For all , and we have
[TABLE]
Proof.
Apply to both sides of (8.15), expand as and use (6.25) and (7.6). ∎
In particular, for we have
[TABLE]
Moreover, by Lemma 4.5 we have , so that (8.22) implies that for ,
[TABLE]
where by abuse of notation we restrict to be a -matrix.
Remark 8.11**.**
Equation (8.23) is a special case of a relation observed in the example of -models in [CG, Sec. 3.3]. The notation used there is related to ours as , where , . In [CG], the matrix of the modular -transformation of pseudo-trace functions in some basis is denoted by . Using these notations, (8.23) reads for , (assuming the conjectures in [GR2, Sec. 5] so that we can express and in terms of modular properties of pseudo-trace functions).
9 Example: symplectic fermions
In this section we consider a family of examples of factorisable finite ribbon categories, namely the so-called symplectic fermion categories . The aim is to compute the modified trace for this class of examples and to illustrate the use of Proposition 8.8.
Let be a non-zero finite-dimensional symplectic vector space over and let satisfy . The -linear finite ribbon category was introduced in [DR1, Ru]. Conjecturally, it is ribbon-equivalent to the representations of the even part of the symplectic fermion vertex operator super-algebra constructed from (hence the name), see [DR3, Conj. 7.3] for the precise statement and further references.
One can show that is ribbon-equivalent to a certain finite-dimensional quasi-triangular quasi-Hopf algebra [GR1, FGR2], but we will not make use of this here. Instead we sketch the construction in [DR1]. We refer to [DR1, Sec. 5.2] or to e.g. [DR2, FGR2] for further details.
Denote the symmetric tensor category of finite-dimensional complex super-vector spaces by . We define
[TABLE]
the exterior algebra of , and consider it as an algebra in by taking the -grading of modulo 2. In particular, and , where denotes the super-dimension. carries the structure of a commutative and cocommutative quasi-triangular Hopf algebra in . We denote the multiplication, comultiplication, unit and counit by , , , and , respectively (see [DR1, Eqn. (5.4)] for the coalgebra structure).
We will make use of a specific cointegral for (see [DR1, Eqn. (5.16)]). To describe it, let and be a symplectic basis of such that , , etc. Then is non-zero only on the top component of , where it takes the value
[TABLE]
The category is the direct sum of two full subcategories,
[TABLE]
where denotes the category of left -modules in . and each contains two simple objects (up to isomorphism), which we denote as
[TABLE]
Here, is the parity-flip functor. The -action on and is trivial.
Since is semi-simple, and are also projective. The projective cover of is itself, with being the projection on the top component of (together with a choice of isomorphism to ). Altogether, the projective covers are
[TABLE]
In particular, the Cartan matrix in (5.23) reads, in the ordering (9.4) of simple objects,
[TABLE]
We write for the tensor product functor and will reserve the notation for the tensor product in and for that of -modules in . The tensor product is -graded, and depending on which sector are chosen from, it is defined to be:
[TABLE]
In sector , the tensor product is that of , with -action given by the coproduct (and using the symmetric braiding of ). In sectors and , the tensor product is that of the underlying super-vector spaces. In sector , the -action is given by the left regular action on .
We refer to [DR1, FGR2] for the associator and braiding isomorphisms in , as well as for the ribbon twist and for the rigid and the pivotal structures. The -dependence enters into these structure maps. We have:
Proposition 9.1** ([DR1, FGR2]).**
* is factorisable.*
The embedding , , where is endowed with the trivial -action, is an embedding of ribbon categories. Every projective object in is isomorphic to for some , embedded in . Note that by definition,
[TABLE]
where are all even endomorphisms of the super-vector space which commute with the -action and are all even endomorphisms of the super-vector space . Write for the super-trace.
Proposition 9.2**.**
Let be a finite-dimensional super-vector space and , . Then, for each ,
[TABLE]
defines a modified trace on .
Proof.
The proof makes use of the explicit associator and rigid structure of . Since we only need them for this proof, we prefer to refer to [DR1, FGR2] for the explicit formulas, rather than repeat them here. We use the simple projective object to start with as in (4.11) and set .
To compute , we note that by (9.7), and that the associator is the identity map for the trivial -module . Here, the dual is as in . We define as
[TABLE]
Here, the boxes indicate in which category the string diagram is to be evaluated. In particular, the right hand side is given in and the crossings are the flips of super-vector spaces. We can now write
[TABLE]
In the last equality we used the partial trace property (4.1), and we implicitly used that is the identity map. Next we write out in terms of the structure morphisms (recall that we omit the tensor product ‘ ’ between objects for better readability):
[TABLE]
where stands for the composition . Finally, we compute using the explicit associator in the 111-sector (see [DR1, Eqn. (2.27)]) and the duality maps from [DR1, Eqns. (3.59), (4.77), (4.78)]:
[TABLE]
where, appears in the associator in the 111-sector. In the equality (*) we used the identities and (for the first identity, use [DR1, Eqns. (3.45), (2.29)] and the fact that in the present case the distinguished group-like element of [DR1] is just , for the second identity use [DR1, Eqns. (4.84), (3.45)]). Altogether, gives the expression in (9.9).
The computation of also uses the partial trace property (4.1): . ∎
Let us evaluate the modified trace from Proposition 9.2 for the four indecomposable projectives in (9.4). Note that all (left) -module maps in are given by right multiplication with an even element . The same holds for -module maps .
[TABLE]
After the computation of the modified trace, we now turn to an application of Proposition 8.8. Recall that . From (9.5) one sees that , so that we can choose . For the matrices , and from (8.14)–(8.16) one finds, for the above order of basis vectors,
[TABLE]
where has been computed in [FGR2] (for ). The constants are (using again the -action from [FGR2] and (8.20))
[TABLE]
Remark 9.3**.**
The matrix can be read off directly from the modular -transformation of characters for projective modules of the even part of the symplectic fermion vertex operator algebra (see [Ka, GK, Ab] and e.g. [DR3, Sec. 2.1] for the modular properties of the characters of simple modules). 2. 2.
Using Proposition 8.10, can be used to compute the modified trace of open Hopf link operators for projective objects. For example, (8.22) with , gives
[TABLE]
Starting from the data in (9.15), Equation (8.17) allows one to compute, for all ,
[TABLE]
Because is degenerate, we only obtain the sum . However, in the present case we can do better. Namely, for finite tensor categories we have
[TABLE]
So in the present example, knowledge of the composition series of the together with the data (9.15) actually fixes completely.
Appendix A Proofs of Lemma 2.1 and Corollary 2.3
Proof of Lemma 2.1.
Consider the product complex
[TABLE]
Since the tensor product functor is biexact by assumption, this complex has exact rows and columns.
Let be the kernel of . It is shown in the proof of [GKPM1, Lem. 2.5.1] that
the morphism , where are the projections on the first and second direct summand, factors through as with surjective. 2. 2.
there are (necessarily unique) morphisms and such that
[TABLE]
Write . Note that
[TABLE]
where in the last step we used the commuting diagram (2.5). Thus factors through , \hat{b}=\big{[}{\bf 1}\xrightarrow{\tilde{b}}K\xrightarrow{\kappa}{}^{*}B\otimes B\big{]}. We have
[TABLE]
Since is injective, we get . Along similar lines – using injectivity of – one shows that . Define the morphism as
[TABLE]
By the above calculation we see that is equal to the right hand side of (2.6). To see that it also equal to the left hand side of (2.6), we make use of the surjection , where is projective. We will now show that , which completes the proof.
Since by point 1 above, is surjective (and since is projective), we can find such that the left square in the following commuting diagram commutes:
[TABLE]
Now compute
[TABLE]
where in (*) we used the commuting diagram (A.6) and the fact that . Since is injective, we can conclude that . Hence
[TABLE]
Similarly one can show and
[TABLE]
Combining (A.8) and (A.9) we finally find
[TABLE]
∎
Proof of Corollary 2.3.
The proof of Part 1 follows that of [GKPM1, Cor. 2.5.2]. We give the details for the first equality, the second one is shown analogously using Remark 2.2 (1).
Let be a decomposition of into indecomposable objects , and write and for the corresponding embedding and projection maps. Writing and applying dinaturality of to , we have
[TABLE]
Now for all simple and we have by assumption on (as ). It is therefore enough to prove the corollary for indecomposable .
Let be indecomposable. By Fitting’s Lemma, every element of is either invertible or nilpotent. By assumption, cannot be invertible, hence it is nilpotent, say . Let and consider the commutative diagram
[TABLE]
with exact rows, i.e. is the cokernel of the kernel of aka. the image of . The map exists and is unique by the universal property of the cokernel. By Lemma 2.1 we have
[TABLE]
We proceed by induction on the degree of nilpotency. For the claim of part 1 of the corollary is clear. Suppose now the claim holds for all maps whose degree of nilpotency is less or equal to . Since , there is such that . But then , and since is surjective, . By our induction assumption, (A.13) is zero.
For part 2, consider the dinatural transformation from to with
[TABLE]
To obtain (2.9), set and use the second equality in part 1 to see . Then use to turn into an endomorphism of , and after using the naturality of the braiding we get (2.9). ∎
Appendix B Proofs of Lemma 5.6 and Proposition 5.8
For this appendix, we fix
- •
: a finite abelian category over ,
- •
: a projective generator of ,
- •
the -algebra of endomorphisms of .
Write for the identity functor on the projective ideal of , and for the algebra of its natural endomorphisms. The following lemma is standard.
Lemma B.1**.**
The map , \eta\mapsto\eta\raisebox{-3.00003pt}{|_{\mathcal{P}roj(\mathcal{A})}} is a -algebra isomorphism.
Proof.
Since is zero iff its values on all projective objects are zero, the map is injective. For surjectivity, let and pick an exact sequence with . Consider the diagram
[TABLE]
The left square commutes by naturality of on projectives. The dashed morphism exists by the universal property of the cokernel of . It is straightforward to verify that is independent of the choice of and that the family is a natural endomorphism of extending from to all of . ∎
Proof of Lemma 5.6.
Let and define the family as in (5.19):
[TABLE]
We will show that . The statement of the lemma then follows from Lemma B.1.
Let and be given. We need the following auxiliary result. Let and in be defined as
[TABLE]
Write and dito for . Then for all we have
[TABLE]
By non-degeneracy of the pairings we obtain . Using this identity, we get
[TABLE]
in other words, the family is natural in . ∎
To prepare the proof of Proposition 5.8, we need some further properties of the maps in (5.19), and we need a description of Calabi-Yau trace maps in terms of their restriction to indecomposable projectives.
Lemma B.2**.**
Let and let , . Then .
Proof.
By an argument similar to that showing in (B.3) above, one obtains that for all and ,
[TABLE]
Then
[TABLE]
where we used (B.6) with both sides composed with . ∎
Corollary B.3**.**
.
Proof.
Let and be given. Pick an such that there is a splitting , , . Let , be the components maps, that is, . Then
[TABLE]
Thus equals the image of under . ∎
For denote by the component of in . Write
[TABLE]
In other words, iff for all , the pairings , are non-degenerate and symmetric in the sense that .
Recall the notation for the set of trace maps which turn into a Calabi-Yau category (Definition 5.4).
Lemma B.4**.**
The map
[TABLE]
is a -linear isomorphism.
Proof.
Injectivity: Let be given. Write and let , , and , be the embedding and projection maps of the individual summands. By cyclicity and linearity of , we have the decomposition
[TABLE]
where we used for the primitive idempotents . Thus is uniquely determined by the values , showing that the map (B.10) is injective.
Surjectivity: Given , we can define all via (B.11). One checks that is independent of the choice of embedding and projection maps. It remains to verify that is non-degenerate and symmetric. Let and let be a realisation of as a direct sum of indecomposable projectives as above, and one of .
- •
(Symmetry) Let , . We compute
[TABLE]
- •
(Non-degeneracy) Let be non-zero. Then there exist such that . By non-degeneracy of the there is such that
[TABLE]
Thus for we get .
This shows that the map (B.10) is surjective. ∎
Proof of Proposition 5.8.
Recall that denotes a choice of projective generator of and . We will show the following statements, which are equivalent to points 1 and 2 in the proposition upon setting and as the right regular representation.
admits a central form turning it into a symmetric algebra if and only if admits a Calabi-Yau structure. 2. 2.
If is symmetric, the map from (5.13) restricts to an isomorphism of algebras .
Part 1: Clearly, if , then turns into a symmetric algebra with the central form .
Suppose conversely that is a symmetric algebra with respect to the central form . Write and write , , , for the embedding and projection maps of the individual summands. For define
[TABLE]
i.e. we only use the component with for each summand. We now claim that (recall the definition (B)). By Lemma B.4 this will prove part 1.
Non-degeneracy: Let be non-zero. We need to find such that . Since is non-zero, so is . By non-degeneracy of , there is such that . Set and compute
[TABLE]
Symmetry: Let , . We compute
[TABLE]
Part 2: By the assumption that is symmetric and by part 1, is Calabi-Yau via some . By Proposition 5.7 (and by a similar argument for ), neither nor depend on the choice of traces. In particular, we may replace whatever central form was originally equipped with by .
In terms of the central form , the map from (5.3) reads
[TABLE]
Using this, we compute
[TABLE]
∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[AAGTV] N. Andruskiewitsch, I. Angiono, A. Garci a Iglesias, B. Torrecillas, C. Vay, From Hopf algebras to tensor categories , Conformal field theories and tensor categories, 1–31, Math. Lect. Peking Univ., Springer, Heidelberg, 2014 [1204.5807 [math.QA]] . · doi ↗
- 2[Ab] T. Abe, A ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} -orbifold model of the symplectic fermionic vertex operator superalgebra , Mathematische Zeitschrift 255 (2007) 755–792 [math.QA/0503472] . · doi ↗
- 3[AN] Y. Arike, K. Nagatomo, Some remarks on pseudo-trace functions for orbifold models associated with symplectic fermions , Int. J. Math. 24 (2013) 1350008 [1104.0068 [math.QA]] . · doi ↗
- 4[BDSPV] B. Bartlett, C.L. Douglas, C.J. Schommer-Pries, J. Vicary, Modular categories as representations of the 3-dimensional bordism 2-category , 1509.06811 [math.AT] .
- 5[BK] B. Bakalov, A.A. Kirillov, Lectures on Tensor Categories and Modular Functors , American Mathematical Society, Providence, 2001.
- 6[Br] M. Broué, Higman’s criterion revisited , Michigan Math. J. 58 (2009) 125–179 . · doi ↗
- 7[CG] T. Creutzig, T. Gannon, Logarithmic conformal field theory, log-modular tensor categories and modular forms , J. Phys. A 50 (2017) 404004 [1605.04630 [math.QA]] . · doi ↗
- 8[CGPM] F. Costantino, N. Geer, B. Patureau-Mirand, Some remarks on the unrolled quantum group of sl(2) , J. Pure and Appl. Algebra 219 (2015) 3238–3262 [1406.0410 [math.QA]] . · doi ↗