$K$-theory of AF-algebras from braided C*-tensor categories

TL;DR

This paper links the K-theory of AF-algebras derived from braided C*-tensor categories to polynomial rings, providing explicit calculations for certain Lie groups and aiming to simplify the realization of Verlinde rings in K-theory.

Contribution

It demonstrates that the multiplication in the K-theory of AF-algebras is induced by a *-homomorphism from a braided tensor category, and provides new explicit calculations for rank two Lie groups.

Findings

K-theory of AF-algebras can be described via polynomial rings with multiplication from braided tensor categories.

Explicit K-theoretic calculations performed for SU(3), Sp(4), and G2.

Verlinde rings can be recovered in specific cases through this approach.

Abstract

Renault, Wassermann, Handelman and Rossmann (early 1980s) and Evans and Gould (1994) explicitly described the $K$-theory of certain unital AF-algebras $A$ as (quotients of) polynomial rings. In this paper, we show that in each case the multiplication in the polynomial ring (quotient) is induced by a $*$-homomorphism $A\otimes A\to A$ arising from a unitary braiding on a C*-tensor category and essentially defined by Erlijman and Wenzl (2007). We also present some new explicit calculations based on the work of Gepner, Fuchs and others. Specifically, we perform computations for the rank two compact Lie groups SU(3), Sp(4) and G$_2$ that are analogous to the Evans-Gould computation for the rank one compact Lie group SU(2). The Verlinde rings are the fusion rings of Wess-Zumino-Witten models in conformal field theory or, equivalently, of certain related C*-tensor categories. Freed, Hopkins and Teleman (early 2000s) realized these rings via twisted equivariant $K$-theory. Inspired by this, our long-term goal is to realize these rings in a simpler $K$-theoretical manner, avoiding the technicalities of loop group analysis. As a step in this direction, we note that the Verlinde rings can be recovered as above in certain special cases.

Tables

Table 1. Table 1: The polynomials Q λ → SU ​ ( 3 ) ​ ( x , y ) subscript superscript 𝑄 SU 3 → 𝜆 𝑥 𝑦 Q^{\mathrm{SU}(3)}_{\vec{\lambda}}(x,y) with λ 1 ≥ λ 2 subscript 𝜆 1 subscript 𝜆 2 \lambda_{1}\geq\lambda_{2} and | λ → | ≤ 6 → 𝜆 6 |\vec{\lambda}|\leq 6

$\overrightarrow{\lambda }$	$Q_{\overrightarrow{\lambda }}^{\mathrm{SU}\left(3\right)}(x,y)$
$(0,0)$	1
$(1,0)$	$x$
$(2,0)$	$x^2-y$
$(1,1)$	$xy-1$
$(3,0)$	$x^3-2xy+1$
$(2,1)$	$x^{2}y-y^2-x$
$(4,0)$	$x^4-3x^{2}y+y^2+2x$
$(3,1)$	$x^{3}y-2x{y}^2-x^2+2y$
$(2,2)$	$x^2{y}^2-x^3-y^3$
$(5,0)$	$x^5-4x^{3}y+3x{y}^2+3x^2-2y$
$(4,1)$	$x^{4}y-3x^2{y}^2+y^3-x^3+4xy-1$
$(3,2)$	$x^3{y}^2-2x{y}^3-x^4+x^{2}y+2y^2-x$
$(6,0)$	$x^6-5x^{4}y+6x^2{y}^2-y^3+4x^3-6xy+1$
$(5,1)$	$x^{5}y-4x^3{y}^2-x^4+3x{y}^3+6x^{2}y-3y^2-2x$
$(4,2)$	$x^4{y}^2-3x^2{y}^3-x^5+y^4+2x^{3}y+3x{y}^2-2x^2-y$
$(3,3)$	$x^3{y}^3-2x^{4}y-2x{y}^4+3x^2{y}^2+2x^3+2y^3-5xy+1$

Table 2. Table 2: The polynomials P λ → SU ​ ( 3 ) ​ ( s , t ) subscript superscript 𝑃 SU 3 → 𝜆 𝑠 𝑡 P^{\mathrm{SU}(3)}_{\vec{\lambda}}(s,t) with λ 1 ≥ λ 2 subscript 𝜆 1 subscript 𝜆 2 \lambda_{1}\geq\lambda_{2} and | λ → | ≤ 6 → 𝜆 6 |\vec{\lambda}|\leq 6

$\overrightarrow{\lambda }$	$P_{\overrightarrow{\lambda }}^{\mathrm{SU}\left(3\right)}(s,t)$
$(0,0)$	1
$(1,0)$	$1$
$(2,0)$	$1-t$
$(1,1)$	$1-st$
$(3,0)$	$1-2t+s{t}^2$
$(2,1)$	$1-t-st$
$(4,0)$	$1-3t+t^2+2s{t}^2$
$(3,1)$	$1-2t-st+2s{t}^2$
$(2,2)$	$1-s-t$
$(5,0)$	$1-4t+3t^2+3s{t}^2-2s{t}^3$
$(4,1)$	$1-3t+t^2-st+4s{t}^2-s^2{t}^3$
$(3,2)$	$1-2t-s+st+2s{t}^2-s^2{t}^2$
$(6,0)$	$1-5t+6t^2-t^3+4s{t}^2-6s{t}^3+s^2{t}^4$
$(5,1)$	$1-4t-st+3t^2+6s{t}^2-3s{t}^3-2s^2{t}^3$
$(4,2)$	$1-3t-s+t^2+2st+3s{t}^2-2s^2{t}^2-s^2{t}^3$
$(3,3)$	$1-2s-2t+3st+2s^{2}t+2s{t}^2-5s^2{t}^2+s^3{t}^3$

Equations

$${\mathrm{Ver}}_k(G)\cong {}^{\tau }{K}_G^{\dim (G)}(G),$$

$$\Vert T^{\ast }T\Vert =\Vert T{\Vert }^2$$

$$\mathrm{End}(\pi )\cong \underset{\lambda \in \Lambda }{⨁}{M}_{N_{\pi }^{\lambda }}(\mathbb{C})$$

$$\mathrm{Mor}(\pi \otimes \rho ,\psi )\cong \mathrm{Mor}(\pi ,\psi \otimes \bar{\rho })\cong \mathrm{Mor}(\rho ,\bar{\pi }\otimes \psi ).$$

$$c_{\pi ,\rho }:\pi \otimes \rho ⟶\rho \otimes \pi$$

$$(T_2\otimes T_1)\circ c_{{\pi }_1,{\rho }_1}=c_{{\pi }_2,{\rho }_2}\circ (T_1\otimes T_2);$$

$$({\mathrm{id}}_{\rho }\otimes c_{\pi ,\sigma })\circ (c_{\pi ,\rho }\otimes {\mathrm{id}}_{\sigma })$$

$$(c_{\pi ,\sigma }\otimes {\mathrm{id}}_{\rho })\circ ({\mathrm{id}}_{\pi }\otimes c_{\rho ,\sigma })$$

$$c_{\pi ,\rho }^{\ast }\circ c_{\pi ,\rho }$$

$$c_{\pi ,\rho }\circ c_{\pi ,\rho }^{\ast }$$

$$S_{\mu ,\nu }={\mathrm{Tr}}_{\mu \otimes \nu }(c_{\nu ,\mu }\circ c_{\mu ,\nu })$$

$$S_{\mu ,\nu }={\mathrm{Tr}}_{\mu \otimes \nu }({\mathrm{id}}_{\mu \otimes \nu })=d(\mu \otimes \nu )=d(\mu )d(\nu ),$$

$$\pi \otimes {\pi }_{\nu }\cong \underset{\mu \in \Lambda }{⨁}{\pi }_{\mu }^{\oplus N_{\pi ,\nu }^{\mu }}$$

$$\lambda \cdot \nu =\sum _{\mu \in \Lambda }{N}_{\lambda ,\nu }^{\mu }\mu$$

$${\pi }_i\otimes {\pi }_j\cong {\pi }_{|i-j|}\oplus {\pi }_{|i-j|+2}\oplus \cdots \oplus {\pi }_{i+j}$$

$${\pi }_i\otimes {\pi }_j\cong \{\begin{array}{ll}{\pi }_{|i-j|}\oplus \cdots \oplus {\pi }_{i+j}& \text{ if }i+j\le k,\\ {\pi }_{|i-j|}\oplus \cdots \oplus {\pi }_{2k-(i+j)}& \text{ if }i+j>k,\end{array}$$

$$V=\{(x_{\vec{\lambda }}^{(1)},\dots ,x_{\vec{\lambda }}^{(r)}):\ell (\vec{\lambda })\le k\}$$

$$J_k(G)=\{p(x_1,\dots ,x_r)\in \mathbb{Z}[x_1,\dots ,x_r]:p(\vec{z})=0\text{ for all }\vec{z}\in V\}$$

$$J_k(\mathrm{SU}(2))=⟨Q_{k+1}^{\mathrm{SU}(2)}(x_1)⟩$$

$$J_k(\mathrm{SU}(3))=⟨Q_{(k+1,0)}^{\mathrm{SU}(3)}(x_1,x_2),Q_{(k+2,0)}^{\mathrm{SU}(3)}(x_1,x_2)⟩,$$

$$J_k(\mathrm{Sp}(4))=⟨\{Q_{\vec{\lambda }}^{\mathrm{Sp}(4)}(x_1,x_2):{\lambda }_1+{\lambda }_2=k+1\}\cup \{Q_{(0,k+2)}^{\mathrm{Sp}(4)}(x_1,x_2)\}⟩,$$

$$J_k({\mathrm{G}}_2)=⟨\{Q_{\vec{\lambda }}^{{\mathrm{G}}_2}(x_1,x_2):{\lambda }_1+2{\lambda }_2=k+1\}\cup \{Q_{(k+2,0)}^{{\mathrm{G}}_2}(x_1,x_2)\}⟩,$$

$${\mathrm{Tr}}_{{\sigma }^{\otimes (n+1)}}({\iota }_n(T))={\mathrm{Tr}}_{{\sigma }^{\otimes n}}(T){\mathrm{Tr}}_{\sigma }({\mathrm{id}}_{\sigma })=d(\sigma ){\mathrm{Tr}}_{{\sigma }^{\otimes n}}(T)$$

$$A(\mathcal{C},\pi )={\text{ind-lim}}_n(A(\mathcal{C},\pi {)}_n,{\iota }_n),$$

$$g\cdot (x_1\otimes x_2\otimes \cdots )=\bar{\pi }(g)x_1\bar{\pi }(g{)}^{\ast }\otimes \pi (g)x_2\pi (g{)}^{\ast }\otimes \cdots .$$

$${\theta }_n:A(\mathcal{C},\pi {)}_n\otimes A(\mathcal{C},\pi {)}_n⟶A(\mathcal{C},\pi {)}_{2n}$$

$${\theta }_n(T_1\otimes T_2)=U_n(T_1\otimes T_2)U_n^{\ast }$$

$$U_{n}=\prod_{j=1}^{n-1}\big{(}\mathrm{id}_{\sigma}^{\otimes(n-j)}\otimes c_{\sigma,\sigma}^{\otimes j}\otimes\mathrm{id}_{\sigma}^{\otimes(n-j)}\big{)}\in\mathcal{U}(A(\mathcal{C},\pi)_{2n}).$$

$${\theta }_{n+1}\circ ({\iota }_n\otimes {\iota }_n)={\iota }_{2n+1}\circ {\iota }_{2n}\circ {\theta }_n.$$

$$\theta :A(\mathcal{C},\pi )\otimes A(\mathcal{C},\pi )\to A(\mathcal{C},\pi ).$$

$$c_{\sigma ,{\sigma }^n}=({\mathrm{id}}_{\sigma }^{\otimes (n-1)}\otimes c_{\sigma ,\sigma })(c_{\sigma ,{\sigma }^{n-1}}\otimes {\mathrm{id}}_{\sigma })$$

Full text

The Verlinde rings are the fusion rings of Wess–Zumino–Witten models in conformal field theory or, equivalently, of certain related C*-tensor categories. Freed, Hopkins and Teleman (early 2000s) realized these rings via twisted equivariant $K$-theory. Inspired by this, our long-term goal is to realize these rings in a simpler $K$-theoretical manner, avoiding the technicalities of loop group analysis. As a step in this direction, we note that the Verlinde rings can be recovered as above in certain special cases.

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 C*-tensor categories

4. 2.1.1 Features of a rigid C*-tensor category

5. 2.1.2 Braided C*-tensor categories

6. 2.2 Fusion rings

7. 3 AF-algebras from braided categories

8. 3.1 Construction and basic examples

9. 3.2 From unitary braidings to $*$-homomorphisms

10. 3.2.1 Definition of the $*$-homomorphism

11. 3.2.2 SU(2) and Temperley–Lieb–Jones algebras

12. 3.3 Induced (ordered) ring structure in $K$-theory

13. 4 Explicit computations

14. 4.1 SU(2)k

15. 4.2 SU(2)

16. 4.3 SU(3)k

17. 4.3.1 On the polynomials $Q_{\vec{\lambda}}^{\mathrm{SU}(3)}(x,y)$ and $P_{\vec{\lambda}}^{\mathrm{SU}(3)}(s,t)$

18. 4.3.2 On certain monomials $m_{\vec{\lambda},n}(s,t)$

19. 4.3.3 Definition, injectivity and positivity of $\psi$

20. 4.3.4 Proof of surjectivity of $\psi$

21. 4.3.5 Conclusion of the proof of Theorem 4.7

22. 4.4 SU(3)

23. 4.5 Sp(4)k

24. 4.6 (G2)k

25. 5 Recovering the Verlinde ring in special cases

26. 5.1 General remarks

27. 5.2 SU(2)k

28. 6 Concluding remarks

1 Introduction

This paper concerns the $K$-theory of certain unital AF-algebras. Recall that AF-algebras are limits of inductive sequences of finite-dimensional C*-algebras with connecting $*$-homomorphisms. They were first introduced by Bratteli in the paper [7]. The $K$-theory $K_{0}(A)$ of an AF-algebra $A$ is an ordered group, i.e., an abelian (additively written) group $G$ together with a positive cone $G_{+}$, i.e., a subset $G_{+}\subset G$ that satisfies $G_{+}+G_{+}\subset G_{+}$, $G_{+}\cap(-G_{+})=\{0\}$ and $G=G_{+}-G_{+}$. If $A$ is a unital AF-algebra then $K_{0}(A)$ is an ordered group with a distinguished order unit (cf. Remark 4.22), and Elliott [15] showed that unital AF-algebras are classified by this data. He also classified non-unital AF-algebras by a slightly more complicated invariant. The ordered groups that arise as $K_{0}(A)$ for some AF-algebra $A$ are called dimension groups. They were abstractly characterized by Effros, Handelman and Shen in [14]. We refer the reader to Effros’ monograph [13] for more information on the $K$-theory of AF-algebras.

Although $K_{0}(A)$ is in general only an ordered group, it also sometimes has a natural ordered ring structure (cf. section 3.3). By the Elliott Classification Theorem [15], if $A$ is a unital AF-algebra such that $K_{0}(A)$ has the structure of an ordered ring then the product on $K_{0}(A)$ is induced by a (non-explicit) $*$-homomorphism $A\otimes A\to A$. In the present paper, we consider certain unital AF-algebras $A$ for which $K_{0}(A)$ has the structure of an ordered ring and the underlying $*$-homomorphism $A\otimes A\to A$ can be defined in terms of a unitary braiding on a rigid C*-tensor category. (These concepts are described in some detail in section 2.1 below, and some historical background and references can be found in Remark 3.7.) We place certain $K$-theory calculations of Renault, Wassermann, Handelman and Rossmann (early 1980s) and Evans–Gould (1994) within this framework and, in some special cases, give new explicit descriptions of the ring $K_{0}(A)$ in terms of generators and relations, i.e., as a quotient of a polynomial ring by an ideal generated by certain polynomials. Below, we motivate our computations and put them into a historical context.

In the early 2000s, Freed, Hopkins and Teleman used $K$-theory to study Wess–Zumino–Witten (WZW) models in 2d conformal field theory. Each WZW model has an underlying rigid C*-tensor category (in fact, a unitary modular tensor category) $\mathbf{Rep}_{k}(G)$ with a unitary braiding. Here, $G$ is a simple, connected, simply connected, compact Lie group and $k$ is a positive integer known as the “level” (cf. Example 2.5 below). The fusion ring of this category, which is known as the Verlinde ring $\mathrm{Ver}_{k}(G)$, describes the so-called operator product expansion of primary fields in the model. Freed, Hopkins and Teleman described $\mathrm{Ver}_{k}(G)$ in terms of twisted equivariant $K$-theory (cf. [22, 23, 24]). In symbols, those authors proved that, as rings,

[TABLE]

where $G$ acts on itself by conjugation, the twist $\tau=\tau(k)$ belongs to the equivariant Čech cohomology group $H^{3}_{G}(G;\mathbb{Z})\cong\mathbb{Z}$, and the ring structure on the right hand side arises from the group product. (The above formula is actually a special case of a more general theorem from [22, 23, 24].) Note that ${}^{\tau}K_{G}^{\dim(G)}(G)$ can also be viewed, in terms of the graded $K$-theory of C*-algebras, as $K_{\dim(G)}^{G}(C_{0}(G,\mathbb{K}_{\tau}))$ or, equivalently, as $K_{\dim(G)}(C_{0}(G,\mathbb{K}_{\tau})\rtimes G)$, where $C_{0}(G,\mathbb{K}_{\tau})$ is the C*-algebra of sections of a certain twisted $G$-equivariant graded bundle $\mathbb{K}_{\tau}$ of compact operators over $G$.

Inspired by this, a long-term goal of ours is to find a simpler $K$-theoretical description of these rings, avoiding the technically complicated analysis of loop groups used in [22, 23, 24] and perhaps even allowing us to describe the underlying rigid C*-tensor category in terms of modules over a “natural” (possibly non-AF) C*-algebra along with a “natural” tensor product on these modules. This served as one impetus for the present paper. Since it was written, we have actually accomplished something along these lines. Namely, in [1] (see Remark 5.6 therein) we use the graphical calculus for $\mathbf{Rep}_{k}(G)$ (see Remark 3.7 below) to realize this category as a braided C*-tensor category of Hilbert C*-modules over a C*-algebra of compact operators. In the present paper, we content ourselves with realizing — in certain special cases — the ring $\mathrm{Ver}_{k}(G)$, equipped with a natural positive cone, as the ordered $K_{0}$-ring of a unital AF-algebra arising from the category $\mathbf{Rep}_{k}(G)$.

$K$-theory has also been used to study certain actions of compact groups on AF-algebras, for instance in the work of Wassermann [66], Handelman–Rossmann [36, 37] and Handelman [34, 35] in the 1980s. As part of their work, those authors explicitly described the $K$-theory of the corresponding fixed point AF-algebras as polynomial rings. For example, inspired by Renault’s computation of $K_{0}(M_{2^{\infty}}^{\mathbb{T}})$ (cf. the appendix to [59]), Wassermann showed that $K_{0}(M_{2^{\infty}}^{\mathrm{SU}(2)})\cong\mathbb{Z}[t]$ as ordered rings (cf. [66], pp. 118–123; see also Theorem 4.5 below), while Handelman and Rossmann showed that $K_{0}(M_{n^{\infty}}^{\mathrm{U}(n)})\cong\mathbb{Z}[t_{1},\ldots,t_{n-1}]$ as rings (cf. Proposition VII.1 in [36]). In these formulae, $M_{n^{\infty}}$ is the infinite tensor product $M_{n}(\mathbb{C})\otimes M_{n}(\mathbb{C})\otimes\cdots$ and the action of $\mathrm{SU}(2)$ on $M_{2^{\infty}}$ (resp. of $\mathrm{U}(n)$ on $M_{n^{\infty}}$) arises from the action of $\mathrm{SU}(2)$ on $M_{2}(\mathbb{C})$ (resp. of $\mathrm{U}(n)$ on $M_{n}(\mathbb{C})$) by conjugation. (See also Example 3.1 below.)

Related to this, Evans and Gould in [17] explicitly described the $K$-theory of certain unital AF-algebras that arise from Jones’ subfactors [41] and from WZW models associated to SU(2) as quotients of polynomial rings (see also Theorem 4.2 below). In the present paper, we note that braided C*-tensor categories, in the sense of Erlijman–Wenzl [16], yield a common framework for the computations of Renault, Wassermann, Handelman and Rossmann on the one hand and of Evans and Gould on the other by showing that in each case the ring structure in $K$-theory is induced by a $*$-homomorphism, essentially defined by Erlijman and Wenzl in [16], that arises from a unitary braiding on an underlying rigid C*-tensor category. (Note that the tensor product of Hilbert C*-modules that appears in [1] is defined via a $*$-homomorphism that is given by a similar formula.) For example, in Wassermann’s computation the underlying category is the category $\mathbf{Rep}(\mathrm{SU}(2))$ of continuous finite-dimensional unitary representations of $\mathrm{SU}(2)$ whereas in the Evans–Gould computation it is $\mathbf{Rep}_{k}(\mathrm{SU}(2))$. Moreover, we perform computations for the rank two Lie groups SU(3), Sp(4) and G2 that are analogous to the Evans–Gould computation for SU(2). Our calculations are based on the work of Gepner [30], Fuchs [26] and others (see section 2.2) in which Verlinde rings were described as quotients of polynomial rings or, equivalently, of representation rings.

Let us now describe the structure of the present paper. In section 2, we discuss preliminary material on rigid C*-tensor categories and unitary braidings, including both the complicated examples $\mathbf{Rep}_{k}(G)$ and simpler examples, such as the representation categories of compact groups. We also define the fusion rings of these categories, and discuss modular $S$-matrices and the aforementioned work of Gepner, Fuchs and others on the Verlinde rings.

Thereafter, in section 3, we define the unital AF-algebras with which we are concerned in the present paper. Specifically, given a rigid C*-tensor category $\mathcal{C}$ and an object $\pi$ in $\mathcal{C}$, we define the unital AF-algebra A(\mathcal{C},\pi)=\text{\mathrm{ind}$-$\mathrm{lim}}_{n}\mathrm{End}((\bar{\pi}\otimes\pi)^{\otimes n}). Following Erlijman–Wenzl [16], we also use a unitary braiding on $\mathcal{C}$ to define a $*$-homomorphism $\theta\colon A(\mathcal{C},\pi)\otimes A(\mathcal{C},\pi)\to A(\mathcal{C},\pi)$. As an example, we describe in detail the situation where $\mathcal{C}=\mathbf{Rep}_{k}(\mathrm{SU}(2))$ and $\pi$ is the fundamental representation, in which case $A(\mathcal{C},\pi)$ may be described in terms of Temperley–Lieb diagrams. Returning to the general setup, we next show that the $*$-homomorphism $\theta$ induces a well-defined ring structure on $K_{0}(A(\mathcal{C},\pi))$ by embedding $K_{0}(A(\mathcal{C},\pi))$ as a subgroup of a localization of the fusion ring of $\mathcal{C}$ in such a way that $K_{0}(\theta)$ is identified with the product on the localization.

In section 4, we compute $K_{0}(A(\mathbf{Rep}_{k}(G),\pi))$, where $G$ is one of the rank two Lie groups SU(3), Sp(4) and G2 and $\pi$ is a fundamental representation. For the convenience of the reader, we also state the aforementioned computations of Wassermann and Evans–Gould. The computation for $G=\mathrm{SU}(3)$ is given in detail, but we omit the similar (but simpler) details of the computations for Sp(4) and G2. We also give a computation of $K_{0}(A(\mathbf{Rep}(\mathrm{SU}(3)),\pi))$, but as this is essentially due to Handelman and Rossmann, we only provide the main ingredients in an “elementary” proof (using only Lie theory and general facts about ordered groups and rings).

In section 5, we show that if $\mathcal{C}=\mathbf{Rep}_{k}(\mathrm{SU}(2))$, where $k-1\notin 3\mathbb{Z}$ or $k\in 2\mathbb{Z}$, then one can find an object $\pi$ in $\mathcal{C}$ (or, more precisely, an explicit isomorphism class) such that $K_{0}(A(\mathcal{C},\pi))$ and $\mathrm{Ver}_{k}(\mathrm{SU}(2))$ are isomorphic as ordered rings. Finally, concluding remarks and open questions may be found in section 6.

We end the present section with a comment on notation. We will use the symbol $\mathbb{N}$ to denote the set of (strictly) positive integers. The set of non-negative integers will be denoted by $\mathbb{N}_{0}$. Given a set $X$ and $r\in\mathbb{N}$, we will denote the Cartesian product $X\times X\times\cdots\times X$ ($r$ factors) by $X^{\times r}$.

Acknowledgements. We would like to thank the following entities for their generous hospitality during the writing of the present paper: The Isaac Newton Institute for Mathematical Sciences in Cambridge, England, during the research program Operator Algebras: Subfactors and their Applications in the spring of 2017; the Hausdorff Research Institute for Mathematics in Bonn, Germany, during the trimester program von Neumann Algebras in the summer of 2016; and the Dublin Institute for Advanced Studies in Dublin, Ireland, during a research visit in December 2017.

We would also like to thank the organizers of the conference Young Mathematicians in C-Algebras 2018* (YMC*A 2018) in Leuven, Belgium, the organizers of the Operator Algebra Seminar at the University of Copenhagen, and the organizers of the Analysis Seminar at Glasgow University for giving the first named author opportunities to present our work.

This research was supported by Engineering and Physical Sciences Research Council (EPSRC) grants EP/K032208/1 and EP/N022432/1.

2 Preliminaries

2.1 C*-tensor categories

In this paper, we will consider examples of so-called rigid C-tensor categories*. A complete definition of such a category, as well as some historical background for this definition, can e.g. be found in Chapter 2 of [52]. (See also e.g. [16], [8], [55].) Here, we content ourselves with an outline of the main features of such a category. (In particular, we are sweeping under the rug the question of strictness of such a category, which is e.g. covered in [52].) As we go over these features, the reader should keep in mind the following simple examples.

• •

The category $\mathbf{Hilb}$, whose objects are finite-dimensional Hilbert spaces and whose morphisms are linear maps.

• •

The category $\mathbf{Rep}(G)$, whose objects are finite-dimensional unitary representations of a finite group $G$ and whose morphisms are intertwining linear maps.

For the convenience of the reader, we give a reminder about conjugation in these categories (which is one of the features of a general rigid C*-tensor category). Given a Hilbert space $H$, the conjugate Hilbert space $\overline{H}$ is defined, as a set, by $\overline{H}=\{\bar{\xi}\,:\,\xi\in H\}$. Its Hilbert space operations are defined by $\bar{\xi}+\bar{\eta}=\overline{\xi+\eta}$, $z\bar{\xi}=\overline{\bar{z}\xi}$ and $\langle\bar{\xi},\bar{\eta}\rangle=\langle\eta,\xi\rangle$ for $\xi,\eta\in H$ and $z\in\mathbb{C}$. Given a linear map $T\colon H\to K$, where $K$ is a Hilbert space, the conjugate operator $\overline{T}\colon\overline{H}\to\overline{K}$ is defined by $\overline{T}(\bar{\xi})=\overline{T(\xi)}$ for $\xi\in H$. Given a unitary representation $\pi\colon G\to\mathcal{U}(H)$, its conjugate (or dual) representation is the unitary representation $\bar{\pi}\colon G\to\mathcal{U}(\overline{H})$ defined by $\bar{\pi}(g)=\overline{\pi(g)}$ for $g\in G$. It is an easy exercise to check that, indeed, $\overline{H}$ is a Hilbert space, $\overline{T}$ is a linear map, and $\bar{\pi}$ is a unitary representation, and that if $T$ intertwines two unitary representations $\pi$ and $\rho$ then $\overline{T}$ intertwines $\bar{\pi}$ and $\bar{\rho}$.

Frobenius Reciprocity, which is mentioned below as one of the features of a rigid C*-tensor category, is a classical fact for representations of finite groups.

2.1.1 Features of a rigid C*-tensor category

A rigid C-tensor category* $\mathcal{C}$ has the following features (where $\pi$ and $\rho$ always denote arbitrary objects of $\mathcal{C}$):

• •

Each morphism set $\mathrm{Mor}(\pi,\rho)$ is a (complex) Banach space.

• •

There is an involutive $*$-operation on morphisms, mapping a morphism $T\colon\pi\to\rho$ to a morphism $T^{*}\colon\rho\to\pi$, such that the C-identity*

[TABLE]

holds for any morphism $T\colon\pi\to\rho$.

Accordingly, two objects $\pi$ and $\rho$ in $\mathcal{C}$ are said to be (unitarily) isomorphic if there exists a morphism $u\colon\pi\to\rho$ such that $u^{*}u=\mathrm{id}_{\pi}$ and $uu^{*}=\mathrm{id}_{\rho}$.

• •

One can take direct sums of objects and of morphisms, and there is a distinguished object $\boldsymbol{0}$, which is a zero object. In particular, $\boldsymbol{0}\oplus\pi\cong\pi\oplus\boldsymbol{0}\cong\pi$.

• •

One can take tensor products of objects and of morphisms, and there is a distinguished object $\boldsymbol{1}$, which is a tensor unit. In particular, $\boldsymbol{1}\otimes\pi\cong\pi\otimes\boldsymbol{1}\cong\pi$.

• •

Every object $\pi$ (resp. morphism $T$) has a conjugate $\bar{\pi}$ (resp. $\overline{T}$).

• •

Semisimplicity: Every object is a (finite) direct sum of simple objects. (By definition, an object $\pi$ is simple if $\mathrm{End}(\pi)\mathrel{\mathop{:}}=\mathrm{Mor}(\pi,\pi)\cong\mathbb{C}$.)

• •

The tensor unit $\boldsymbol{1}$ is a simple object.

• •

Each endomorphism space $\mathrm{End}(\pi)$ is a finite-dimensional unital C*-algebra with a canonical faithful positive trace $\mathrm{Tr}_{\pi}$ that satisfies $\mathrm{Tr}_{\pi\otimes\rho}(T_{1}\otimes T_{2})=\mathrm{Tr}_{\pi}(T_{1})\mathrm{Tr}_{\rho}(T_{2})$ for all $T_{1}\in\mathrm{End}(\pi)$ and $T_{2}\in\mathrm{End}(\rho)$. (These traces were constructed in this general setting by Longo and Roberts [49].)

Moreover, the following is true. Denote by $\Lambda$ the set of isomorphism classes of simple objects in $\mathcal{C}$ and choose a representative $\pi_{\lambda}$ in $\lambda$ for each $\lambda\in\Lambda$. If $\pi\cong\bigoplus_{\lambda\in\Lambda}\pi_{\lambda}^{\oplus N_{\pi}^{\lambda}}$ for some multiplicities $N_{\pi}^{\lambda}\in\mathbb{N}_{0}$ then

[TABLE]

as C*-algebras and $N_{\pi}^{\lambda}=\dim_{\mathbb{C}}\mathrm{Mor}(\pi_{\lambda},\pi)$ for all $\lambda\in\Lambda$.

• •

Each object $\pi$ has a quantum dimension $d(\pi)=\mathrm{Tr}_{\pi}(\mathrm{id}_{\pi})$, which satisfies $d(\pi\otimes\rho)=d(\pi)d(\rho)$ and $d(\pi\oplus\rho)=d(\pi)+d(\rho)$ (cf. [49]).

• •

Frobenius Reciprocity, that is, certain isomorphisms

[TABLE]

In particular, $\boldsymbol{1}$ is always a direct summand of $\pi\otimes\bar{\pi}$.

Moreover, the operations and objects ∗, \bar{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\cdot}}, $\otimes$, $\oplus$, $\boldsymbol{0}$ and $\boldsymbol{1}$ satisfy certain natural compatibility conditions, which we will not state here.

Example 2.1*.*

In addition to the examples that we mentioned above, the category $\mathbf{Rep}(G)$ of continuous finite-dimensional unitary representations of a compact (quantum) group $G$ is also a rigid C*-tensor category (cf. e.g. [52]).

Let us end this section by stating a fact that we will use later.

Remark 2.2*.*

Let $\pi$ be an object in a rigid C*-tensor category $\mathcal{C}$. Then the unitary equivalence classes of projections in the finite-dimensional C*-algebra $\mathrm{End}(\pi)$ are indexed by the (isomorphism classes of) direct summands $\rho$ of $\pi$. More precisely, if we denote the class corresponding to $\rho$ by $X_{\rho}$ then a projection $p\in\mathrm{End}(\pi)$ belongs to $X_{\rho}$ if and only if there exists a morphism $v\colon\rho\to\pi$ such that $v^{*}v=\mathrm{id}_{\rho}$ and $vv^{*}=p$. This is essentially the statement that $\mathcal{C}$ is a C*-category that admits direct sums and is semisimple.

2.1.2 Braided C*-tensor categories

The examples that we are most interested in have an additional feature, namely a unitary braiding, which we proceed to define following [16].

A unitary braiding $c_{-,-}$ on a (strict) rigid C*-tensor category $\mathcal{C}$ is an assignment of an isomorphism

[TABLE]

to every pair $(\pi,\rho)$ of objects in $\mathcal{C}$, which

• •

is natural, i.e., given morphisms $T_{j}\colon\pi_{j}\to\rho_{j}$ ($j=1,2$), we have that

[TABLE]

• •

satisfies the hexagon identities, i.e.,

[TABLE]

• •

is unitary, i.e.,

[TABLE]

Following [16], we call a rigid C*-tensor category with a distinguished unitary braiding a braided C-tensor category*.

Example 2.3*.*

The categories $\mathbf{Hilb}$ and $\mathbf{Rep}(G)$, where $G$ is a (genuine, i.e., not “quantum”) compact group, have a unitary braiding. Namely, to a pair $(\pi,\rho)$ of unitary representations, $\pi\colon G\to\mathcal{U}(H)$ and $\rho\colon G\to\mathcal{U}(K)$, one associates the isomorphism $c_{\pi,\rho}\colon H\otimes K\to K\otimes H$ defined by $\xi\otimes\eta\mapsto\eta\otimes\xi$. This braiding is symmetric (or trivial) in the sense that $c_{\rho,\pi}\circ c_{\pi,\rho}=\mathrm{id}_{\pi\otimes\rho}$. (The categories $\mathbf{Rep}(G)$, where $G$ ranges over all compact groups, were shown to be characterized by the existence of such a braiding by Doplicher and Roberts in [10].)

We are interested in certain rigid C*-tensor categories that have finitely many (isomorphism classes of) simple objects (i.e., they are what are often called fusion categories) as well as an asymmetric, even non-degenerate, unitary braiding (i.e., they are what are often called unitary modular tensor categories). Non-degeneracy of a braiding is defined as follows. To a rigid C*-tensor category $\mathcal{C}$ with a finite set $\Lambda$ of simple objects and a unitary braiding $c_{-,-}$, one can associate a matrix $S\in M_{\Lambda}(\mathbb{C})$, called the modular $S$-matrix, by setting

[TABLE]

for $\mu,\nu\in\Lambda$. The braiding $c_{-,-}$ is said to be non-degenerate, and the category $\mathcal{C}$ to be modular, if $S$ is an invertible matrix. (See e.g. [28] for more information.)

Example 2.4*.*

For $\mathcal{C}=\mathbf{Rep}(G)$, where $G$ is a finite group, the modular $S$-matrix is given by

[TABLE]

whereby it is of rank one. Thus, this category is not modular unless $G$ is the trivial group, in which case the category in question is $\mathbf{Hilb}$.

Example 2.5*.*

It is rather more difficult to construct examples ($\neq\mathbf{Hilb}$) of modular braided C*-tensor categories. One class of examples arises from WZW models in 2d conformal field theory or, in other words, as categories of integrable highest-weight modules over certain affine Lie algebras or vertex operator algebras. (See e.g. [40] and the references therein.) These categories, which we will denote by $\mathbf{Rep}_{k}(G)$, are parameterized by pairs $(G,k)$, where $G$ is a simple, connected, simply connected, compact Lie group (such as $\mathrm{SU}(n)$ for $n\geq 2$) and $k$ is a positive integer that is known as the “level”. We will use the symbol $G_{k}$ to denote the quantum group (see below) or WZW model that underlies $\mathbf{Rep}_{k}(G)$.

The category $\mathbf{Rep}_{k}(G)$ can also be realized using level $k$ positive energy representations of the loop group $LG$ of $G$, which is the group of smooth maps $S^{1}\to G$ under pointwise multiplication. Note, however, that the tensor product in $\mathbf{Rep}_{k}(G)$ does not arise from the usual tensor product of representations (see [56], [67]). Moreover, the category $\mathbf{Rep}_{k}(G)$ also arises from the representation theory of a certain quantum group $U_{q}(G)$ at a root of unity $q$, which is a Hopf algebra associated to $G$ (see [69]). For an overview of the various realizations of $\mathbf{Rep}_{k}(G)$, we refer the reader to [38].

Below, we will explain relevant aspects of these categories, but we will not provide the reader with a detailed construction. For this, we direct the reader to the references in [38]. (See also Remark 3.7 below.)

2.2 Fusion rings

Let $\mathcal{C}$ be a rigid C*-tensor category. We retain the notation of the previous section. In particular, $\Lambda$ is the set of (isomorphism classes of) simple objects in $\mathcal{C}$. Given an object $\pi$ in $\mathcal{C}$ and $\mu,\nu\in\Lambda$, we have that

[TABLE]

for certain multiplicities $N_{\pi,\nu}^{\mu}=N_{\pi\otimes\nu}^{\mu}$, which are called the fusion rules of the category and are organized into fusion matrices $N_{\pi}=(N_{\pi,\nu}^{\mu})_{\nu,\mu}\in M_{\Lambda}(\mathbb{C})$. It is sometimes useful to consider the fusion graph $\Gamma_{\pi}$ of $\mathcal{C}$ with respect to $\pi$, whose vertex set is $\Lambda$ and whose adjacency matrix is $N_{\pi}$. In other words, in $\Gamma_{\pi}$ there are $N_{\pi,\nu}^{\mu}$ edges from the vertex $\nu\in\Lambda$ to the vertex $\mu\in\Lambda$.

Remark 2.6*.*

If $\Lambda$ is a finite set then the quantum dimension $d(\pi)$ of $\pi$ is precisely $\|N_{\pi}\|$, i.e., the Perron–Frobenius eigenvalue of $N_{\pi}$. The corresponding Perron–Frobenius eigenvector is $(d(\mu))_{\mu\in\Lambda}$. (See e.g. [28] for more information.)

The fusion ring $F_{\mathcal{C}}$ of $\mathcal{C}$ is defined as follows. As a group, it is the free abelian group $\mathbb{Z}^{(\Lambda)}=\bigoplus_{\mu\in\Lambda}\mathbb{Z}\mu$, which we turn into an associative unital ring by imposing the product

[TABLE]

for $\lambda,\nu\in\Lambda$. (See e.g. [26], [28] for information on fusion rings.)

Throughout this paper, we restrict attention to categories whose fusion ring is commutative. Braided C*-tensor categories of course have this property. However, there are many examples of rigid C*-tensor categories that have a commutative fusion ring but no unitary braiding. For instance, as mentioned in [16], Drinfeld–Jimbo quantized universal enveloping algebras $U_{q}(G)$ with $q>1$ are such examples. Moreover, it is easy to find examples of rigid C*-tensor categories whose fusion ring is non-commutative. For instance, given a non-abelian discrete group $\Gamma$, the category $\mathbf{Rep}(\hat{\Gamma})$ is such an example. (Here, $\hat{\Gamma}$ is the compact quantum group $(C_{r}^{*}(\Gamma),\Delta)$, where $C_{r}^{*}(\Gamma)$ is the reduced group C*-algebra of $\Gamma$ and the co-multiplication $\Delta$ is defined by $u_{g}\mapsto u_{g}\otimes u_{g}$ for $g\in\Gamma$, where $\{u_{g}\}_{g\in\Gamma}$ are the canonical unitary elements in $C_{r}^{*}(\Gamma)$.)

The fusion ring of the category $\mathbf{Rep}_{k}(G)$ is called the Verlinde ring of $G$ at level $k$ (after Verlinde, who studied these rings in [64]) and we will denote it by $\mathrm{Ver}_{k}(G)$. (Here, and for the remainder of this section, $G$ denotes a simple, connected, simply connected, compact Lie group.) The fusion rules of $\mathbf{Rep}_{k}(G)$ may be viewed as a truncation of the fusion rules of $\mathbf{Rep}(G)$. They are given by the Kač–Walton Formula (cf. e.g. [28] and the references therein).

It is a classical fact that the simple objects $\{\pi_{\vec{\lambda}}\}_{\vec{\lambda}}$ in the category $\mathbf{Rep}(G)$ are naturally parameterized by the set $\mathbb{N}_{0}^{\times r}$ of Dynkin labels $\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{r})$, where $r$ is the rank of $G$, in such a way that the zero vector $\vec{0}$ corresponds to the trivial representation and the standard basis vectors $\vec{e}_{j}$ correspond to the fundamental representations. Meanwhile, the simple objects $\{\pi_{\vec{\lambda}}\}_{\vec{\lambda}}$ in the category $\mathbf{Rep}_{k}(G)$ are naturally indexed by a certain finite subset of $\mathbb{N}_{0}^{\times r}$ corresponding to integrable highest weights at level $k$. Explicitly, the Dynkin labels $\vec{\lambda}$ that correspond to simple objects in $\mathbf{Rep}_{k}(G)$ are exactly those for which the “level” $\ell(\vec{\lambda})\mathrel{\mathop{:}}=\sum_{j=1}^{r}a_{j}^{\vee}\lambda_{j}$ of $\vec{\lambda}$ is at most $k$. Here, $a_{1}^{\vee},\ldots,a_{r}^{\vee}$ are the colabels of the group $G$, which are e.g. explicitly given in [27].

Example 2.7*.*

In the category $\mathbf{Rep}(\mathrm{SU}(2))$, the simple objects are indexed by the set $\Lambda=\mathbb{N}_{0}$ and the fusion rules are given by

[TABLE]

for $i,j\in\mathbb{N}_{0}$. Meanwhile, in the category $\mathbf{Rep}_{k}(\mathrm{SU}(2))$, the simple objects are indexed by the set $\Lambda=\{0,1,\ldots,k\}$ and the fusion rules are given by

[TABLE]

for $0\leq i,j\leq k$ (as shown by Gepner and Witten in [32]).

Example 2.8*.*

In the category $\mathbf{Rep}(\mathrm{SU}(3))$, the simple objects are indexed by the set $\Lambda=\mathbb{N}_{0}^{\times 2}$ and the fusion rules are determined by the fusion graph $\Gamma_{\pi_{(1,0)}}$ of $\mathbf{Rep}(\mathrm{SU}(3))$ with respect to the fundamental representation $\pi_{(1,0)}$, which is shown in figure 1. Note that the fusion graph of $\mathbf{Rep}(\mathrm{SU}(3))$ with respect to $\pi_{(0,1)}=\bar{\pi}_{(1,0)}$ is obtained from $\Gamma_{\pi_{(1,0)}}$ by reversing all of the edges.

In the category $\mathbf{Rep}_{k}(\mathrm{SU}(3))$, the simple objects are indexed by the set $\Lambda=\{\vec{\lambda}\in\mathbb{N}_{0}^{\times 2}\,:\,\ell(\vec{\lambda})\leq k\}$, where $\ell(\vec{\lambda})=\lambda_{1}+\lambda_{2}$ for $\vec{\lambda}=(\lambda_{1},\lambda_{2})\in\mathbb{N}_{0}^{\times 2}$. The fusion rules are determined by the fusion graph $\Gamma_{\pi_{(1,0)}}$ of $\mathbf{Rep}_{k}(\mathrm{SU}(3))$ with respect to the simple object $\pi_{(1,0)}$, which is obtained from the graph in figure 1 by discarding all vertices $\vec{\lambda}$ for which $\ell(\vec{\lambda})>k$ (as well as any edges that are connected to them). Again, the fusion graph of $\mathbf{Rep}_{k}(\mathrm{SU}(3))$ with respect to $\pi_{(0,1)}=\bar{\pi}_{(1,0)}$ is obtained from $\Gamma_{\pi_{(1,0)}}$ by reversing all edges. The fusion rules of $\mathbf{Rep}_{k}(\mathrm{SU}(3))$ were explicitly calculated at all “levels” $k$ in [4].

In our $K$-theory computations, we will use a result of Gepner [30], which was transferred from the setting of $\mathrm{Ver}_{k}(G)\otimes\mathbb{C}$ (the Verlinde algebra) to the setting of $\mathrm{Ver}_{k}(G)$ by Fuchs [26]. To state it, we first recall some facts and introduce some terminology and notation. It is a classical theorem that when $\mathcal{C}=\mathbf{Rep}(G)$, in which case $F_{\mathcal{C}}$ is the representation ring $R(G)$ of $G$, there is a ring isomorphism $F_{\mathcal{C}}\to\mathbb{Z}[x_{1},\ldots,x_{r}]$ that maps (the isomorphism class of) the $j$’th fundamental representation to the variable $x_{j}$ and the trivial representation to $1$. (As above, $r$ is the rank of $G$.) We will denote the image of $\pi_{\vec{\lambda}}$ under this isomorphism by $Q_{\vec{\lambda}}^{G}(x_{1},\ldots,x_{r})$. The fusion variety $V$ associated to the category $\mathbf{Rep}_{k}(G)$ is the subset

[TABLE]

of $\mathbb{C}^{r}$. In this formula, $x_{\vec{\lambda}}^{(j)}=S_{\vec{e}_{j},\vec{\lambda}}/S_{\vec{0},\vec{\lambda}}$ for $j=1,\ldots,r$, where $S$ is the modular $S$-matrix for $\mathbf{Rep}_{k}(G)$, which is given by the Kač–Peterson Formula [45]. (The formula is e.g. spelled out for each $G$ in [27] and for $G=\mathrm{SU}(3)$ in [19], but note that the formula for the modular $S$-matrix on page 400 of [19] contains a typographical error: The ‘$+$’ in the second term should be ‘$-$’.) The fusion ideal $J_{k}(G)$ associated to the category $\mathbf{Rep}_{k}(G)$ is the ideal

[TABLE]

of $\mathbb{Z}[x_{1},\ldots,x_{r}]$. In particular, $(x_{\vec{0}}^{(1)},\ldots,x_{\vec{0}}^{(r)})=(S_{\vec{e}_{1},\vec{0}}/S_{\vec{0},\vec{0}},\ldots,S_{\vec{e}_{r},\vec{0}}/S_{\vec{0},\vec{0}})=(d(\pi_{\vec{e}_{1}}),\ldots,d(\pi_{\vec{e}_{r}}))$ (cf. e.g. [28]) is a common zero of the polynomials in $J_{k}(G)$. (As usual, $d$ denotes the quantum dimension.)

Theorem 2.9** (Gepner, Fuchs).**

The fusion ring of $\mathbf{Rep}_{k}(G)$, that is, the Verlinde ring $\mathrm{Ver}_{k}(G)$, is isomorphic to $\mathbb{Z}[x_{1},\ldots,x_{r}]/J_{k}(G)$ in such a way that the simple object $\pi_{\vec{\lambda}}$ corresponds to the coset $[Q^{G}_{\vec{\lambda}}(x_{1},\ldots,x_{r})]$.

In [30], Gepner also proved that

[TABLE]

and

[TABLE]

where, given elements $r_{1},\ldots,r_{n}$ (resp. a subset $X$) in a ring $R$, we denote by $\langle r_{1},\ldots,r_{n}\rangle$ (resp. $\langle X\rangle$) the (two-sided) ideal generated by the set $\{r_{1},\ldots,r_{n}\}$ (resp. $X$). Below, we will use these facts as well as the formula

[TABLE]

which follows from the work of Bourdeau–Mlawer–Riggs–Schnitzer [5] and Gepner–Schwimmer [31], and the formula

[TABLE]

which may e.g. be deduced from the work of Douglas (cf. Theorem 1.1 in [11]). For an in-depth discussion of explicit generating sets for the fusion ideals, we refer the reader to the papers [6] and [12].

3 AF-algebras from braided categories

3.1 Construction and basic examples

Let $\mathcal{C}$ be a rigid C*-tensor category and fix an object $\pi$ in $\mathcal{C}$. Put $\sigma=\bar{\pi}\otimes\pi$ and, for each $n\in\mathbb{N}_{0}$, $A(\mathcal{C},\pi)_{n}=\mathrm{End}(\sigma^{\otimes n})$. Define, for each $n\in\mathbb{N}_{0}$, a $*$-homomorphism $\iota_{n}\colon A(\mathcal{C},\pi)_{n}\to A(\mathcal{C},\pi)_{n+1}$ by $\iota_{n}(T)=T\otimes\mathrm{id}_{\sigma}$ for $T\colon\sigma^{\otimes n}\to\sigma^{\otimes n}$. Since

[TABLE]

for all $T\colon\sigma^{\otimes n}\to\sigma^{\otimes n}$, it follows that each $\iota_{n}$ is injective, hence isometric. Thus, we may define the inductive limit C*-algebra

[TABLE]

which is a unital AF-algebra. Note that, up to $*$-isomorphism, $A(\mathcal{C},\pi)$ only depends on the fusion rules for $\pi$, i.e., on the matrix $N_{\pi}$.

Example 3.1*.*

Unsurprisingly, the easiest examples arise from the category $\mathbf{Hilb}$. If $V$ is a Hilbert space of dimension $N$ then $A(\mathbf{Hilb},V)\cong M_{N^{\infty}}$, where $M_{N^{\infty}}$ is the infinite tensor product $M_{N}(\mathbb{C})\otimes M_{N}(\mathbb{C})\otimes\cdots$, i.e., the UHF-algebra associated to the “supernatural number” $N^{\infty}$. In particular, $A(\mathbf{Hilb},\mathbb{C})\cong\mathbb{C}$ while $A(\mathbf{Hilb},\mathbb{C}^{2})$ is $*$-isomorphic to the CAR-algebra.

To get a slightly more complicated example, let $\pi$ be an object of $\mathbf{Rep}(G)$, where $G$ is a compact group. Suppose that $\pi$ is an $N$-dimensional representation. Then $A(\mathbf{Rep}(G),\pi)$ is $*$-isomorphic to the fixed point algebra $M_{N^{\infty}}^{G}$ of the following action of $G$ on $M_{N^{\infty}}$ by automorphisms. For each $g\in G$, view $\pi(g)$ and $\bar{\pi}(g)$ as unitary matrices in $M_{N}(\mathbb{C})$. Then $G$ acts via the formula

[TABLE]

As we have mentioned before, such fixed point algebras were much studied by Wassermann [66], Handelman and Rossmann [36, 37], and Handelman [34, 35] in the early 1980s, and some of their $K$-theory computations will be presented later in this paper (cf. Theorem 4.5, Theorem 4.20 and Corollary 4.23).

Remark 3.2*.*

As e.g. explained in Remark 8.2 of [55], it follows from the work of many people, including Jones [41], Wenzl [68, 69], Popa [54], Banica [2] and Xu [70], that $A(\mathcal{C},\pi)$ is the inductive limit of a tower of higher relative commutants arising from a certain subfactor associated to the pair $(\mathcal{C},\pi)$.

3.2 From unitary braidings to $*$-homomorphisms

We will next, following [16], define a $*$-homomorphism $A(\mathcal{C},\pi)\otimes A(\mathcal{C},\pi)\to A(\mathcal{C},\pi)$ that will turn out to induce a multiplication map on $K_{0}(A(\mathcal{C},\pi))$.

3.2.1 Definition of the $*$-homomorphism

Assume now moreover that $\mathcal{C}$ is a braided C*-tensor category with unitary braiding $c_{-,-}$. For each $n\in\mathbb{N}$, Erlijman and Wenzl in [16] defined a $*$-homomorphism

[TABLE]

by

[TABLE]

for $T_{1},T_{2}\in A(\mathcal{C},\pi)_{n}=\mathrm{End}(\sigma^{\otimes n})$, where

[TABLE]

Using a well-known graphical calculus for braided C*-tensor categories, $\theta_{n}(T_{1}\otimes T_{2})$ may be depicted as follows when $n=3$.

\theta_{n}(T_{1}\otimes T_{2})=$$T_{1}$$T_{2}

The following lemma was proved in [16].

Lemma 3.3** (Erlijman–Wenzl).**

For each $n\in\mathbb{N}$,

[TABLE]

In fact, Erlijman and Wenzl used the maps $\theta_{n}$ to construct new examples of subfactors. In the present paper, we only use the above lemma (and the fact that each $\theta_{n}$ is isometric) to get an induced $*$-homomorphism

[TABLE]

For the convenience of the reader, we give an algebraic proof of the lemma.

Proof of Lemma 3.3.

(For simplicity, we give the proof in the case where $\mathcal{C}$ is a strict category, but this is not an actual restriction.) Note first that

[TABLE]

for all $n\geq 1$ by the hexagon identities.

We next claim that

[TABLE]

for all $n\geq 1$. The statement is clear for $n=1$. Assume that the statement holds for $n=k-1$. Then, writing $\mathrm{id}$ for $\mathrm{id}_{\sigma}$ and using the fact that

[TABLE]

for all $n\geq 1$ by definition, we get that

[TABLE]

which is exactly the statement for $n=k$.

We are now ready to prove that

[TABLE]

for all $n\geq 1$. Setting \gamma=\big{(}\theta_{n+1}\circ(\iota_{n}\otimes\iota_{n})\big{)}(T_{1}\otimes T_{2})), equation (1) and the naturality of the braiding imply that

[TABLE]

which proves what we wanted. ∎

Example 3.4*.*

Let $\mathcal{C}$ be the category $\mathbf{Rep}(G)$, where $G$ is a compact group. Then, as we saw in Example 3.1, $A(\mathbf{Rep}(G),\pi)$ is $*$-isomorphic to a fixed point algebra $M_{N^{\infty}}^{G}$, where $N$ is the dimension of $\pi$. Under this identification (recalling the unitary braiding on $\mathbf{Rep}(G)$ from Example 2.3), the above $*$-homomorphism $\theta\colon A(\mathcal{C},\pi)\otimes A(\mathcal{C},\pi)\to A(\mathcal{C},\pi)$ corresponds to the restriction $M_{N^{\infty}}^{G}\otimes M_{N^{\infty}}^{G}\to M_{N^{\infty}}^{G}$ of the $*$-isomorphism $M_{N^{\infty}}\otimes M_{N^{\infty}}\to M_{N^{\infty}}$ that interlaces the tensor factors.

3.2.2 SU(2) and Temperley–Lieb–Jones

algebras

We will next explain how the $*$-homomorphism $\theta$ is defined when $\mathcal{C}\!=\!\mathbf{Rep}_{k}(\mathrm{SU}(2))$ and $\pi\!=\!\pi_{1}$. We recall first the notion of an $(m,n)$-Temperley–Lieb diagram (for $m,n\in\mathbb{N}_{0}$ of equal parity), which first appeared in the work of Kauffman [46]. Such a diagram consists of $(n+m)/2$ non-crossing smooth strands inside a rectangle with $m$ marked points on the upper edge and $n$ marked points on the lower edge, each marked point being connected to a unique strand. For example, the following is a $(4,6)$-Temperley–Lieb diagram.

We next use Temperley-Lieb diagrams to define some (complex) algebras. As a vector space, the $n$’th Temperley–Lieb algebra with parameter $\delta\in\mathbb{C}$ (for $n\in\mathbb{N}_{0}$) is the formal (complex) linear span of all $(n,n)$-Temperley–Lieb diagrams, i.e.,

[TABLE]

We define the product of two diagrams by stacking them, aligning marked points, smoothing strands, and removing all closed loops at the cost of multiplying by $\delta^{N}$, where $N$ is the total number of closed loops. The following picture illustrates the product of two $(4,4)$-Temperley–Lieb diagrams in the algebra $\mathrm{TL}_{4}(\delta)$.

$=\,\,\delta$

The $n$’th Temperley–Lieb–Jones algebra with parameter $\delta$ is defined as

[TABLE]

where the trace $\mathrm{Tr}$ (often called a Markov trace) is defined on diagrams by

Note that $\widetilde{\mathrm{TL}}_{n}(\delta)$ is a $*$-algebra under the $*$-operation that reflects diagrams about a horizontal axis and that the map $i_{n}\colon\widetilde{\mathrm{TL}}_{n}(\delta)\to\widetilde{\mathrm{TL}}_{n+1}(\delta)$, defined on diagrams by placing a vertical strand on the right side, is an injective $*$-homomorphism.

We believe that the following result may be ascribed to Jones [41], Kauffman [46] and Goodman–de la Harpe–Jones [33].

Theorem 3.5** (Jones, Kauffman, Goodman–de la Harpe–Jones).**

Let $k\in\mathbb{N}$ be given and put $\delta=2\cos(\pi/(k+2))$. Then $\widetilde{\mathrm{TL}}_{n}(\delta)$ is a C-algebra for all $n$, and there is a $*$-isomorphism*

[TABLE]

Remark 3.6*.*

Put $\mathscr{J}=\{2\cos(\pi/(k+2))\,:\,k\in\mathbb{N}\}\cup[2,\infty)$. A version of Jones’ Index Rigidity Theorem from [41] states that, given a parameter $\delta$, $\mathrm{Tr}$ is a positive linear functional on $\mathrm{TL}_{n}(\delta)$ for all $n$ if and only if $\delta\in\mathscr{J}$. In particular, $\widetilde{\mathrm{TL}}_{n}(\delta)$ is a C*-algebra for all $n$ whenever $\delta\in\mathscr{J}$.

Under the identification in the preceding theorem, $\theta$ is given by superposition of diagrams. For instance, if we define the $\theta_{n}$ and $U_{n}$ in terms of the self-conjugate object $\pi=\pi_{1}$ rather than $\sigma$ then $\theta_{3}$ may be depicted in the following way.

\bigotimes$$\mapsto$$=

Here, the diagram in the top-most rectangle on the right represents the unitary element $U_{3}$ and the crossings are to be interpreted as follows.

\,\,=ie^{\frac{\pi i}{2(k+2)}}$$\,-\,ie^{-\frac{\pi i}{2(k+2)}}

This formula for a crossing arose from Kauffman’s work on knot invariants [46].

Remark 3.7*.*

The Temperley–Lieb algebras first appeared in the work of Temperley and Lieb [61] on Potts and ice-type models in statistical mechanics, in which they were defined in terms of generators and relations. These relations reappeared in the work of Jones [41], in which the Temperley–Lieb–Jones algebras manifested as subalgebras of higher relative commutants of subfactors (see also [33]). Soon, in a paper [46] about a knot invariant introduced by Jones [42], Kauffman described Temperley–Lieb algebras in terms of Temperley–Lieb diagrams (see also [44]).

Later, it was realized that a diagrammatic description could also be given for standard invariants of subfactors (cf. Jones’ introduction of subfactor planar algebras [43] based on work of Popa [54]) and tensor categories (cf. [63] as well as e.g. [16], [8]). For example, the category $\mathbf{Rep}_{k}(\mathrm{SU}(2))$ was described in terms of Temperley–Lieb diagrams (cf. [63] as well as e.g. [71], [9], [20]) while the category $\mathbf{Rep}_{k}(\mathrm{SU}(3))$ was described in terms of so-called A2-Temperley–Lieb diagrams (cf. [48], in which A2-Temperley–Lieb diagrams were first introduced, as well as e.g. [68], [9], [18], [20]). In fact, it was observed that rigid C*-tensor categories are in some sense the same as so-called factor planar algebras (cf. [51], [8]).

3.3 Induced (ordered) ring structure in $K$-theory

Using the “external product” $K_{0}(A(\mathcal{C},\pi))\otimes_{\mathbb{Z}}K_{0}(A(\mathcal{C},\pi))\to K_{0}(A(\mathcal{C},\pi)\otimes A(\mathcal{C},\pi))$ in $K$-theory (cf. e.g. section 4.7 of [39]), which is in fact a group isomorphism in this case, the $*$-homomorphism $\theta$ induces a binary operation

[TABLE]

which will turn out to be an associative product on $K_{0}(A(\mathcal{C},\pi))$ that endows it with the structure of a unital ring. We will prove this by identifying $K_{0}(A(\mathcal{C},\pi))$ with a subgroup of the localization $F_{\mathcal{C}}[\sigma^{-1}]$ in such a way that $K_{0}(\theta)$ corresponds to the product on $F_{\mathcal{C}}[\sigma^{-1}]$. (Recall that $F_{\mathcal{C}}[\sigma^{-1}]$ consists of equivalence classes of formal fractions $x/\sigma^{n}$ with $x\in F_{\mathcal{C}}$ and $n\in\mathbb{N}_{0}$ under the equivalence relation $\sim$ defined as follows: $x/\sigma^{n}\sim y/\sigma^{m}$ if $\sigma^{N}(\sigma^{m}x-\sigma^{n}y)=0$ in $F_{\mathcal{C}}$ for some $N\in\mathbb{N}_{0}$.)

Identifying $K_{0}(A(\mathcal{C},\pi)_{n})$ with $\bigoplus_{\mu\prec\sigma^{\otimes n}}\mathbb{Z}\mu$, where $\mu\prec\sigma^{\otimes n}$ is to be read as“$\mu$ occurs as a direct summand of $\sigma^{\otimes n}$”, we have that $K_{0}(A(\mathcal{C},\pi))$ is isomorphic (as an ordered group) to the limit of the inductive sequence

[TABLE]

where $M_{\sigma}$ is defined on basis vectors by

[TABLE]

and the positive cone in $\bigoplus_{\mu\prec\sigma^{\otimes n}}\mathbb{Z}\mu$ is $\{\sum_{\mu\prec\sigma^{\otimes n}}v_{\mu}\mu\,:\,v_{\mu}\geq 0\text{ for all }\mu\}$. Note that if we view $\bigoplus_{\mu\prec\sigma^{\otimes n}}\mathbb{Z}\mu$ as a subset of $F_{\mathcal{C}}$ then $M_{\sigma}$ just multiplies by $\sigma$. Define, for each $n\in\mathbb{N}_{0}$, a group homomorphism

[TABLE]

by $\phi_{n}(\mu)=\mu/\sigma^{n}$ for $\mu\prec\sigma^{\otimes n}$. Then clearly $\phi_{n+1}\circ M_{\sigma}=\phi_{n}$ for all $n$, whereby we get an induced group homomorphism

[TABLE]

It is straightforward to verify that $\phi$ is injective. We claim that $m\circ(\phi_{n}\otimes\phi_{n})=\phi_{2n}\circ K_{0}(\theta_{n})$, i.e., that the following diagram commutes, for all $n$, where $m$ is the product on $F_{\mathcal{C}}[\sigma^{-1}]$.

[TABLE]

By Remark 2.2, a projection $p\in\mathrm{End}(\sigma^{\otimes n})$ belongs to the (unitary equivalence) class corresponding to the direct summand $\mu$ of $\sigma^{\otimes n}$ if and only if there exists an element $v\in\mathrm{Mor}(\mu,\sigma^{\otimes n})$ such that $v^{*}v=\mathrm{id}_{\mu}$ and $vv^{*}=p$. It follows that if $p\in\mathrm{End}(\sigma^{\otimes n})$ belongs to the class corresponding to $\mu$ and $q\in\mathrm{End}(\sigma^{\otimes n})$ to the one corresponding to $\nu$ then $\theta_{n}(p\otimes q)=U_{n}(p\otimes q)U_{n}^{*}\in\mathrm{End}(\sigma^{\otimes 2n})$ belongs to the class corresponding to the direct summand $\mu\otimes\nu$ of $\sigma^{\otimes 2n}$. This proves the claim. We conclude that $m\circ(\phi\otimes\phi)=\phi\circ K_{0}(\theta)$, whereby $K_{0}(\theta)$ has the properties that we asserted at the beginning of this section.

Remark 3.8*.*

Denote by $\mathcal{C}_{1}$ the full C*-tensor subcategory of $\mathcal{C}$ generated by the simple objects that occur as direct summands in tensor powers of $\sigma$. Then $\phi$ maps into the subring $\mathrm{F}_{\mathcal{C}_{1}}[\sigma^{-1}]$ of $\mathrm{F}_{\mathcal{C}}[\sigma^{-1}]$.

By an ordered ring, we shall mean a commutative ring $R$ with identity equipped with a positive cone $R_{+}$, i.e., a subset $R_{+}\subset R$ satisfying $R_{+}+R_{+}\subset R_{+}$, $R_{+}\cap(-R_{+})=\{0\}$ and $R=R_{+}-R_{+}$, for which $R_{+}\cdot R_{+}\subset R_{+}$ and in which the identity element $1$ is an order unit for the ordered group $(R,R_{+})$ under addition (cf. Remark 4.22). In particular, we have the following. Define a positive cone in $F_{\mathcal{C}}[\sigma^{-1}]$ by

[TABLE]

where $d$ denotes the ring homomorphism $d\colon\mathrm{F}_{\mathcal{C}}[\sigma^{-1}]\to\mathbb{R}$ defined by $d(\mu/\sigma^{n})=d(\mu)/d(\sigma)^{n}$ for $\mu\in\Lambda$ and $n\in\mathbb{N}_{0}$. (On the right hand side, $d$ denotes the quantum dimension in $\mathcal{C}$.) Then $K_{0}(A(\mathcal{C},\pi))$ and $\mathrm{F}_{\mathcal{C}}[\sigma^{-1}]$ are ordered rings and $\phi$ is a positive map, i.e., $\phi(K_{0}(A(\mathcal{C},\pi))_{+})\subset\mathrm{F}_{\mathcal{C}}[\sigma^{-1}]_{+}$.

Remark 3.9*.*

Assume now that $\mathcal{C}$ has only finitely many simple objects. We claim that $\phi$ is an ordered ring isomorphism onto the ordered subring $\mathrm{F}_{\mathcal{C}_{1}}[\sigma^{-1}]$ of $\mathrm{F}_{\mathcal{C}}[\sigma^{-1}]$. This follows from the fact that, in this case, the directed system in equation (2) is stationary in the sense of Chapter 6 in [13], and is closely related to the fact that, by Theorem 6.1 in [13], $A(\mathcal{C},\pi)$ has a unique (normalized) faithful positive trace. (Note also that $A(\mathcal{C},\pi)$ is a simple C*-algebra.)

Let us provide some further details. It is easy to show that $\phi$ is a ring isomorphism onto $\mathrm{F}_{\mathcal{C}_{1}}[\sigma^{-1}]$. (In particular, $K_{0}(A(\mathcal{C},\pi))$ is a finitely generated ring.) We will prove that it maps the positive cone of $K_{0}(A(\mathcal{C},\pi))$ onto that of $\mathrm{F}_{\mathcal{C}_{1}}[\sigma^{-1}]$