
TL;DR
This paper characterizes when skew polynomial extensions of Hopf algebras can be given a compatible Hopf structure, simplifying the conditions on the comultiplication map after a change of variables.
Contribution
It shows that, under certain conditions, the comultiplication in skew polynomial extensions can be simplified, providing a complete characterization of such Hopf algebra extensions.
Findings
Simplification of the comultiplication map after change of variables
Complete characterization of Hopf structures on skew polynomial extensions
Applicable to noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension
Abstract
Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism and a -derivation of a Hopf -algebra for when the skew polynomial extension of admits a Hopf algebra structure that is compatible with that of . In fact, they gave a complete characterization of which and can occur under the hypothesis that , with and , where is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that , with is a grouplike element in and when is a domain and is noetherian. In particular, this completely characterizes skew polynomial extensions of aβ¦
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.
Hopf Ore Extensions
Hongdi Huang
Department of Pure Mathematics
University of Waterloo
Waterloo, ON N2L 3G1
Canada
Abstract.
Brown, OβHagan, Zhang, and Zhuang gave a set of conditions on an automorphism and a -derivation of a Hopf -algebra for when the skew polynomial extension of admits a Hopf algebra structure that is compatible with that of . In fact, they gave a complete characterization of which and can occur under the hypothesis that , with and , where is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that , with is a grouplike element in and when is a domain and is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.
Key words and phrases:
Hopf algebras, Ore Extensions, Crossed products
2010 Mathematics Subject Classification:
16T05, 18G20, 16S35.
The author acknowledges support from the National Sciences and Engineering Research Council of Canada.
1. Introduction
Throughout this paper, we take to be a field and all algebras are over . Given a -algebra , a -algebra automorphism of , and a -linear -derivation of that is, for , one can form a skew polynomial extension , which is the -algebra generated by and the indeterminate , subject to the relations for all . Let be a Hopf algebra with multiplication , comultiplication , unit , counit and antipode . One can refer to the excellent books [Swe69, Mon93] for further background on Hopf algebras. Panov [Pan03] asked the following natural question: Given a Hopf algebra , for which automorphisms and -derivations does the skew polynomial ring have a Hopf algebra structure extending the given structure on ? Panov answered this question under this hypothesis that is a skew primitive element of ; i.e., , with . This does not give a complete answer, however, as there are examples of Hopf algebra extensions where this hypothesis does not hold. In particular, Brown, OβHagan, Zhang, and Zhuang [BOZZ15] gave such an example; namely, if we take to be the coordinate ring of the group of upper-triangular unipotent complex matrices then is generated as a -algebra by the coordinate functions , where evaluating and at an element of corresponds to taking respectively the -, -, and -entry of the element. Then with coefficient Hopf algebra , but is not skew primitive, since a straightforward computation shows that . To deal with such examples, [BOZZ15] relaxed Panovβs hypothesis and studied skew polynomial extensions of Hopf algebras in which is of the form , where and . In addition, Brown et al. [BOZZ15] gave the following definition.
Definition 1.1**.**
Let be a Hopf -algebra. A Hopf Ore Extension (HOE) of is a -algebra such that:
- (i)
is a Hopf -algebra with Hopf subalgebra ; 2. (ii)
there exists an algebra automorphism and a -derivation of such that ; 3. (iii)
there are and such that
Using this definition [BOZZ15] gave a complete list of conditions on and for when is a HOE of . In addition, they asked the following question.
Question 1.2**.**
Does the third condition in Definition 1.1 follow from the first two, after a change of the variable ?
This is an important question, as it is unclear in general whether a skew polynomial extension of a Hopf algebra that is itself a Hopf algebra should have the property that the indeterminate is so well-behaved under the comultiplication map.
If one could answer their question, one can remove this final hypothesis and give a complete answer to Panovβs original question. We note that in the case when is a connected Hopf algebra (that is, a Hopf algebra whose coradical is the base field), Brown, OβHagan, Zhang, and Zhuang [BOZZ15] showed that the answer to their question is affirmative and that after a change of variables one can have for some . Recently, it was shown in the case that is a commutative affine Hopf -algebra and is algebraically closed and of characteristic zero then after a change of variables one has [BSM18]. This, then, gives another instance of when the question of Brown, OβHagan, Zhang, and Zhuang [BOZZ15] has a positive answer. In this paper, we will show that under the hypotheses that is a domain and is noetherian then after a change of variables we have , with and a grouplike element of . We are unaware of any Hopf algebra that is a domain for which is not a domain and in light of our work, it would be interesting if one could prove that is a domain whenever is, when is a Hopf algebra. Using this result, along with earlier work of [BOZZ15], we obtain the following theorem.
Theorem 1.3**.**
Let be a noetherian Hopf -algebra and let be an Ore extension of . Suppose that is a domain. Then has a Hopf algebra structure extending that of if and only if after a change of variables we have the following:
- (i)
there exists a grouplike element of and such that , and and ; 2. (ii)
there is a character such that
[TABLE]
for all , where is a left winding automorphism , and is the composition of the corresponding right winding automorphism with conjugation by ; 3. (iii)
the -derivation satisfies the relation
[TABLE]
and
[TABLE]
In the above theorem, the hypothesis that is noetherian is needed to ensure that the antipode is bijective [Skr06], which allows us to use work of [BOZZ15] to get that has a linear form. The hypothesis that is a domain plays a more significant role. However, it appears to be difficult to show that is a domain when is a Hopf algebra that is a domain. Rowen and Saltman [RS13] exhibit division -algebras and , both finite-dimensional over their centres and each containing an algebraically closed field of characteristic 0, such that not a domain. Their construction is non-trivial and it does not obviously lend itself to produce a counterexample in the Hopf algebra case. In this paper, we shall show that is a domain when is algebraically closed of characteristic zero and is a noetherian cocomutative Hopf algebra of finite Gelfand-Kirillov dimension that is a domain. In this case, one has that is isomorphic to the smash product of the enveloping algebra of a finite-dimensional Lie algebra and a finitely generated nilpotent-by-finite group. The underlying Lie algebra is generated by the primitive elements in , and the nilpotent-by-finite group is just the group of grouplike elements of , which acts on via -algebra automorphisms, giving the smash product structure. In this case, we prove the following theorem:
Theorem 1.4**.**
Let be an algebraically closed field of characteristic zero and let be a noetherian cocommutative -Hopf algebra of finite Gelfand-Kirillov dimension that is a domain. Then is a domain. In particular, the results of Theorem 1.3 apply in this setting.
The outline of this paper is as follows. In Β§2, we prove Theorem 1.3 and then in Β§3, we give the proof of Theorem 1.4. We conclude in Β§4 with some pertinent remarks and questions.
2. Proof of Theorem 1.3
In this section, we give a proof of our main result. Throughout this section, we take to be a Hopf -algebra and to be a skew polynomial extension of . Suppose that admits a Hopf algebra structure extending that of . Recall that is a free left -module with basis and is a left -module with basis . Thus we have that
[TABLE]
with . In fact, the hypothesis that is a domain gives that
[TABLE]
with (see [BOZZ15, Lemma 1, Β§2.2]). After a change of variables and corresponding adjustments to , we may assume that For if , then let and so , . Let . Then a straighforward computation shows that , whence is a -derivation. Therefore, .
In the following Lemmas, we will show much more: after a change of variables we have , where is a grouplike element of . This is a significant step, as it shows that can be assumed to have a much simpler form, which gives an explicit Hopf algebra structure on the Ore extension that is compatible with the Hopf structure on .
To begin, we list the following facts which are useful in the proof of the subsequent Lemmas. Using coassociativity of and the form given in Equation (2.0.1) and then comparing the coefficients of all relevant terms (e.g., , ) on both sides of the equation , we obtain the following equations:
[TABLE]
We use these equations to derive additional useful equations. Throughout, we use Sweedler notation to make things more compact, that is, we simply write for an element of , with the understanding that this is actually a sum of pure tensors. We note that we may always assume, in addition, that when we choose an expression for an element , that and are -linearly independent sets. We set
[TABLE]
and
[TABLE]
Observe that applying to Equation (2.0.4), we obtain on the left side
[TABLE]
which is
[TABLE]
and on the right side, we obtain , which is
[TABLE]
Thus we obtain the new equation
[TABLE]
We do not give the complete details of the following computations, as they can be done in a similar manner. We apply to Equation (2.0.2) and we obtain
[TABLE]
By a result of Skryabin [Skr06, Corollary 1], is bijective on and so we must have with and a unit in . Notice that
[TABLE]
The coefficient of in the right side of Equation (2.0.11) is , and the coefficient of on the left side of Equation (2.0.11) is [math]. Since is a unit and is an automorphism, we see that and after the standard fact that then obtain that
[TABLE]
Now we apply the to Equation (2.0.4). We obtain on the left side
[TABLE]
and on the right side
[TABLE]
So we can see that
[TABLE]
Lemma 2.1**.**
Let be a noetherian Hopf -algebra and let admit a Hopf algebra structure with a Hopf subalgebra. Suppose that is a domain. Then after a change of the variables with the property that and corresponding adjustments to and , we can ensure that in Equation (2.0.1); namely, that with
Proof.
Suppose is a domain. As argued in the above part, we have Equation (2.0.1)
[TABLE]
where Using the fact that and that in Equation (2.0.1) gives ; that is,
[TABLE]
Equations (2.0.7), (2.0.8) and (2.1.1) tell us that
[TABLE]
so in particular and are nonzero. Thus, Equations (2.0.9) and (2.0.12) give
[TABLE]
Further, Equation (2.0.10) tells that
[TABLE]
Thus by Equation (2.0.13), we see that Since , and is a domain, we see that . Thus we have shown that . β
Lemma 2.2**.**
Let be a noetherian Hopf -algebra and suppose that admits a Hopf algebra structure with a Hopf subalgebra. Suppose that is a domain and , with . Then after a change of the variable , we can assume that , where is a grouplike element in and . Moreover, and .
Proof.
By the assumption that has the form of Equation (2.0.1) with , we get
[TABLE]
from Equation (2.0.5). Applying to Equation (2.2.1), we obtain that
[TABLE]
By the associativity of the multiplication map, i.e., , if we apply to both sides, then we obtain on the left side
[TABLE]
and on the right side
[TABLE]
Therefore, we have
[TABLE]
Since is a domain and is left invertible, is invertible and . Applying to Equation , and using Equations (2.0.7) and (2.0.8), we see that
[TABLE]
Note that is a unit, and thus
[TABLE]
Combining Equations and , we have
[TABLE]
By Equation (2.2.1), we have
[TABLE]
Applying to Equation (2.2.4) and using the fact that , it results that
[TABLE]
Note again that is a domain and . Cancelling from both sides of Equation (2.2.5), we have
[TABLE]
Then
[TABLE]
Replace , and by , and , respectively. Then we have that
[TABLE]
Using the fact that along with Equation , if we compare the coefficients of , then we obtain the equation: Hence is a grouplike element and thus has inverse. Notice that
[TABLE]
To get a simpler form of later, one can replace by and after a change of variables, we can assume that
[TABLE]
Using the identity that and Equation (2.2.8), a direct computation shows that and . β
As a consequence, we have the following corollary.
Corollary 2.3**.**
Let be a noetherian Hopf -algebra and suppose that admits a Hopf algebra structure extending that of . Suppose that is a domain. Then after a change of variables for the variable , we have , where is a grouplike element in and and thus condition in Definition 1.1 follows from conditions and . In particular, the Question 1.2 has an affirmative answer under the above hypotheses.
This corollary allows us to immediately obtain our main result.
Proof of Theorem 1.3.
Suppose that is a domain. Let be a Hopf algebra with a Hopf structure extending that of the Hopf algebra . Then we have follows from Lemmas 2.1 and 2.2.
The maps , and of must preserve the relation . In particular, we have the following equations:
[TABLE]
Using arguments from [Pan03, Theorem 1.3] and [BOZZ15, Theorem, Β§2.4], we obtain and .
Conversely, a similar argument to that used in [Pan03, Theorem 1.3] and [BOZZ15, Theorem, Β§2.4] shows that , , and imply that is a Hopf algebra with as a Hopf subalgebra. β
3. Cocommutative Hopf algebras
In light of Theorem 1.3, it becomes natural to ask when is a domain. Obviously, a necessary condition is that be a domain, but this is not sufficient in general, even in the case of an algebraically closed base field (see work of Rowen and Saltman [RS13]). We focus on the special case: when is a cocomutative noetherian Hopf algebra of finite Gelfand-Kirillov dimension over an algebraically closed field of characteristic zero that is a domain, and prove in this case that is also a domain. To complete the proof of Theorem 1.4, we will need a result describing when crossed products are domains. The following theorem, whose proof can be found in the book of Passman [Pas89, Corollary 37.11]
Theorem 3.1**.**
Let be an Ore domain and let let be a group and suppose that has a finite subnormal series
[TABLE]
with each quotient locally polycyclic-by-finite. If is torsion-free then the crossed product is an Ore domain. In particular if is an Ore domain and is a torsion-free polyclic-by-finite group then the smash product is a domain.
Using this result, we can give the proof of Theorem 1.4.
Proof of Theorem 1.4.
By a refinement of a result of Kostant (see Bell and Leung [BL14, Proposition 2.1]), we have that where is a finite-dimension Lie algebra over and is a finitely generated nilpotent-by-finite group that acts on . Hence, we have . Let denote the Lie algebra and let denote . Then , where acts on in the natural way induced from the action of on .
Since is a domain, is torsion-free, and thus is also torsion-free. Moreover, is also finitely generated and nilpotent-by-finite, since is. Since is finite-dimensional, we have that is an Ore domain; moreover is a torsion-free polycyclic-by-finite group, and so we see that is a domain from Theorem 3.1. β
Corollary 3.2**.**
Let be an algebraically closed field of characteristic zero and let be a noetherian cocommutative Hopf algebra of finite Gelfand-Kirillov dimension over which is a domain. Let be an Ore extension over . Then has a Hopf algebra structure extending that of if and only if after a change of variables we have the following:
- (i)
there exists a grouplike element of and such that , and and ; 2. (ii)
there is a character such that
[TABLE]
for all , where is a left winding automorphism , and is the composition of the corresponding right winding automorphism with conjugation by ; 3. (iii)
the -derivation satisfies the relation
[TABLE]
and
[TABLE]
Proof.
Theorem 1.4 tells us that in this case is a domain. Then the claim immediately follows from Theorem 1.3. β
4. Concluding remarks
We note that in the paper [BSM18], a version of Corollary 3.2 was proved for finitely generated commutative Hopf algebras that are domains over an algebraically closed field . We note that this follows immediately from our Theorem 1.3, as such an algebra is of the form for an irreducible affine algebraic group over and so is just , which is again a domain. In Corollary 3.2, we have the hypothesis that the ring have finite Gelfand-Kirillov dimension. Conjecturally, the result should hold for noetherian cocommutative Hopf algebras over an algebraically closed field of characteristic zero, since such algebras are isomorphic to algebras of the form ; since is faithfully flat over both and the group algebra , if is noetherian, then so must these two subalgebras. Conjecturally, enveloping algebras are noetherian if and only if is finite-dimensional and is noetherian if and only if is polycyclic-by-finite. Hence Theorem 3.1 can be applied to give that is a domain if is a domain, since is necessarily torsion-free. In light of this, we ask the following questions.
Question 4.1**.**
Let be a cocommutative noetherian Hopf algebra over an algebraically closed field of characteristic zero. Is a domain if is a domain?
If the reader feels like being more ambitious, we raise the following question, which, combined with Theorem 1.3, would give an affirmative answer to Question 1.2 in the case when is a domain over an algebraically closed field if it could be answered affirmatively.
Question 4.2**.**
Let be an algebraically closed field and let be a noetherian Hopf algebra that is a domain. Is a domain?
Acknowledgments
The author gratefully acknowledges her advisor Jason Bell for his constant encouragement and advice. The author also thanks Ken Brown for useful comments and thanks the referee for suggesting an improvement to the proofs of Lemmas 2.1 and 2.2.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[BSM 18] J. Bell, O. SΓ‘nchez and R. Moosa, D-groups and the DixmierβMoeglin equivalence, Algebra Number Theory 12 (2018), no. 2, 343β378.
- 2[BL 14] J. Bell, W. Leung, The Dixmier-Moeglin equivalence for cocommutative Hopf algebras of finite Gelfand-Kirillov dimension, Algebr. Represent. Theory 17 (2014), no. 6, 1843β1852.
- 3[BOZZ 15] K.A. Brown, S. OβHagan, J.J Zhang, G. Zhuang, Connected Hopf algebras and iterated ore extension, J. Pure Appl. Algebra 219 (2015), no. 6, 2405β2433.
- 4[Pan 03] A. N. Panov, Ore extensions of Hopf algebras, Mat. Zametki 74 (2003), 425β434.
- 5[Mon 93] S. Montgomery, Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82 . Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993.
- 6[Pas 89] D. Passman, Infinite Crossed Products , Pure and Applied Mathematics, 135. Academic Press, Inc., Boston, MA, 1989.
- 7[RS 13] L. Rowen and D. Saltman, Tensor products of division algebras and fields, J. Algebra 394 , (2013), 296β309.
- 8[Skr 06] S. Skryabin, New results on the bijectivity of antipode of a Hopf algebra, J. Algebra 306 (2006), 622β633.
