Commutative-by-finite Hopf algebras

Commutative-by-finite Hopf algebras Kenny Brown University of Glasgow

Recent developments in noncommutative algebra and related areas Conference at University of Washington 17- 19 March 2018

Kenny Brown (University of Glasgow)

Commutative-by-finite Hopf

Recent developments in noncommutative alge / 14

FRONT NOTES

These are slides as given in the talk, except for adjustment of one of the open questions following a correction supplied by James Zhang. Apologies for the totally inadequate quality of my referencing. All queries, corrections, comments are welcome, to [email protected]




Aim and Plan

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AIMS:




Aim and Plan

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AIMS:To explain the title, state some well-known results, and ask a lot of questions.

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PLAN: Objects of study Properties, examples, non-example, questions... The antipode




Aim and Plan

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ASSUMPTIONS: Throughout, k is an algebraically closed field.




Aim and Plan

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ASSUMPTIONS: Throughout, k is an algebraically closed field. {H, ∆, S, } denotes a Hopf k-algebra, H + = ker.




§1. Objects of study

Definition 1

A subalgebra A of a Hopf algebra H is normal if it is invariant under the left and right adjoint actions of H; that is, for all a ∈ A and h ∈ H, X X (adl h)(a) = h1 aS(h2 ) ∈ K , (adr h)(a) = S(h1 )ah2 ∈ K .




§1. Objects of study

Definition 1

A subalgebra A of a Hopf algebra H is normal if it is invariant under the left and right adjoint actions of H; that is, for all a ∈ A and h ∈ H, X X (adl h)(a) = h1 aS(h2 ) ∈ K , (adr h)(a) = S(h1 )ah2 ∈ K .

2

A Hopf k-algebra H is commutative-by-finite if it is a finite (left or right) module over a commutative normal Hopf subalgebra A.




§2. Basic properties

Proposition Let A ⊆ H be a commutative-by-finite Hopf algebra. (i) A+ H is a Hopf ideal of H,





Proposition Let A ⊆ H be a commutative-by-finite Hopf algebra. (i) A+ H is a Hopf ideal of H, so H

:=

H/A+ H

is a finite dimensional Hopf algebra.






:=

H/A+ H

is a finite dimensional Hopf algebra. (ii) H is a faithfully flat right and left A-module.






:=

H/A+ H

is a finite dimensional Hopf algebra. (ii) H is a faithfully flat right and left A-module. (iii) (Skryabin) The antipode S of H is bijective.




§2.Finiteness conditions Proposition Let A ⊆ H be a commutative-by-finite Hopf algebra.




§2.Finiteness conditions Proposition Let A ⊆ H be a commutative-by-finite Hopf algebra. (i) The following are equivalent: H is noetherian. H is affine. A is noetherian. A is affine.





(ii) Suppose that H is affine. Then GKdim(H) = GKdim(A) = Kdim(H) = Kdim(A) =: n < ∞.





(ii) Suppose that H is affine. Then GKdim(H) = GKdim(A) = Kdim(H) = Kdim(A) =: n < ∞.

(iii) Suppose that H is affine. Then H is a finite Z (H)-module. Kenny Brown (University of Glasgow)



§2. Finiteness questions

Questions Let U be a Hopf k-algebra satisfying a polynomial identity. (i) Is U a finite Z (U)-module?





Questions Let U be a Hopf k-algebra satisfying a polynomial identity. (i) Is U a finite Z (U)-module? What about if U is in addition affine and/or noetherian? (ii) Is U affine ⇔ noetherian?





Questions Let U be a Hopf k-algebra satisfying a polynomial identity. (i) Is U a finite Z (U)-module? What about if U is in addition affine and/or noetherian? (ii) Is U affine ⇔ noetherian? Answer is yes if U is commutative (Molnar).




§2. Homological properties.

Theorem Let A ⊆ H be an affine commutative-by-finite Hopf algebra. Let GKdim(H) = n.





Theorem Let A ⊆ H be an affine commutative-by-finite Hopf algebra. Let GKdim(H) = n. (i) H is AS-Gorenstein and Auslander-Gorenstein, with inj.dim(H) = n. (ii) H is GK-Cohen-Macaulay and injectively homogeneous.





Theorem Let A ⊆ H be an affine commutative-by-finite Hopf algebra. Let GKdim(H) = n. (i) H is AS-Gorenstein and Auslander-Gorenstein, with inj.dim(H) = n. (ii) H is GK-Cohen-Macaulay and injectively homogeneous. (iii) Suppose H has finite global dimension. Then H is AS-regular and hom.hom.,





Theorem Let A ⊆ H be an affine commutative-by-finite Hopf algebra. Let GKdim(H) = n. (i) H is AS-Gorenstein and Auslander-Gorenstein, with inj.dim(H) = n. (ii) H is GK-Cohen-Macaulay and injectively homogeneous. (iii) Suppose H has finite global dimension. Then H is AS-regular and hom.hom., and so is a finite direct sum of prime hom.hom. algebras, each of global and GK-dimension n.




§3.Examples The following are examples of affine commutative-by-finite Hopf k-algebras.




§3.Examples The following are examples of affine commutative-by-finite Hopf k-algebras. 1 Enveloping algebras of f.d. restricted Lie algebras, chark = p > 0.




§3.Examples The following are examples of affine commutative-by-finite Hopf k-algebras. 1 Enveloping algebras of f.d. restricted Lie algebras, chark = p > 0. 2 Quantised enveloping algebras and quantised coordinate algebras at a root of unity.




§3.Examples The following are examples of affine commutative-by-finite Hopf k-algebras. 1 Enveloping algebras of f.d. restricted Lie algebras, chark = p > 0. 2 Quantised enveloping algebras and quantised coordinate algebras at a root of unity. 3 Group algebras of f.g. abelian-by-finite groups.




§3.Examples The following are examples of affine commutative-by-finite Hopf k-algebras. 1 Enveloping algebras of f.d. restricted Lie algebras, chark = p > 0. 2 Quantised enveloping algebras and quantised coordinate algebras at a root of unity. 3 Group algebras of f.g. abelian-by-finite groups. 4 Prime regular affine Hopf algebras H with GKdim(H) = 1, chark = 0.




§3.Examples The following are examples of affine commutative-by-finite Hopf k-algebras. 1 Enveloping algebras of f.d. restricted Lie algebras, chark = p > 0. 2 Quantised enveloping algebras and quantised coordinate algebras at a root of unity. 3 Group algebras of f.g. abelian-by-finite groups. 4 Prime regular affine Hopf algebras H with GKdim(H) = 1, chark = 0. These were classified by Lu-Q.Wu-Zhang (2007), B-Zhang (2010), Ding-Liu-J.Wu (2014). There are 2 commutative and 1 non-commutative cocommutative examples, and 3 infinite families. All are commutative-by-finite.




§3.Questions and (more) Examples Questions (See [B-Zhang, 2010].) 1 Classify prime regular affine Hopf k-algebras of GK-dimension 1 when chark > 0.




§3.Questions and (more) Examples Questions (See [B-Zhang, 2010].) 1 Classify prime regular affine Hopf k-algebras of GK-dimension 1 when chark > 0. 2 Classify prime affine Hopf k-algebras of GK-dimension 1 when chark = 0.




§3.Questions and (more) Examples Questions (See [B-Zhang, 2010].) 1 Classify prime regular affine Hopf k-algebras of GK-dimension 1 when chark > 0. 2 Classify prime affine Hopf k-algebras of GK-dimension 1 when chark = 0. More examples: 1 Noetherian Hopf domains H in char 0, with GKdim(H) = 2.




§3.Questions and (more) Examples Questions (See [B-Zhang, 2010].) 1 Classify prime regular affine Hopf k-algebras of GK-dimension 1 when chark > 0. 2 Classify prime affine Hopf k-algebras of GK-dimension 1 when chark = 0. More examples: 1 Noetherian Hopf domains H in char 0, with GKdim(H) = 2. 2 group algebras, 2 enveloping algebras, and 3 infinite families found by Goodearl-Zhang (2010);




§3.Questions and (more) Examples Questions (See [B-Zhang, 2010].) 1 Classify prime regular affine Hopf k-algebras of GK-dimension 1 when chark > 0. 2 Classify prime affine Hopf k-algebras of GK-dimension 1 when chark = 0. More examples: 1 Noetherian Hopf domains H in char 0, with GKdim(H) = 2. 2 group algebras, 2 enveloping algebras, and 3 infinite families found by Goodearl-Zhang (2010); a further infinite family found by Wang-Zhang-Zhuang (2013).




§3.Questions and (more) Examples Questions (See [B-Zhang, 2010].) 1 Classify prime regular affine Hopf k-algebras of GK-dimension 1 when chark > 0. 2 Classify prime affine Hopf k-algebras of GK-dimension 1 when chark = 0. More examples: 1 Noetherian Hopf domains H in char 0, with GKdim(H) = 2. 2 group algebras, 2 enveloping algebras, and 3 infinite families found by Goodearl-Zhang (2010); a further infinite family found by Wang-Zhang-Zhuang (2013). The group algebras, the commutative enveloping algebra, and 2 12 of the infinite families are PI - all the PI families are commutative-by-finite. Kenny Brown (University of Glasgow)



§3.Questions and (more) Examples Questions (See [B-Zhang, 2010].) 1 Classify prime regular affine Hopf k-algebras of GK-dimension 1 when chark > 0. 2 Classify prime affine Hopf k-algebras of GK-dimension 1 when chark = 0. More examples: 1 Noetherian Hopf domains H in char 0, with GKdim(H) = 2. 2 group algebras, 2 enveloping algebras, and 3 infinite families found by Goodearl-Zhang (2010); a further infinite family found by Wang-Zhang-Zhuang (2013). The group algebras, the commutative enveloping algebra, and 2 12 of the infinite families are PI - all the PI families are commutative-by-finite. Two of the infinite commutative-by-finite families are not regular. Kenny Brown (University of Glasgow)



§3.Non-example and more questions Question (Wang, Zhang, Zhuang, 2013) Is the above list complete, at least if H is pointed?





Example (Gelaki-Letzter, 2003) There exists a prime affine noetherian Hopf algebra U in char0, GKdim(U) = 2, U a finite Z (U)-module;





Example (Gelaki-Letzter, 2003) There exists a prime affine noetherian Hopf algebra U in char0, GKdim(U) = 2, U a finite Z (U)-module; but U is not commutative-by-finite. U is the bosonisation of the enveloping algebra of the Lie superalgebra pl(1, 1).





Example (Gelaki-Letzter, 2003) There exists a prime affine noetherian Hopf algebra U in char0, GKdim(U) = 2, U a finite Z (U)-module; but U is not commutative-by-finite. U is the bosonisation of the enveloping algebra of the Lie superalgebra pl(1, 1). As such, it contains an element u, forming part of a PBW basis for U, with u 2 = 0.





Example (Gelaki-Letzter, 2003) There exists a prime affine noetherian Hopf algebra U in char0, GKdim(U) = 2, U a finite Z (U)-module; but U is not commutative-by-finite. U is the bosonisation of the enveloping algebra of the Lie superalgebra pl(1, 1). As such, it contains an element u, forming part of a PBW basis for U, with u 2 = 0. So, since U is a free khui-module, gl.dim(U) = pr.dimU (k) ≥ pr.dimkhui (k) = ∞. Kenny Brown (University of Glasgow)



§3. Yet more questions

Hence we ask:

Question 1

Is every regular affine or noetherian PI Hopf algebra commutative-by-finite?




§3. Yet more questions

Hence we ask:

Question 1

2

Is every regular affine or noetherian PI Hopf algebra commutative-by-finite? Is every affine or noetherian PI Hopf domain commutative-by-finite?




§4. The antipode We start from an old result:





Theorem (B-Zhang, 2008) Let T be a noetherian AS-Gorenstein Hopf algebra with inj.dim(T ) = d and bijective antipode S. Then ` S 4 = γ ◦ τχr ◦ τ−χ ,

where χ is the character of the integral ExtdT (k, T ), the τ s are winding automorphisms, and γ is an inner automorphism.






where χ is the character of the integral ExtdT (k, T ), the τ s are winding automorphisms, and γ is an inner automorphism. γ is known (Radford, 1976) when dimk (T ) < ∞.






where χ is the character of the integral ExtdT (k, T ), the τ s are winding automorphisms, and γ is an inner automorphism. γ is known (Radford, 1976) when dimk (T ) < ∞. Every noetherian AS-Gorenstein Hopf algebra has bijective antipode, (Lu, Oh, Wang, Yu, 2017). Kenny Brown (University of Glasgow)



In light of previous slide, we ask:





Questions Let T be noetherian AS-Gorenstein. Keep notation of last slide.





Questions Let T be noetherian AS-Gorenstein. Keep notation of last slide. 1 What is γ?





Questions Let T be noetherian AS-Gorenstein. Keep notation of last slide. 1 What is γ? Not known even when T commutative-by-finite.





Questions Let T be noetherian AS-Gorenstein. Keep notation of last slide. 1 What is γ? Not known even when T commutative-by-finite. 2 Does S have finite order when T is PI, or (even) when T is commutative-by-finite?





Questions Let T be noetherian AS-Gorenstein. Keep notation of last slide. 1 What is γ? Not known even when T commutative-by-finite. 2 Does S have finite order when T is PI, or (even) when T is commutative-by-finite?

THANKS!


