Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken...

Hopf algebras and Homological Algebra

Ken Brown

University of Glasgow

BIRS WorkshopBanff, Canada

10 Sept 2015

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 1 / 17

1 Finite injective dimension

2 Finite global dimension

3 Applications

4 Quantum homogeneous spaces

k a field and (H ,∆, S , ε) a Hopf k-algebra, which is either(1) affine of finite GK-dimension; or(2) noetherian.We assume throughout that S is bijective.

Question(a) Does (1)⇒ (2)? (b) Does each of (1) or (2) imply S bijective?

3 Applications

1. Finite injective dimension

Theorem(Larson, Sweedler, 1969) Let H be a finite dimensional Hopfk-algebra. Then H has a nondegenerate associative bilinear formα : H ⊗ H → k .

That is, H is a Frobenius algebra; in particular, H is an injective(right and left) H-module - we write injdimH = 0.

Theorem(Larson, Sweedler, 1969) Let H be a finite dimensional Hopfk-algebra. Then H has a nondegenerate associative bilinear formα : H ⊗ H → k .That is, H is a Frobenius algebra;

in particular, H is an injective(right and left) H-module - we write injdimH = 0.

Theorem(Larson, Sweedler, 1969) Let H be a finite dimensional Hopfk-algebra. Then H has a nondegenerate associative bilinear formα : H ⊗ H → k .That is, H is a Frobenius algebra; in particular, H is an injective(right and left) H-module - we write injdimH = 0.

1.Finite injective dimension

Definition(B-Zhang, 2008) Let A be noetherian augmented k-algebra, viaε : A→ k . Then A is AS-Gorenstein if

1 injdimAA = d <∞;

2 dimk ExtdA(k ,A|A) = 1; and ExtiA(k ,A|A) = 0 if i 6= d ;

3 the left-sided versions of (1) and (2) also hold.

Examples1 H a finite dim Hopf algebra, then H is AS-Gorenstein, with

d = 0.

2 A a connected graded affine commutative k-algebra, theninjdimA <∞⇔ A is AS-Gorenstein; and then d = GKdimA.

3 The algebra R of 2× 2 upper triangular matrices over k hasinjdimR = 1, but R is not AS-Gorenstein.

d = 0.

2 A a connected graded affine commutative k-algebra, theninjdimA <∞⇔ A is AS-Gorenstein;

and then d = GKdimA.

d = 0.

DefinitionLet A be a noetherian k-algebra.

1 Let M be a non-zero f.g. A-module. The grade j(M) of M isthe least integer j such that ExtjA(M ,A) 6= 0 (or ∞ if there isno such j).

2 Suppose that GKdimA <∞. Then A is GK-Cohen Macaulay if

j(M) + GKdimM = GKdimA

for every non-zero f.g. (right or left) A-module M .

DefinitionLet A be a noetherian k-algebra.

1 Let M be a non-zero f.g. A-module. The grade j(M) of M isthe least integer j such that ExtjA(M ,A) 6= 0 (or ∞ if there isno such j).

2 Suppose that GKdimA <∞. Then A is GK-Cohen Macaulay if

j(M) + GKdimM = GKdimA

for every non-zero f.g. (right or left) A-module M .

Remarks1 (Levasseur, 1992) When A is Auslander-Gorenstein,δ(M) := injdimA− j(M) defines a finitely partitive dimensionfunction on f.g A-modules M .

2 Then the GK-Cohen Macaulay condition is telling us that δ(−)is “nothing new”.

3 The GK-Cohen Macaulay condition is “wrong” in general forHopf algebras: e.g. for noetherian group algebras kG ,GKdimkG does not exist in general; and even for G f.g.nilpotent, so that GKdimkG exists, kG is not in generalGK-Cohen Macaulay.

4 So the “real point” of the GK-Cohen Macaulay condition is totell us that δ(−) is a symmetric dimension function.

3 The GK-Cohen Macaulay condition is “wrong” in general forHopf algebras: e.g. for noetherian group algebras kG ,GKdimkG does not exist in general;

and even for G f.g.nilpotent, so that GKdimkG exists, kG is not in generalGK-Cohen Macaulay.

Metatheorem“All known noetherian Hopf k-algebras are AS-Gorenstein.”

Remarks1 For details of all the “easy” cases, see [Brown-Zhang, 2008,§6.2].

2 First non-trivial general case: (Wu, Zhang, 2003) Noetherianaffine PI Hopf algebras are AS-Gorenstein and GK-CohenMacaulay.

3 Second non-trivial general case: (Zhuang, 2013) If H is aconnected Hopf algebra of finite GK-dimension, then H isAS-Gorenstein and GK-Cohen Macaulay.

Regarding the Wu-Zhang PI theorem, they ask the obvious question:

Question(Wu, Zhang, 2003) Is every noetherian (PI) Hopf algebra affine?

See Molnar’s theorem (PAMS, 1975) for the commutative case. Thequestion seems to be open even with “PI” omitted.And the old question remains....

QuestionLet H be a noetherian Hopf algebra. Is H AS-Gorenstein?

QuestionDo such H satisfy a “form” of the GK-Cohen Macaulay condition?

For example, what about the case of pointed H?

See Molnar’s theorem (PAMS, 1975) for the commutative case. Thequestion seems to be open even with “PI” omitted.

And the old question remains....

For example, what about the case of pointed H?Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 8 / 17

2. Finite global dimension

DefinitionA noetherian augmented algebra A is called AS-regular if it isAS-Gorenstein and has finite global dimension.

Examples1 Finite dimensional Hopf algebras have finite global dimension ⇔

semisimple.

2 Commutative affine Hopf k-algebras (k char 0) are AS-regular.

3 Enveloping algebras (g f.d.), quantised env. algebras, quantisedfunction algebras are AS-regular.

4 Group algebra kG , (G polycyclic-by-finite) is AS-regular ⇔ Ghas no element of order chark .

semisimple.

Given the above examples, it looked as if finite global dimension of Hhung on its finite dimensional Hopf subalgebras. But....

[Goodearl-Zhang, 2010] There is a family of noetherian Hopf algebradomains of GK-dim 2 which are AS-Gorenstein, not AS-regular.

Hence:

QuestionIs there an (easily checkable) structural property of a Hopf algebra(assumed affine of finite GKdim or noetherian), which is equivalent tofinite global dimension?

Hence:

3. Applications:(I)Homological

Definition (Lu-Wu-Zhang, 2007)

Let H be an AS-Gorenstein Hopf algebra with injdimH = d . The(left) homological integral

∫ `H

of H is the 1-dimensional H-bimodule

Extd(k ,H).

So when dimkH <∞,∫ `H

= HomH(k ,H) is just the “traditional”Hopf integral, the copy of the trivial module inside H .

Theorem (B-Zhang, 2008)

Let H be an AS-Gorenstein Hopf algebra with injdimA = d . Let χbe the character of the right structure on

∫ `H

1 H has a rigid dualising complex R ∼= νH1[d ].

2 The Nakayama automorphism ν is S2τ `χ, where τ `χ denotes theleft winding automorphism of H got from χ.

∫ `H

Extd(k ,H).

∫ `H

Extd(k ,H).

∫ `H

3. Applications: (I)Homological

Combining the dualising complex theorem with [Van den Bergh,1998;2002] gives:

Corollary

Poincare duality/twisted Calabi-Yau: Let H be an AS-regular Hopfalgebra with injdimH = d . Keep the notation as in the theorem. Forevery H-bimodule M and for every i , H i(H , νM1) ∼= Hd−i(H ,M).

3. Applications: (I)Homological

Combining the dualising complex theorem with [Van den Bergh,1998;2002] gives:

Corollary

Poincare duality/twisted Calabi-Yau: Let H be an AS-regular Hopfalgebra with injdimH = d . Keep the notation as in the theorem. Forevery H-bimodule M and for every i , H i(H , νM1) ∼= Hd−i(H ,M).

3. Applications: (II) Antipode

Since we can also apply the dualising complex theorem to(H ,∆op, S−1, ε), giving ν = S−2τ rχ,

and ν is unique up to an innerauto of H ,we can equate the 2 answers for ν and deduce:

CorollaryLet H be an AS-Gorenstein Hopf algebra, notation as before. Then,for some inner auto γ of H ,

S4 = γ ◦ τ `−χ ◦ τ rχ.

For finite dimensional H , this is [Radford, 1976], with a known γ.Namely, γ is conjugation by the group-like of H which is the rightcharacter of

∫ `H∗ .

QuestionFor H AS-Gorenstein, what is γ in the formula for S4?

Since we can also apply the dualising complex theorem to(H ,∆op, S−1, ε), giving ν = S−2τ rχ,and ν is unique up to an innerauto of H ,

we can equate the 2 answers for ν and deduce:

S4 = γ ◦ τ `−χ ◦ τ rχ.

∫ `H∗ .

Since we can also apply the dualising complex theorem to(H ,∆op, S−1, ε), giving ν = S−2τ rχ,and ν is unique up to an innerauto of H ,we can equate the 2 answers for ν and deduce:

S4 = γ ◦ τ `−χ ◦ τ rχ.

∫ `H∗ .

S4 = γ ◦ τ `−χ ◦ τ rχ.

∫ `H∗ .

S4 = γ ◦ τ `−χ ◦ τ rχ.

For finite dimensional H , this is [Radford, 1976], with a known γ.

Namely, γ is conjugation by the group-like of H which is the rightcharacter of

∫ `H∗ .

S4 = γ ◦ τ `−χ ◦ τ rχ.

∫ `H∗ .

S4 = γ ◦ τ `−χ ◦ τ rχ.

∫ `H∗ .

3. Applications: (III) Structure

Recall that finite dimensional Hopf algebras of finite global dimensionare semisimple.

The following generalisation ought to be the shadowof more general results:

Theorem(B-Goodearl, 1997) Let H be a noetherian affine Hopf algebra whichsatisfies a polynomial identity. Suppose that H has finite globaldimension n. Then H is a finite direct sum of prime algebras, eachsummand having the same GK-dimension, namely n.

Idea of proof: show that H is Auslander-regular and GK-CohenMacaulay, from which it is ”homologically homogeneous”, and oldwork of B-Hajarnavis-Stafford-Zhang kicks in.

QuestionSimilar for H pointed noetherian?

Recall that finite dimensional Hopf algebras of finite global dimensionare semisimple.The following generalisation ought to be the shadowof more general results:

4. Quantum homogeneous spaces

DefinitionA left quantum homogeneous space of the Hopf algebra H is a leftcoideal subalgebra T of H such that H is faithfully flat as left and aright T -module. Analogously for right...

For the basics, see [Schneider,1990], [Masuoka,1991].Results parallel to the above should apply to the quantumhomogeneous spaces of Hopf algebras H which are noetherian orhave finite GK-dimension.See [B-Gilmartin, arXiv2015] for the case of H connected.Also work by a number of people (Kraehmer, Liu, Wu...) on thetwisted Calabi-Yau property of various quantum homogeneous spaces.Let’s look at the case of the Podles spheres....

For the basics, see [Schneider,1990], [Masuoka,1991].

Results parallel to the above should apply to the quantumhomogeneous spaces of Hopf algebras H which are noetherian orhave finite GK-dimension.See [B-Gilmartin, arXiv2015] for the case of H connected.Also work by a number of people (Kraehmer, Liu, Wu...) on thetwisted Calabi-Yau property of various quantum homogeneous spaces.Let’s look at the case of the Podles spheres....

For the basics, see [Schneider,1990], [Masuoka,1991].Results parallel to the above should apply to the quantumhomogeneous spaces of Hopf algebras H which are noetherian orhave finite GK-dimension.

See [B-Gilmartin, arXiv2015] for the case of H connected.Also work by a number of people (Kraehmer, Liu, Wu...) on thetwisted Calabi-Yau property of various quantum homogeneous spaces.Let’s look at the case of the Podles spheres....

For the basics, see [Schneider,1990], [Masuoka,1991].Results parallel to the above should apply to the quantumhomogeneous spaces of Hopf algebras H which are noetherian orhave finite GK-dimension.See [B-Gilmartin, arXiv2015] for the case of H connected.Also work by a number of people (Kraehmer, Liu, Wu...) on thetwisted Calabi-Yau property of various quantum homogeneous spaces.

Let’s look at the case of the Podles spheres....

Definition(Following [Muller, Schneider,1999]) Let α, β, γ ∈ k and consider the(K−1, 1)-primitive element

X := αEK−1 + βF + γ(K−1 − 1) ∈ Uq(sl(2)) \ {0}.

Let B = Bα,β,γ := {h ∈ Oq(SL(2)) : h � X = 0}, where � is theusual right hit action. Then B is a rt. coideal subalgebra ofOq(SL(2)) and is a quantum hom. space ⇔ there is no n ∈ Z≥0 s.t.

αβq(qn + q−n

q − q−1)2 + γ2 = 0.

In this case B is called a Podles quantum sphere. If(α, β, γ) = (0, 0, 1) then B is the standard Podles quantum sphere.

X := αEK−1 + βF + γ(K−1 − 1) ∈ Uq(sl(2)) \ {0}.

Let B = Bα,β,γ := {h ∈ Oq(SL(2)) : h � X = 0}, where � is theusual right hit action.

Then B is a rt. coideal subalgebra ofOq(SL(2)) and is a quantum hom. space ⇔ there is no n ∈ Z≥0 s.t.

αβq(qn + q−n

q − q−1)2 + γ2 = 0.

X := αEK−1 + βF + γ(K−1 − 1) ∈ Uq(sl(2)) \ {0}.

αβq(qn + q−n

q − q−1)2 + γ2 = 0.

X := αEK−1 + βF + γ(K−1 − 1) ∈ Uq(sl(2)) \ {0}.

αβq(qn + q−n

q − q−1)2 + γ2 = 0.

Theorem(Kraehmer, 2012; Liu, Shen, Wu, 2014) Let B be a Podles quantumsphere.

1 B is Auslander-regular, GK-Cohen Macaulay and AS-regular ofdim. 2.

2 If αβ = 0, then B satisfies twisted Poincare duality, (with νgiven by B-Zhang-type formula).

Question(Kraehmer; Liu, Shen, Wu) Does twisted Calabi-Yau also hold for Bwhen αβ 6= 0?

Theorem(Kraehmer, 2012; Liu, Shen, Wu, 2014) Let B be a Podles quantumsphere.

1 B is Auslander-regular, GK-Cohen Macaulay and AS-regular ofdim. 2.

2 If αβ = 0, then B satisfies twisted Poincare duality, (with νgiven by B-Zhang-type formula).

Question(Kraehmer; Liu, Shen, Wu) Does twisted Calabi-Yau also hold for Bwhen αβ 6= 0?

Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken...

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