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

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

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

Page 1: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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

Page 2: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

Plan

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 2 / 17

Page 3: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

Plan

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 2 / 17

Page 4: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

Plan

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 2 / 17

Page 5: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 3 / 17

Page 6: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 3 / 17

Page 7: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 3 / 17

Page 8: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 4 / 17

Page 9: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 4 / 17

Page 10: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 4 / 17

Page 11: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 4 / 17

Page 12: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 4 / 17

Page 13: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 4 / 17

Page 14: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1.Finite injective dimension

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 .

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 5 / 17

Page 15: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1.Finite injective dimension

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 .

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 5 / 17

Page 16: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1.Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 6 / 17

Page 17: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1.Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 6 / 17

Page 18: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1.Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 6 / 17

Page 19: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1.Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 6 / 17

Page 20: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1.Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 6 / 17

Page 21: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 7 / 17

Page 22: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 7 / 17

Page 23: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 7 / 17

Page 24: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 7 / 17

Page 25: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 8 / 17

Page 26: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 8 / 17

Page 27: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 8 / 17

Page 28: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 8 / 17

Page 29: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

1. Finite injective dimension

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?Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 8 / 17

Page 30: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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 .

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 9 / 17

Page 31: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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 .

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 9 / 17

Page 32: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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 .

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 9 / 17

Page 33: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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 .

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 9 / 17

Page 34: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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 .

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 9 / 17

Page 35: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

2. Finite global dimension

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?

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

Page 36: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

2. Finite global dimension

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?

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

Page 37: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

2. Finite global dimension

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?

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

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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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 11 / 17

Page 39: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 11 / 17

Page 40: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 11 / 17

Page 41: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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).

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 12 / 17

Page 42: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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).

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 12 / 17

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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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 13 / 17

Page 44: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 13 / 17

Page 45: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 13 / 17

Page 46: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 13 / 17

Page 47: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 13 / 17

Page 48: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 13 / 17

Page 49: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 13 / 17

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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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 14 / 17

Page 51: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 14 / 17

Page 52: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 14 / 17

Page 53: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 14 / 17

Page 54: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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?

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 14 / 17

Page 55: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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....

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 15 / 17

Page 56: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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....

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 15 / 17

Page 57: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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....

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 15 / 17

Page 58: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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....

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 15 / 17

Page 59: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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....

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 15 / 17

Page 60: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

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....

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 15 / 17

Page 61: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

4. Quantum homogeneous spaces

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 16 / 17

Page 62: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

4. Quantum homogeneous spaces

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 16 / 17

Page 63: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

4. Quantum homogeneous spaces

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 16 / 17

Page 64: Hopf algebras and Homological Algebra · 2015-09-10 · Hopf algebras and Homological Algebra Ken Brown University of Glasgow BIRS Workshop Ban , Canada 10 Sept 2015 Ken Brown (University

4. Quantum homogeneous spaces

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.

Ken Brown (University of Glasgow BIRS Workshop Banff, Canada )Hom. Hopf 10 Sept 2015 16 / 17

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4. Quantum homogeneous spaces

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?

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

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4. Quantum homogeneous spaces

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?

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