Homology (of) Hopf algebras Christian Kassel ´ ´ Institut de Recherche Mathematique Avancee CNRS - Universite´ Louis Pasteur Strasbourg, France

Rencontre franco-britannique et ´ en l’honneur des soixante ans de Jacques Alev Journees Universite´ de Reims 28 novembre 2008

Introduction

I Report on joint work with Julien Bichon (Clermont-Ferrand):

The lazy homology of a Hopf algebra, arXiv:0807.1651

I Original motivation: The classification of Hopf Galois extensions,

which are noncommutative analogues of principal fiber bundles. For these noncommutative principal fiber bundles, the structural group is a Hopf algebra

Introduction

I Report on joint work with Julien Bichon (Clermont-Ferrand):

The lazy homology of a Hopf algebra, arXiv:0807.1651

I Original motivation: The classification of Hopf Galois extensions,

which are noncommutative analogues of principal fiber bundles. For these noncommutative principal fiber bundles, the structural group is a Hopf algebra

Previous work

I In joint work with Eli Aljadeff (Polynomial identities and noncommutative

versal torsors, Adv. Math. 218 (2008), 1453–1495), we concentrated on a class of Hopf Galois extensions obtained from a given Hopf algebra by twisting its product using a two-cocycle, and we constructed “universal spaces” using polynomial identities I The special case where the Hopf algebra is a group algebra had been

worked out by E. Aljadeff, D. Haile, M. Natapov (Graded identities of matrix algebras and the universal graded algebra, to appear in Trans. Amer. Math. Soc., 2008)

Previous work

I In joint work with Eli Aljadeff (Polynomial identities and noncommutative

versal torsors, Adv. Math. 218 (2008), 1453–1495), we concentrated on a class of Hopf Galois extensions obtained from a given Hopf algebra by twisting its product using a two-cocycle, and we constructed “universal spaces” using polynomial identities I The special case where the Hopf algebra is a group algebra had been

worked out by E. Aljadeff, D. Haile, M. Natapov (Graded identities of matrix algebras and the universal graded algebra, to appear in Trans. Amer. Math. Soc., 2008)

Cohomology group and universal coefficient theorem

I In their work Aljadeff, Haile, and Natapov make use of the second

cohomology group H 2 (G, k × ) of a group G and the universal coefficient theorem relating the cohomology of G to the integral homology of G via an exact sequence of the form 1 → Ext1 (H1 (G), k × ) → H 2 (G, k × ) → Hom(H2 (G), k × ) → 1 I The concept of two-cocycles and cohomologous two-cocycles can be

defined for any Hopf algebra, but in general cocycles do not form a group

Cohomology group and universal coefficient theorem

I In their work Aljadeff, Haile, and Natapov make use of the second

cohomology group H 2 (G, k × ) of a group G and the universal coefficient theorem relating the cohomology of G to the integral homology of G via an exact sequence of the form 1 → Ext1 (H1 (G), k × ) → H 2 (G, k × ) → Hom(H2 (G), k × ) → 1 I The concept of two-cocycles and cohomologous two-cocycles can be

defined for any Hopf algebra, but in general cocycles do not form a group

Restricting cocycles

• Restricting to a class of two-cocycles, called lazy cocycles, Schauenburg, Chen, Bichon, Carnovale et al. managed to define two groups H`1 (H, k) and

H`2 (H, k )

for any Hopf algebra H, extending group cohomology i × H`i (H, k ) ∼ = H (G, k )

(i = 1, 2)

when H = k [G] is a group algebra • The groups H`1 (H, k ) and H`2 (H, k) are called the lazy cohomology groups of H

Summary of joint work with Julien Bichon I For any Hopf algebra H we construct two commutative Hopf algebras

(NOT groups!) H1` (H) and H2` (H) I (i) together with a group isomorphism

H`1 (H, k ) −→ Alg(H1` (H), k ) I (ii) and an exact sequence of groups (UCT) κ

1 −→ Ext1 (H, k ) −→ H`2 (H, k ) −→ Alg(H2` (H), k) When the ground field k is algebraically closed, the homomorphism κ : H`2 (H, k ) −→ Alg(H2` (H), k) is an isomorphism I Moreover, if H = k [G] is a group algebra, then

Hi` (H) ∼ = k [Hi (G)]

(i = 1, 2)

Summary of joint work with Julien Bichon I For any Hopf algebra H we construct two commutative Hopf algebras

(NOT groups!) H1` (H) and H2` (H) I (i) together with a group isomorphism

H`1 (H, k ) −→ Alg(H1` (H), k ) I (ii) and an exact sequence of groups (UCT) κ

1 −→ Ext1 (H, k ) −→ H`2 (H, k ) −→ Alg(H2` (H), k) When the ground field k is algebraically closed, the homomorphism κ : H`2 (H, k ) −→ Alg(H2` (H), k) is an isomorphism I Moreover, if H = k [G] is a group algebra, then

Hi` (H) ∼ = k [Hi (G)]

(i = 1, 2)

Summary of joint work with Julien Bichon I For any Hopf algebra H we construct two commutative Hopf algebras

(NOT groups!) H1` (H) and H2` (H) I (i) together with a group isomorphism

H`1 (H, k ) −→ Alg(H1` (H), k ) I (ii) and an exact sequence of groups (UCT) κ

1 −→ Ext1 (H, k ) −→ H`2 (H, k ) −→ Alg(H2` (H), k) When the ground field k is algebraically closed, the homomorphism κ : H`2 (H, k ) −→ Alg(H2` (H), k) is an isomorphism I Moreover, if H = k [G] is a group algebra, then

Hi` (H) ∼ = k [Hi (G)]

(i = 1, 2)

Summary of joint work with Julien Bichon I For any Hopf algebra H we construct two commutative Hopf algebras

(NOT groups!) H1` (H) and H2` (H) I (i) together with a group isomorphism

H`1 (H, k ) −→ Alg(H1` (H), k ) I (ii) and an exact sequence of groups (UCT) κ

1 −→ Ext1 (H, k ) −→ H`2 (H, k ) −→ Alg(H2` (H), k) When the ground field k is algebraically closed, the homomorphism κ : H`2 (H, k ) −→ Alg(H2` (H), k) is an isomorphism I Moreover, if H = k [G] is a group algebra, then

Hi` (H) ∼ = k [Hi (G)]

(i = 1, 2)

Plan

• Part One: Lazy cohomology

• Part Two: Universal constructions

• Part Three: Lazy homology

• References

• Appendix: Cocycle-twisting of Hopf algebras

Convolution groups I We fix a ground field k and a coalgebra C

with coproduct ∆ : C → C ⊗ C and counit ε : C → k I The dual vector space Hom(C, k ) is an associative unital algebra

whose product is the convolution product given for f , g ∈ Hom(C, k ) and x ∈ H by X (f ∗ g)(x) = f (x 0 ) g(x 00 ) (x)

where ∆(x) =

P

(x)

x 0 ⊗ x 00 (Sweedler’s sigma notation)

I The counit ε : C → k is the unit for the convolution product

ε∗f =f =f ∗ε I Let Reg(C) be the group of convolution-invertible elements

of Hom(C, k) Since C ⊗ C is a coalgebra, we may also consider Reg(C ⊗ C)

Convolution groups I We fix a ground field k and a coalgebra C

with coproduct ∆ : C → C ⊗ C and counit ε : C → k I The dual vector space Hom(C, k ) is an associative unital algebra

whose product is the convolution product given for f , g ∈ Hom(C, k ) and x ∈ H by X (f ∗ g)(x) = f (x 0 ) g(x 00 ) (x)

where ∆(x) =

P

(x)

x 0 ⊗ x 00 (Sweedler’s sigma notation)

I The counit ε : C → k is the unit for the convolution product

ε∗f =f =f ∗ε I Let Reg(C) be the group of convolution-invertible elements

of Hom(C, k) Since C ⊗ C is a coalgebra, we may also consider Reg(C ⊗ C)

Convolution groups I We fix a ground field k and a coalgebra C

with coproduct ∆ : C → C ⊗ C and counit ε : C → k I The dual vector space Hom(C, k ) is an associative unital algebra

whose product is the convolution product given for f , g ∈ Hom(C, k ) and x ∈ H by X (f ∗ g)(x) = f (x 0 ) g(x 00 ) (x)

where ∆(x) =

P

(x)

x 0 ⊗ x 00 (Sweedler’s sigma notation)

I The counit ε : C → k is the unit for the convolution product

ε∗f =f =f ∗ε I Let Reg(C) be the group of convolution-invertible elements

of Hom(C, k) Since C ⊗ C is a coalgebra, we may also consider Reg(C ⊗ C)

Convolution groups I We fix a ground field k and a coalgebra C

with coproduct ∆ : C → C ⊗ C and counit ε : C → k I The dual vector space Hom(C, k ) is an associative unital algebra

whose product is the convolution product given for f , g ∈ Hom(C, k ) and x ∈ H by X (f ∗ g)(x) = f (x 0 ) g(x 00 ) (x)

where ∆(x) =

P

(x)

x 0 ⊗ x 00 (Sweedler’s sigma notation)

I The counit ε : C → k is the unit for the convolution product

ε∗f =f =f ∗ε I Let Reg(C) be the group of convolution-invertible elements

of Hom(C, k) Since C ⊗ C is a coalgebra, we may also consider Reg(C ⊗ C)

Lazy forms Let H be a Hopf algebra I A linear form µ ∈ Reg(H) is lazy if for all x ∈ H,

X

µ(x 0 ) x 00 =

(x)

X

µ(x 00 ) x 0 ∈ H

(x)

They form an abelian (central) subgroup Reg` (H) ⊂ Reg(H) I A bilinear form σ ∈ Reg(H ⊗ H) is lazy if for all x, y ∈ H,

X

σ(x 0 ⊗ y 0 ) x 00 y 00 =

(x)(y)

X

σ(x 00 ⊗ y 00 ) x 0 y 0 ∈ H

(x)(y) (2)

They form a subgroup Reg` (H) ⊂ Reg(H ⊗ H) I Remark. All (bi)linear forms are lazy on a cocommutative Hopf algebra,

i.e., such that X (x)

x 0 ⊗ x 00 =

X (x)

x 00 ⊗ x 0

Lazy forms Let H be a Hopf algebra I A linear form µ ∈ Reg(H) is lazy if for all x ∈ H,

X

µ(x 0 ) x 00 =

(x)

X

µ(x 00 ) x 0 ∈ H

(x)

They form an abelian (central) subgroup Reg` (H) ⊂ Reg(H) I A bilinear form σ ∈ Reg(H ⊗ H) is lazy if for all x, y ∈ H,

X

σ(x 0 ⊗ y 0 ) x 00 y 00 =

(x)(y)

X

σ(x 00 ⊗ y 00 ) x 0 y 0 ∈ H

(x)(y) (2)

They form a subgroup Reg` (H) ⊂ Reg(H ⊗ H) I Remark. All (bi)linear forms are lazy on a cocommutative Hopf algebra,

i.e., such that X (x)

x 0 ⊗ x 00 =

X (x)

x 00 ⊗ x 0

Lazy forms Let H be a Hopf algebra I A linear form µ ∈ Reg(H) is lazy if for all x ∈ H,

X

µ(x 0 ) x 00 =

(x)

X

µ(x 00 ) x 0 ∈ H

(x)

They form an abelian (central) subgroup Reg` (H) ⊂ Reg(H) I A bilinear form σ ∈ Reg(H ⊗ H) is lazy if for all x, y ∈ H,

X

σ(x 0 ⊗ y 0 ) x 00 y 00 =

(x)(y)

X

σ(x 00 ⊗ y 00 ) x 0 y 0 ∈ H

(x)(y) (2)

They form a subgroup Reg` (H) ⊂ Reg(H ⊗ H) I Remark. All (bi)linear forms are lazy on a cocommutative Hopf algebra,

i.e., such that X (x)

x 0 ⊗ x 00 =

X (x)

x 00 ⊗ x 0

Lazy cocycles I An element σ ∈ Reg(2) ` (H) is a lazy (two-)cocycle if

X

σ(x 0 , y 0 ) σ(x 00 y 00 , z) =

(x),(y )

X

σ(y 0 , z 0 ) σ(x, y 00 z 00 )

(y),(z) (2)

for all x, y, z ∈ H. They form a subgroup Z`2 (H) ⊂ Reg` (H) I There is a group homomorphism ∂ : Reg` (H) → Reg(2) ` (H) defined for

all µ ∈ Reg` (H) and x, y ∈ H by X ∂(µ)(x ⊗ y) = µ(x 0 ) µ(y 0 ) µ−1 (x 00 y 00 ) (x)(y)

2 I The image of ∂ : Reg` (H) → Reg(2) ` (H) is a central subgroup of Z` (H)

Lazy cocycles I An element σ ∈ Reg(2) ` (H) is a lazy (two-)cocycle if

X

σ(x 0 , y 0 ) σ(x 00 y 00 , z) =

(x),(y )

X

σ(y 0 , z 0 ) σ(x, y 00 z 00 )

(y),(z) (2)

for all x, y, z ∈ H. They form a subgroup Z`2 (H) ⊂ Reg` (H) I There is a group homomorphism ∂ : Reg` (H) → Reg(2) ` (H) defined for

all µ ∈ Reg` (H) and x, y ∈ H by X ∂(µ)(x ⊗ y) = µ(x 0 ) µ(y 0 ) µ−1 (x 00 y 00 ) (x)(y)

2 I The image of ∂ : Reg` (H) → Reg(2) ` (H) is a central subgroup of Z` (H)

Lazy cocycles I An element σ ∈ Reg(2) ` (H) is a lazy (two-)cocycle if

X

σ(x 0 , y 0 ) σ(x 00 y 00 , z) =

(x),(y )

X

σ(y 0 , z 0 ) σ(x, y 00 z 00 )

(y),(z) (2)

for all x, y, z ∈ H. They form a subgroup Z`2 (H) ⊂ Reg` (H) I There is a group homomorphism ∂ : Reg` (H) → Reg(2) ` (H) defined for

all µ ∈ Reg` (H) and x, y ∈ H by X ∂(µ)(x ⊗ y) = µ(x 0 ) µ(y 0 ) µ−1 (x 00 y 00 ) (x)(y)

2 I The image of ∂ : Reg` (H) → Reg(2) ` (H) is a central subgroup of Z` (H)

Lazy cohomology groups: definition

I The lazy cohomology groups H`1 (H, k ) and H`2 (H, k ) are defined by the

exact sequence of groups ∂

1 −→ H`1 (H, k ) −→ Reg` (H) −→ Z`2 (H) −→ H`2 (H, k ) −→ 1

I The group H`1 (H, k ) is abelian since Reg` (H) is abelian

I Open question: Is H`2 (H, k ) always abelian?

[It is so in all computed examples, but there is no a priori reason for it]

Lazy cohomology groups: definition

I The lazy cohomology groups H`1 (H, k ) and H`2 (H, k ) are defined by the

exact sequence of groups ∂

1 −→ H`1 (H, k ) −→ Reg` (H) −→ Z`2 (H) −→ H`2 (H, k ) −→ 1

I The group H`1 (H, k ) is abelian since Reg` (H) is abelian

I Open question: Is H`2 (H, k ) always abelian?

[It is so in all computed examples, but there is no a priori reason for it]

Lazy cohomology groups: definition

I The lazy cohomology groups H`1 (H, k ) and H`2 (H, k ) are defined by the

exact sequence of groups ∂

1 −→ H`1 (H, k ) −→ Reg` (H) −→ Z`2 (H) −→ H`2 (H, k ) −→ 1

I The group H`1 (H, k ) is abelian since Reg` (H) is abelian

I Open question: Is H`2 (H, k ) always abelian?

[It is so in all computed examples, but there is no a priori reason for it]

Lazy cohomology groups: some computations I Proposition. If H = k [G] is a group algebra, then i × H`i (H, k ) ∼ = H (G, k )

(i = 1, 2)

where G acts trivially on k × = k − {0}. In particular, H`1 (k [G], k ) = Hom(Gab , k × ) where Gab = H1 (G) is the abelianization of G

I Proposition. If H = O(G) is the Hopf algebra of k-valued functions on

a finite group G, then H`1 (H, k ) = Z (G)

(the center of G)

Remark. The Hopf algebra O(G) is dual to the Hopf algebra k [G]

Lazy cohomology groups: some computations I Proposition. If H = k [G] is a group algebra, then i × H`i (H, k ) ∼ = H (G, k )

(i = 1, 2)

where G acts trivially on k × = k − {0}. In particular, H`1 (k [G], k ) = Hom(Gab , k × ) where Gab = H1 (G) is the abelianization of G

I Proposition. If H = O(G) is the Hopf algebra of k-valued functions on

a finite group G, then H`1 (H, k ) = Z (G)

(the center of G)

Remark. The Hopf algebra O(G) is dual to the Hopf algebra k [G]

The lazy cohomology of the Sweedler algebra Assume that k has characteristic 6= 2 I The Sweedler algebra is the four-dimensional algebra

H4 = k h g, x | g 2 = 1 , gx + xg = 0 , x 2 = 0 i It is the smallest noncommutative noncocommutative Hopf algebra with Coproduct:

∆(g) = g ⊗ g,

∆(x) = 1 ⊗ x + x ⊗ g

Counit: ¨

ε(g) = 1,

ε(x) = 0

Antipode:

S(g) = g,

S(x) = gx

I Lazy cohomology:

H`1 (H4 , k) ∼ = {1}

and

H`2 (H4 , k) ∼ = (k , +)

The lazy cohomology of the Sweedler algebra Assume that k has characteristic 6= 2 I The Sweedler algebra is the four-dimensional algebra

H4 = k h g, x | g 2 = 1 , gx + xg = 0 , x 2 = 0 i It is the smallest noncommutative noncocommutative Hopf algebra with Coproduct:

∆(g) = g ⊗ g,

∆(x) = 1 ⊗ x + x ⊗ g

Counit: ¨

ε(g) = 1,

ε(x) = 0

Antipode:

S(g) = g,

S(x) = gx

I Lazy cohomology:

H`1 (H4 , k) ∼ = {1}

and

H`2 (H4 , k) ∼ = (k , +)

Plan

• Part One: Lazy cohomology

• Part Two: Universal constructions

• Part Three: Lazy homology

• References

• Appendix: Cocycle-twisting of Hopf algebras

How to construct lazy homology: general idea To construct lazy homology and to express the lazy cohomology groups as groups of algebra maps, I we attach to any Hopf algebra H

• a commutative Hopf algebra F (H) such that Reg(H) ∼ = Alg(F (H), k ) • and two coalgebras H [1] and H [2] such that [1] Reg` (H) ∼ = Reg(H ) and

[2] Reg` (H) ∼ = Reg(H ) (2)

I and we must define short exact sequences of commutative Hopf

algebras k → H 0 → H → H 00 → k and determine when the induced sequence of groups 1 → Alg(H 00 , k ) → Alg(H, k ) → Alg(H 0 , k ) → 1 is exact

How to construct lazy homology: general idea To construct lazy homology and to express the lazy cohomology groups as groups of algebra maps, I we attach to any Hopf algebra H

• a commutative Hopf algebra F (H) such that Reg(H) ∼ = Alg(F (H), k ) • and two coalgebras H [1] and H [2] such that [1] Reg` (H) ∼ = Reg(H ) and

[2] Reg` (H) ∼ = Reg(H ) (2)

I and we must define short exact sequences of commutative Hopf

algebras k → H 0 → H → H 00 → k and determine when the induced sequence of groups 1 → Alg(H 00 , k ) → Alg(H, k ) → Alg(H 0 , k ) → 1 is exact

Takeuchi’s free commutative Hopf algebra I To a coalgebra C with coproduct ∆ : C → C ⊗ C and counit ε : C → k

we associate the commutative algebra F (C) generated by symbols tx and tx−1 (x ∈ C) submitted to the relations (a) The maps x 7→ tx and x 7→ tx−1 are linear (b) For all x ∈ C, X X −1 tx 0 tx−1 tx 0 tx 00 00 = ε(x) 1 = (x)

(x)

I The algebra F (C) is a commutative Hopf algebra with coproduct ∆,

counit ε, and (involutive) antipode S given by X ∆(tx ) = tx 0 ⊗ tx 00 , ε(tx ) = ε(x) ,

S(tx ) = tx−1

(x)

I Proposition. The map f 7→ f ◦ t induces a group isomorphism

Alg(F (C), k) ∼ = Reg(C)

Takeuchi’s free commutative Hopf algebra I To a coalgebra C with coproduct ∆ : C → C ⊗ C and counit ε : C → k

we associate the commutative algebra F (C) generated by symbols tx and tx−1 (x ∈ C) submitted to the relations (a) The maps x 7→ tx and x 7→ tx−1 are linear (b) For all x ∈ C, X X −1 tx 0 tx−1 tx 0 tx 00 00 = ε(x) 1 = (x)

(x)

I The algebra F (C) is a commutative Hopf algebra with coproduct ∆,

counit ε, and (involutive) antipode S given by X ∆(tx ) = tx 0 ⊗ tx 00 , ε(tx ) = ε(x) ,

S(tx ) = tx−1

(x)

I Proposition. The map f 7→ f ◦ t induces a group isomorphism

Alg(F (C), k) ∼ = Reg(C)

Takeuchi’s free commutative Hopf algebra I To a coalgebra C with coproduct ∆ : C → C ⊗ C and counit ε : C → k

we associate the commutative algebra F (C) generated by symbols tx and tx−1 (x ∈ C) submitted to the relations (a) The maps x 7→ tx and x 7→ tx−1 are linear (b) For all x ∈ C, X X −1 tx 0 tx−1 tx 0 tx 00 00 = ε(x) 1 = (x)

(x)

I The algebra F (C) is a commutative Hopf algebra with coproduct ∆,

counit ε, and (involutive) antipode S given by X ∆(tx ) = tx 0 ⊗ tx 00 , ε(tx ) = ε(x) ,

S(tx ) = tx−1

(x)

I Proposition. The map f 7→ f ◦ t induces a group isomorphism

Alg(F (C), k) ∼ = Reg(C)

The first lazy quotient I Given a coalgebra C, let C [1] be the quotient of C by the subspace

spanned by the elements X X ϕ(x 0 ) x 00 − ϕ(x 00 ) x 0 (x)

(x ∈ C, ϕ ∈ Hom(C, k ))

(x)

The projection C → C [1] turns C [1] into a cocommutative coalgebra I Proposition. The projection C → C [1] induces a group isomorphism

Reg(C [1] ) ∼ = Reg` (C) Corollary. There is a group isomorphism [1] Reg` (C) ∼ = Alg(F (C ), k)

The first lazy quotient I Given a coalgebra C, let C [1] be the quotient of C by the subspace

spanned by the elements X X ϕ(x 0 ) x 00 − ϕ(x 00 ) x 0 (x)

(x ∈ C, ϕ ∈ Hom(C, k ))

(x)

The projection C → C [1] turns C [1] into a cocommutative coalgebra I Proposition. The projection C → C [1] induces a group isomorphism

Reg(C [1] ) ∼ = Reg` (C) Corollary. There is a group isomorphism [1] Reg` (C) ∼ = Alg(F (C ), k)

The second lazy quotient I Given a Hopf algebra H, let H [2] be the quotient of H ⊗ H by the

subspace spanned by the elements X X ϕ(x 0 y 0 ) x 00 ⊗y 00 − ϕ(x 00 y 00 ) x 0 ⊗y 0 (x)(y )

(x, y ∈ H, ϕ ∈ Hom(H, k ))

(x)(y)

The projection H → H [2] turns H [2] into a coalgebra I Proposition. The projection H → H [2] induces a group isomorphism

Reg(H [2] ) ∼ = Reg` (H) (2)

Corollary. There is a group isomorphism [2] Reg` (H) ∼ = Alg(F (H ), k ) (2)

The second lazy quotient I Given a Hopf algebra H, let H [2] be the quotient of H ⊗ H by the

subspace spanned by the elements X X ϕ(x 0 y 0 ) x 00 ⊗y 00 − ϕ(x 00 y 00 ) x 0 ⊗y 0 (x)(y )

(x, y ∈ H, ϕ ∈ Hom(H, k ))

(x)(y)

The projection H → H [2] turns H [2] into a coalgebra I Proposition. The projection H → H [2] induces a group isomorphism

Reg(H [2] ) ∼ = Reg` (H) (2)

Corollary. There is a group isomorphism [2] Reg` (H) ∼ = Alg(F (H ), k ) (2)

Hopf kernels I A Hopf algebra morphism π : H → H 0 is normal if

X X ¯ ˘ ¯ ˘ x ∈H| π(x 0 ) ⊗ x 00 = 1 ⊗ x = x ∈ H | π(x 00 ) ⊗ x 0 = 1 ⊗ x (x)

(x)

(1) (Condition (1) is always satisfied if H is cocommutative) When π is normal, then we denote both sides of (1) by HKer(π): it is the Hopf kernel of π. I Properties. (a) The Hopf kernel HKer(π) is a Hopf subalgebra of H

(b) If u : G → G0 is a group homomorphism, then the induced Hopf algebra morphism k [u] : k [G] → k [G0 ] is normal and HKer(k [u]) = k [Ker(u)]

Hopf kernels I A Hopf algebra morphism π : H → H 0 is normal if

X X ¯ ˘ ¯ ˘ x ∈H| π(x 0 ) ⊗ x 00 = 1 ⊗ x = x ∈ H | π(x 00 ) ⊗ x 0 = 1 ⊗ x (x)

(x)

(1) (Condition (1) is always satisfied if H is cocommutative) When π is normal, then we denote both sides of (1) by HKer(π): it is the Hopf kernel of π. I Properties. (a) The Hopf kernel HKer(π) is a Hopf subalgebra of H

(b) If u : G → G0 is a group homomorphism, then the induced Hopf algebra morphism k [u] : k [G] → k [G0 ] is normal and HKer(k [u]) = k [Ker(u)]

Hopf quotients I Let H0 ⊂ H be a Hopf subalgebra and let H0+ = Ker(ε : H0 → k ) be the

augmentation ideal of H0 If H0+ H = H H0+ (always satisfied if H is commutative), then we define the Hopf quotient to be H//H0 = H/H0+ H I Properties. (a) The quotient H//H0 is a Hopf algebra

(b) If G0 is a normal subgroup of G, then k [G0 ] is a Hopf subalgebra of k [G] such that k [G0 ]+ k [G] = k [G] k [G0 ]+ , and k [G]//k [G0 ] = k [G/G0 ] (c) (Takeuchi) The category of bicommutative Hopf algebras is abelian with Hopf kernels and Hopf quotients as kernels and quotients

Hopf quotients I Let H0 ⊂ H be a Hopf subalgebra and let H0+ = Ker(ε : H0 → k ) be the

augmentation ideal of H0 If H0+ H = H H0+ (always satisfied if H is commutative), then we define the Hopf quotient to be H//H0 = H/H0+ H I Properties. (a) The quotient H//H0 is a Hopf algebra

(b) If G0 is a normal subgroup of G, then k [G0 ] is a Hopf subalgebra of k [G] such that k [G0 ]+ k [G] = k [G] k [G0 ]+ , and k [G]//k [G0 ] = k [G/G0 ] (c) (Takeuchi) The category of bicommutative Hopf algebras is abelian with Hopf kernels and Hopf quotients as kernels and quotients

Short exact sequences I Let H0 ⊂ H be a Hopf subalgebra such that H0+ H = H H0+ so that the

Hopf quotient H//H0 makes sense An exact sequence of Hopf algebras is a sequence of the form k −→ H0 −→ H −→ H//H0 −→ k

(2)

I Theorem. (a) The sequence (2) induces an exact sequence of groups

1 −→ Alg(H//H0 , k ) −→ Alg(H, k ) −→ Alg(H0 , k) (b) If furthermore H is commutative and k is algebraically closed, then Alg(H, k ) −→ Alg(H0 , k ) is surjective

Short exact sequences I Let H0 ⊂ H be a Hopf subalgebra such that H0+ H = H H0+ so that the

Hopf quotient H//H0 makes sense An exact sequence of Hopf algebras is a sequence of the form k −→ H0 −→ H −→ H//H0 −→ k

(2)

I Theorem. (a) The sequence (2) induces an exact sequence of groups

1 −→ Alg(H//H0 , k ) −→ Alg(H, k ) −→ Alg(H0 , k) (b) If furthermore H is commutative and k is algebraically closed, then Alg(H, k ) −→ Alg(H0 , k ) is surjective

Plan

• Part One: Lazy cohomology

• Part Two: Universal constructions

• Part Three: Lazy homology

• References

• Appendix: Cocycle-twisting of Hopf algebras

The first lazy homology Hopf algebra Let H be a Hopf algebra I Proposition. There are morphisms of commutative Hopf algebras d

d

2 1 F (H [2] ) −→ F (H [1] ) −→ F (k )

respectively induced by X d2 : tx⊗y 7→ tx 0 ty 0 tx−1 00 y 00

and

d1 : tx 7→ tε(x)

(x)(y)

I Definition. The first lazy homology Hopf algebra of H is given by

H1` (H) = HKer(d1 )// Im(d2 ) I Theorem. The Hopf algebra H1` (H) is bicommutative and there is a

natural isomorphism of abelian groups ∼ =

H`1 (H, k ) −→ Alg(H1` (H), k )

The first lazy homology Hopf algebra Let H be a Hopf algebra I Proposition. There are morphisms of commutative Hopf algebras d

d

2 1 F (H [2] ) −→ F (H [1] ) −→ F (k )

respectively induced by X d2 : tx⊗y 7→ tx 0 ty 0 tx−1 00 y 00

and

d1 : tx 7→ tε(x)

(x)(y)

I Definition. The first lazy homology Hopf algebra of H is given by

H1` (H) = HKer(d1 )// Im(d2 ) I Theorem. The Hopf algebra H1` (H) is bicommutative and there is a

natural isomorphism of abelian groups ∼ =

H`1 (H, k ) −→ Alg(H1` (H), k )

The first lazy homology Hopf algebra Let H be a Hopf algebra I Proposition. There are morphisms of commutative Hopf algebras d

d

2 1 F (H [2] ) −→ F (H [1] ) −→ F (k )

respectively induced by X d2 : tx⊗y 7→ tx 0 ty 0 tx−1 00 y 00

and

d1 : tx 7→ tε(x)

(x)(y)

I Definition. The first lazy homology Hopf algebra of H is given by

H1` (H) = HKer(d1 )// Im(d2 ) I Theorem. The Hopf algebra H1` (H) is bicommutative and there is a

natural isomorphism of abelian groups ∼ =

H`1 (H, k ) −→ Alg(H1` (H), k )

Some computations of H1`

I Proposition. If H = k [G] is the Hopf algebra of a group G, then

H1` (k [G]) ∼ = k [H1 (G)] = k [Gab ]

I Proposition. If H = O(G) is the Hopf algebra of k-valued functions on

a finite group G, then H1` (O(G)) ∼ = O(Z (G)) (the Hopf algebra of functions on the center of G) Remark. When k is algebraically closed of characteristic zero, then O(G) is a cosemisimple Hopf algebra

Some computations of H1`

I Proposition. If H = k [G] is the Hopf algebra of a group G, then

H1` (k [G]) ∼ = k [H1 (G)] = k [Gab ]

I Proposition. If H = O(G) is the Hopf algebra of k-valued functions on

a finite group G, then H1` (O(G)) ∼ = O(Z (G)) (the Hopf algebra of functions on the center of G) Remark. When k is algebraically closed of characteristic zero, then O(G) is a cosemisimple Hopf algebra

The cosemisimple case Assume that the ground field k is algebraically closed of characteristic zero I Let H be a cosemisimple Hopf algebra: the category of H-comodules is

semisimple. Let Irrep(H) be the set of isomorphism classes of finite-dimensional simple H-comodules. ¨ Definition (Baumgartel & Lledo, Muger, ¨ Gelaki & Nikshych, Petit). The universal abelian grading group ΓH is the multiplicative abelian group generated by the elements V ∈ Irrep(H) and the relations whenever W ⊂ U ⊗ V

UV =W

I Theorem. If H is a cosemisimple Hopf algebra, then

H1` (H) ∼ = k [ΓH ] Remark. If G is a finite group, then H = O(G) is cosemisimple and × ΓH ∼ = Hom(Z (G), k )

(Pontryagin dual of the center)

We recover H1` (H) ∼ = k [ΓH ] ∼ = O(Z (G))

The cosemisimple case Assume that the ground field k is algebraically closed of characteristic zero I Let H be a cosemisimple Hopf algebra: the category of H-comodules is

semisimple. Let Irrep(H) be the set of isomorphism classes of finite-dimensional simple H-comodules. ¨ Definition (Baumgartel & Lledo, Muger, ¨ Gelaki & Nikshych, Petit). The universal abelian grading group ΓH is the multiplicative abelian group generated by the elements V ∈ Irrep(H) and the relations whenever W ⊂ U ⊗ V

UV =W

I Theorem. If H is a cosemisimple Hopf algebra, then

H1` (H) ∼ = k [ΓH ] Remark. If G is a finite group, then H = O(G) is cosemisimple and × ΓH ∼ = Hom(Z (G), k )

(Pontryagin dual of the center)

We recover H1` (H) ∼ = k [ΓH ] ∼ = O(Z (G))

Towards H2` (H) I Given a Hopf algebra H, consider the sequence of maps d

d

d

3 2 1 H ⊗ H ⊗ H −→ F (H [2] ) −→ F (H [1] ) −→ F (k )

where d3 (x ⊗ y ⊗ z) is the image in F (H [2] ) of −1 ty 0 ⊗z 0 tx 0 ⊗y 00 z 00 tx−1 00 y 000 ⊗z 000 tx 000 ⊗y 0000

I Lemma (a) The Hopf algebra morphism d2 is normal

(b) For all x, y , z ∈ H, d3 (x ⊗ y ⊗ z) belongs to the Hopf kernel HKer(d2 ) (c) The ideal B2` (H) of HKer(d2 ) generated by the elements ` ´ d3 (x ⊗ y ⊗ z) − ε(xyz) 1 and S d3 (x ⊗ y ⊗ z) − ε(xyz) is a Hopf ideal of the Hopf algebra HKer(d2 )

Towards H2` (H) I Given a Hopf algebra H, consider the sequence of maps d

d

d

3 2 1 H ⊗ H ⊗ H −→ F (H [2] ) −→ F (H [1] ) −→ F (k )

where d3 (x ⊗ y ⊗ z) is the image in F (H [2] ) of −1 ty 0 ⊗z 0 tx 0 ⊗y 00 z 00 tx−1 00 y 000 ⊗z 000 tx 000 ⊗y 0000

I Lemma (a) The Hopf algebra morphism d2 is normal

(b) For all x, y , z ∈ H, d3 (x ⊗ y ⊗ z) belongs to the Hopf kernel HKer(d2 ) (c) The ideal B2` (H) of HKer(d2 ) generated by the elements ` ´ d3 (x ⊗ y ⊗ z) − ε(xyz) 1 and S d3 (x ⊗ y ⊗ z) − ε(xyz) is a Hopf ideal of the Hopf algebra HKer(d2 )

The second lazy homology Hopf algebra

I Definition. The second lazy homology Hopf algebra of a Hopf

algebra H is given by H2` (H) = HKer(d2 )/B2` (H)

I Proposition. If H = k [G] is the Hopf algebra of a group G, then

H2` (k [G]) ∼ = k [H2 (G)] Remark. H2` (k [G]) is a group algebra from which we can recover the homology group H2 (G)

The second lazy homology Hopf algebra

I Definition. The second lazy homology Hopf algebra of a Hopf

algebra H is given by H2` (H) = HKer(d2 )/B2` (H)

I Proposition. If H = k [G] is the Hopf algebra of a group G, then

H2` (k [G]) ∼ = k [H2 (G)] Remark. H2` (k [G]) is a group algebra from which we can recover the homology group H2 (G)

A universal coefficient theorem I Theorem. For any Hopf algebra H there is an exact sequence of groups κ

1 −→ Ext1 (H, k ) −→ H`2 (H, k ) −→ Alg(H2` (H), k) If in addition k is algebraically closed, then κ is an isomorphism: ∼ =

κ : H`2 (H, k ) −→ Alg(H2` (H), k) I The group Ext1 (H, k ) is defined by the following exact sequence:

1 → Alg(H1` (H), k ) → Alg(HKer(d1 ), k ) → Alg(Im(d2 ), k ) → Ext1 (H, k ) → 1 induced by the exact sequence of commutative Hopf algebras k −→ Im(d2 ) −→ HKer(d1 ) −→ H1` (H) −→ k defining the first lazy homology group H1` (H)

A universal coefficient theorem I Theorem. For any Hopf algebra H there is an exact sequence of groups κ

1 −→ Ext1 (H, k ) −→ H`2 (H, k ) −→ Alg(H2` (H), k) If in addition k is algebraically closed, then κ is an isomorphism: ∼ =

κ : H`2 (H, k ) −→ Alg(H2` (H), k) I The group Ext1 (H, k ) is defined by the following exact sequence:

1 → Alg(H1` (H), k ) → Alg(HKer(d1 ), k ) → Alg(Im(d2 ), k ) → Ext1 (H, k ) → 1 induced by the exact sequence of commutative Hopf algebras k −→ Im(d2 ) −→ HKer(d1 ) −→ H1` (H) −→ k defining the first lazy homology group H1` (H)

The lazy homology of the Sweedler algebra

Theorem. For the Sweedler algebra H4 we have H1` (H4 ) ∼ =k

and

H2` (H4 ) ∼ = k [X ]

where X is a primitive element, i.e., ∆(X ) = 1 ⊗ X + X ⊗ 1

Remark. In this case H2` (H) is not a group algebra and the group of group-like elements is trivial

The lazy homology of the Sweedler algebra: proof I

[1]

We have H4

[2]

= k and H4 is a five-dimensional coalgebra: [2]

H4

= ky0 ⊕ ky1 ⊕ ky2 ⊕ ky3 ⊕ ky4

with ∆(y0 ) = y0 ⊗ y0 Therefore,

[1] F (H4 )

= k[T , T

−1

and ∆(yi ) = y0 ⊗ yi + yi ⊗ y0

] and [2]

−1

F (H4 ) = k [Y0 , Y0

I

(i = 1, 2, 3, 4)

, Y1 , Y2 , Y3 , Y4 ]

The differential d2 is the Hopf algebra map [2]

−1

d2 : F (H4 ) = k[Y0 , Y0

[1]

, Y1 , Y2 , Y3 , Y4 ] −→ F (H4 ) = k [T , T

−1

given by d2 (Y0 ) = T and d2 (Yi ) = 0 if i = 1, 2, 3, 4 One deduces the first lazy cohomology: H1` (H4 ) = k and the computation of HKer(d2 ) = k [X1 , X2 , X3 , X4 ] where X1 , X2 , X3 , X4 are the primitive elements Xi = Yi /Y0 (i = 1, 2, 3, 4)

I

Computing the values of d3 : H4⊗3 → HKer(d2 ), one finds the second lazy cohomology: ` H2 (H4 ) = k [X1 , X2 , X3 , X4 ]/(X1 = X2 = −X3 = −X4 ) ∼ = k [X ]

]

The lazy homology of the Sweedler algebra: proof I

[1]

We have H4

[2]

= k and H4 is a five-dimensional coalgebra: [2]

H4

= ky0 ⊕ ky1 ⊕ ky2 ⊕ ky3 ⊕ ky4

with ∆(y0 ) = y0 ⊗ y0 Therefore,

[1] F (H4 )

= k[T , T

−1

and ∆(yi ) = y0 ⊗ yi + yi ⊗ y0

] and [2]

−1

F (H4 ) = k [Y0 , Y0

I

(i = 1, 2, 3, 4)

, Y1 , Y2 , Y3 , Y4 ]

The differential d2 is the Hopf algebra map [2]

−1

d2 : F (H4 ) = k[Y0 , Y0

[1]

, Y1 , Y2 , Y3 , Y4 ] −→ F (H4 ) = k [T , T

−1

given by d2 (Y0 ) = T and d2 (Yi ) = 0 if i = 1, 2, 3, 4 One deduces the first lazy cohomology: H1` (H4 ) = k and the computation of HKer(d2 ) = k [X1 , X2 , X3 , X4 ] where X1 , X2 , X3 , X4 are the primitive elements Xi = Yi /Y0 (i = 1, 2, 3, 4)

I

Computing the values of d3 : H4⊗3 → HKer(d2 ), one finds the second lazy cohomology: ` H2 (H4 ) = k[X1 , X2 , X3 , X4 ]/(X1 = X2 = −X3 = −X4 ) ∼ = k [X ]

]

The lazy homology of the Sweedler algebra: proof I

[1]

We have H4

[2]

= k and H4 is a five-dimensional coalgebra: [2]

H4

= ky0 ⊕ ky1 ⊕ ky2 ⊕ ky3 ⊕ ky4

with ∆(y0 ) = y0 ⊗ y0 Therefore,

[1] F (H4 )

= k[T , T

−1

and ∆(yi ) = y0 ⊗ yi + yi ⊗ y0

] and [2]

−1

F (H4 ) = k [Y0 , Y0

I

(i = 1, 2, 3, 4)

, Y1 , Y2 , Y3 , Y4 ]

The differential d2 is the Hopf algebra map [2]

−1

d2 : F (H4 ) = k[Y0 , Y0

[1]

, Y1 , Y2 , Y3 , Y4 ] −→ F (H4 ) = k [T , T

−1

given by d2 (Y0 ) = T and d2 (Yi ) = 0 if i = 1, 2, 3, 4 One deduces the first lazy cohomology: H1` (H4 ) = k and the computation of HKer(d2 ) = k [X1 , X2 , X3 , X4 ] where X1 , X2 , X3 , X4 are the primitive elements Xi = Yi /Y0 (i = 1, 2, 3, 4)

I

Computing the values of d3 : H4⊗3 → HKer(d2 ), one finds the second lazy cohomology: ` H2 (H4 ) = k[X1 , X2 , X3 , X4 ]/(X1 = X2 = −X3 = −X4 ) ∼ = k [X ]

]

References E. Aljadeff, D. Haile, M. Natapov, Graded identities of matrix algebras and the universal graded algebra, arXiv:0710.5568, Trans. Amer. Math. Soc. (2008). E. Aljadeff, C. Kassel, Polynomial identities and noncommutative versal torsors, arXiv:0708.4108, Adv. Math. 218 (2008), 1453–1495. J. Bichon, G. Carnovale, Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras, J. Pure Appl. Algebra 204 (2006), 627–665. J. Bichon, C. Kassel, The lazy homology of a Hopf algebra, arXiv:0807.1651. C. Kassel, Generic Hopf Galois extensions, arXiv:0809.0638. P. Schauenburg, Hopf bimodules, coquasibialgebras, and an exact sequence of Kac, Adv. Math. 165 (2002), 194–263. M. E. Sweedler, Cohomology of algebras over Hopf algebras, Trans. Amer. Math. Soc. 133 (1968), 205–239. M. Takeuchi, Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561–582. M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251–270.

THANK YOU FOR YOUR ATTENTION

Plan

• Part One: Lazy cohomology

• Part Two: Universal constructions

• Part Three: Lazy homology

• References

• Appendix: Cocycle-twisting of Hopf algebras

Cocycle-twisting of Hopf algebras

• Let H be a Hopf algebra and α : H × H → k be an invertible (two-)cocycle • Out of the pair (H, α) we construct a new Hopf algebra H[α], which is H as a coalgebra (unchanged coproduct) and with new product ∗α defined for x, y ∈ H by X x ∗α y = α(x 0 , y 0 ) x 00 y 00 α−1 (x 000 , y 000 ) (x),(y )

Remark. If α is lazy, then H[α] = H as Hopf algebras (it is called “lazy” because it does not alter H)

Cohomologous cocycles

• Lemma (Doi). Let α, β be invertible cocycles. Then H[α] ∼ = H[β] as Hopf algebras if and only if α and β are cohomologous in the following sense • Two-cocycles α and β of H are said to be cohomologous (we write α ∼ β) if there is an invertible linear form λ : H → k with convolution inverse λ−1 : H → k such that for all x, y ∈ H, X β(x, y ) = λ(x 0 ) λ(y 0 ) α(x 00 , y 00 ) λ−1 (x 000 y 000 ) (x),(y)

• By Doi’s lemma, there is a bijection ˘ ¯ ˘ ¯ Hopf algebras H[α] /(isomorphisms) ∼ = invertible cocycles on H /∼

Cocycles on group algebras • Let G be a group and H = k [G] the group algebra with standard coproduct ∆(g) = g ⊗ g

(g ∈ G)

Then the restriction to G of an invertible cocycle α on H is a group-cocycle with values in k × = k − {0} Therefore, α represents an element of the cohomology group H 2 (G, k × ) • Two cocycles α, β on H = k [G] are cohomologous in the sense of the previous slide if and only if for all g, h ∈ G, β(g, h) =

λ(g)λ(h) α(g, h) λ(gh)

for some mapping λ : G → k , i.e., if and only if α and β represent the same element of H 2 (G, k × ) • Therefore, the set {invertible cocycles on k [G]}/∼ has a group structure

Convolution product of cocycles • If H is a cocommutative Hopf algebra, then the convolution product of two cocycles on H is again a cocycle and the set of cohomology classes ˘ ¯ invertible cocycles on H /∼ 2 becomes a group, denoted HSw (H, k ), first considered by Sweedler

Remark. To a cocommutative Hopf algebra Sweedler associated i cohomology groups HSw (H, k ) in any degree i ≥ 1 • For a general Hopf algebra H, the convolution product of two cocycles is no longer a cocycle and the set of cohomology classes ˘ ¯ invertible cocycles on H /∼ is not a group. That is why lazy cohomology was introduced. We have i H`i (H) = HSw (H, k )

for any cocommutative Hopf algebra H

(i = 1, 2)