## Invariant Drinfeld twists on group algebras

### Christian Kassel

Institut de Recherche Math ´ematique Avanc ´ee CNRS - Universit ´e de Strasbourg

Strasbourg, France

International Workshop “Groups and Hopf Algebras”

Memorial University of Newfoundland 5 June 2009

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

I Our original motivation was to compute the secondlazy cohomology
group H^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

the Hopf algebrasOk(G)offunctions on finite non-abelian groups I We reformulate the problem in terms ofinvariant Drinfeld twists

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

I The answer involves theabelian normal subgroups of central typeofG as well as the group ofclass-preserving outer automorphismsofG The proof uses tools fromquantum group theory, mainlyR-matrices I I shall illustrate all this with severalexamples

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

I Our original motivation was to compute the secondlazy cohomology
group H^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

the Hopf algebrasOk(G)offunctions on finite non-abelian groups I We reformulate the problem in terms ofinvariant Drinfeld twists

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

I The answer involves theabelian normal subgroups of central typeofG as well as the group ofclass-preserving outer automorphismsofG The proof uses tools fromquantum group theory, mainlyR-matrices I I shall illustrate all this with severalexamples

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

I Our original motivation was to compute the secondlazy cohomology
group H^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

the Hopf algebrasOk(G)offunctions on finite non-abelian groups I We reformulate the problem in terms ofinvariant Drinfeld twists

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

I The answer involves theabelian normal subgroups of central typeofG as well as the group ofclass-preserving outer automorphismsofG The proof uses tools fromquantum group theory, mainlyR-matrices I I shall illustrate all this with severalexamples

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

## Introduction

I Report on joint work withPierre Guillot(Strasbourg):

Cohomology of invariant Drinfeld twists on group algebras arXiv:0903.2807

^{2}_{`}(H)of Hopf algebras that areneither cocommutative, nor
pointedsuch as

and obtain amethod to computeH^{2}_{`}(H)whenH=Ok(G)

## Plan

### Drinfeld twists and cohomology

### The main theorem

### On the proof

### Examples

### Rationality issues

### References

## Plan

### Drinfeld twists and cohomology

### The main theorem

### On the proof

### Examples

### Rationality issues

### References

## Drinfeld twists

I LetHbe aHopf algebraover some fixed fieldk

**Definition.** ADrinfeld twiston H is an invertible element F of H⊗H
satisfying the condition

(F⊗1) (∆⊗idH)(F) = (1⊗F) (idH⊗∆)(F) (1) where∆ :H→H⊗H is the coproduct of H

Twists were introduced byDrinfeldin his work on quasi-Hopf algebras, in order to “twist” the coproduct ofHwithout changing its product. They have become an important tool in theclassificationof finite-dimensional Hopf algebras. There is now anabundant literatureon twists

I **“Trivial” solutions of**(1) :F = (a⊗a) ∆(a^{−1})
whereais an invertible element ofH

**Problem.**Findmore (all?) solutionsof (1)

We shall look forspecialsolutions of (1) as follows. . .

## Drinfeld twists

I LetHbe aHopf algebraover some fixed fieldk

**Definition.** ADrinfeld twiston H is an invertible element F of H⊗H
satisfying the condition

(F⊗1) (∆⊗idH)(F) = (1⊗F) (idH⊗∆)(F) (1) where∆ :H→H⊗H is the coproduct of H

Twists were introduced byDrinfeldin his work on quasi-Hopf algebras, in order to “twist” the coproduct ofHwithout changing its product. They have become an important tool in theclassificationof finite-dimensional Hopf algebras. There is now anabundant literatureon twists

I **“Trivial” solutions of**(1) :F = (a⊗a) ∆(a^{−1})
whereais an invertible element ofH

**Problem.**Findmore (all?) solutionsof (1)

We shall look forspecialsolutions of (1) as follows. . .

## Invariant twists

I **Definition.** A twist F isinvariant

∆(a)F =F∆(a) for alla∈H In general the product of two twists is not a twist, but. . .

I **Proposition.** (a) Invariant twists form agroupunder multiplication
(b) The group of invariant twists contains as acentral subgroupthe
group oftrivial twists, where a twist F is calledtrivialif

F= (a⊗a) ∆(a^{−1})
for somecentralinvertible element a∈H

We can thus consider the quotient group

{invariant twists onH}/{trivial twists}. . .

## Invariant twists

I **Definition.** A twist F isinvariant

∆(a)F =F∆(a) for alla∈H In general the product of two twists is not a twist, but. . .

I **Proposition.** (a) Invariant twists form agroupunder multiplication
(b) The group of invariant twists contains as acentral subgroupthe
group oftrivial twists, where a twist F is calledtrivialif

F= (a⊗a) ∆(a^{−1})
for somecentralinvertible element a∈H

We can thus consider the quotient group

{invariant twists onH}/{trivial twists}. . .

## Invariant twists

I **Definition.** A twist F isinvariant

∆(a)F =F∆(a) for alla∈H In general the product of two twists is not a twist, but. . .

I **Proposition.** (a) Invariant twists form agroupunder multiplication
(b) The group of invariant twists contains as acentral subgroupthe
group oftrivial twists, where a twist F is calledtrivialif

F= (a⊗a) ∆(a^{−1})
for somecentralinvertible element a∈H

We can thus consider the quotient group

{invariant twists onH}/{trivial twists}. . .

## Invariant twists

I **Definition.** A twist F isinvariant

∆(a)F =F∆(a) for alla∈H In general the product of two twists is not a twist, but. . .

**Proposition.** (a) Invariant twists form agroupunder multiplication
(b) The group of invariant twists contains as acentral subgroupthe
group oftrivial twists, where a twist F is calledtrivialif

F= (a⊗a) ∆(a^{−1})
for somecentralinvertible element a∈H

We can thus consider the quotient group

{invariant twists onH}/{trivial twists}. . .

## Invariant twists

I **Definition.** A twist F isinvariant

∆(a)F =F∆(a) for alla∈H In general the product of two twists is not a twist, but. . .

**Proposition.** (a) Invariant twists form agroupunder multiplication
(b) The group of invariant twists contains as acentral subgroupthe
group oftrivial twists, where a twist F is calledtrivialif

F= (a⊗a) ∆(a^{−1})
for somecentralinvertible element a∈H

We can thus consider the quotient group

{invariant twists onH}/{trivial twists}. . .

## Invariant twists

I **Definition.** A twist F isinvariant

∆(a)F =F∆(a) for alla∈H In general the product of two twists is not a twist, but. . .

**Proposition.** (a) Invariant twists form agroupunder multiplication
(b) The group of invariant twists contains as acentral subgroupthe
group oftrivial twists, where a twist F is calledtrivialif

F= (a⊗a) ∆(a^{−1})
for somecentralinvertible element a∈H

We can thus consider the quotient group

{invariant twists onH}/{trivial twists}. . .

## The group H

^{2}

_{`}

## (G)

I . . . which we consider in the special case whenH=k[G]is thealgebra of a groupG

**Definition.**Given agroupG, set

H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

**Our aim.**Determinethe group H^{2}_{`}(G)for anyfinitegroupG

Whyconsider H^{2}_{`}(G)?

I **Because**it is a first step infinding all invariant twistsonk[G]and. . .
. . . the group H^{2}`(G)is isomorphic to thesecond lazy cohomology group
of the Hopf algebraOk(G)ofk-valued functions onG:

H^{2}_{`}(G)∼=H^{2}_{`}(Ok(G))

## The group H

^{2}

_{`}

## (G)

I . . . which we consider in the special case whenH=k[G]is thealgebra of a groupG

**Definition.**Given agroupG, set

H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

**Our aim.**Determinethe group H^{2}_{`}(G)for anyfinitegroupG

Whyconsider H^{2}_{`}(G)?

I **Because**it is a first step infinding all invariant twistsonk[G]and. . .
. . . the group H^{2}`(G)is isomorphic to thesecond lazy cohomology group
of the Hopf algebraOk(G)ofk-valued functions onG:

H^{2}_{`}(G)∼=H^{2}_{`}(Ok(G))

## The group H

^{2}

_{`}

## (G)

I . . . which we consider in the special case whenH=k[G]is thealgebra of a groupG

**Definition.**Given agroupG, set

H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

**Our aim.**Determinethe group H^{2}_{`}(G)for anyfinitegroupG

Whyconsider H^{2}_{`}(G)?

I **Because**it is a first step infinding all invariant twistsonk[G]and. . .
. . . the group H^{2}`(G)is isomorphic to thesecond lazy cohomology group
of the Hopf algebraOk(G)ofk-valued functions onG:

H^{2}_{`}(G)∼=H^{2}_{`}(Ok(G))

## The group H

^{2}

_{`}

## (G)

I . . . which we consider in the special case whenH=k[G]is thealgebra of a groupG

**Definition.**Given agroupG, set

H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

**Our aim.**Determinethe group H^{2}_{`}(G)for anyfinitegroupG

Whyconsider H^{2}_{`}(G)?

**Because**it is a first step infinding all invariant twistsonk[G]and. . .
. . . the group H^{2}`(G)is isomorphic to thesecond lazy cohomology group
of the Hopf algebraOk(G)ofk-valued functions onG:

H^{2}_{`}(G)∼=H^{2}_{`}(Ok(G))

## The group H

^{2}

_{`}

## (G)

I . . . which we consider in the special case whenH=k[G]is thealgebra of a groupG

**Definition.**Given agroupG, set

H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

**Our aim.**Determinethe group H^{2}_{`}(G)for anyfinitegroupG

Whyconsider H^{2}_{`}(G)?

**Because**it is a first step infinding all invariant twistsonk[G]and. . .
. . . the group H^{2}`(G)is isomorphic to thesecond lazy cohomology group
of the Hopf algebraOk(G)ofk-valued functions onG:

H^{2}_{`}(G)∼=H^{2}_{`}(Ok(G))

## Lazy cohomology

I In the last decade Schauenburg, Bichon, Carnovaleet al. defined
cohomology groups H^{1}_{`}(H)and H^{2}_{`}(H)for anarbitraryHopf algebraH.

These groups are called thelazy cohomologygroups ofH

•On the class ofcocommutativeHopf algebras the lazy cohomology groupscoincidewith the cohomology groups introduced bySweedler (1968)

I In particular, ifH=k[G]is thealgebra of a groupG, then lazy cohomology coincides withgroup cohomology

H^{i}_{`}(H)∼=H^{i}(G,k^{×}) (i=1,2)
whereGacts trivially onk^{×}=k− {0}

I Thefirstlazy cohomology group for ageneralHopf algebra is given by
H^{1}_{`}(H) ={µ∈Alg(H,k)|µ∗idH =idH∗µ}

## Lazy cohomology

I In the last decade Schauenburg, Bichon, Carnovaleet al. defined
cohomology groups H^{1}_{`}(H)and H^{2}_{`}(H)for anarbitraryHopf algebraH.

These groups are called thelazy cohomologygroups ofH

•On the class ofcocommutativeHopf algebras the lazy cohomology groupscoincidewith the cohomology groups introduced bySweedler (1968)

I In particular, ifH=k[G]is thealgebra of a groupG, then lazy cohomology coincides withgroup cohomology

H^{i}_{`}(H)∼=H^{i}(G,k^{×}) (i=1,2)
whereGacts trivially onk^{×}=k− {0}

I Thefirstlazy cohomology group for ageneralHopf algebra is given by
H^{1}_{`}(H) ={µ∈Alg(H,k)|µ∗idH =idH∗µ}

## Lazy cohomology

I In the last decade Schauenburg, Bichon, Carnovaleet al. defined
cohomology groups H^{1}_{`}(H)and H^{2}_{`}(H)for anarbitraryHopf algebraH.

These groups are called thelazy cohomologygroups ofH

•On the class ofcocommutativeHopf algebras the lazy cohomology groupscoincidewith the cohomology groups introduced bySweedler (1968)

I In particular, ifH=k[G]is thealgebra of a groupG, then lazy cohomology coincides withgroup cohomology

H^{i}_{`}(H)∼=H^{i}(G,k^{×}) (i=1,2)
whereGacts trivially onk^{×}=k− {0}

I Thefirstlazy cohomology group for ageneralHopf algebra is given by
H^{1}_{`}(H) ={µ∈Alg(H,k)|µ∗idH =idH∗µ}

## The second lazy cohomology group

I For thesecondlazy cohomology group H^{2}_{`}(H)onlyfew computations
(for non-cocommutative Hopf algebras) had been done

**Our original motivation.**Compute H^{2}`(H)formoreHopf algebras
We decided to try to compute H^{2}_{`}(H)for the Hopf algebrasOk(G)
I **Proposition.**For anyfinite groupG,

H^{2}`(Ok(G))∼=H^{2}`(G)

where H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

I **Proof.**The Hopf algebraOk(G)is dual of the Hopf algebrak[G]

For afinite-dimensionalHopf algebraHwithdualHopf algebraH^{∗}, we
have theidentifications

two-cocycle onH ←→ twist onH^{∗}

lazy two-cocycle onH ←→ invariant twist onH^{∗}
cohomologically trivial two-cocycle onH ←→ trivial twist onH^{∗}

## The second lazy cohomology group

I For thesecondlazy cohomology group H^{2}_{`}(H)onlyfew computations
(for non-cocommutative Hopf algebras) had been done

**Our original motivation.**Compute H^{2}`(H)formoreHopf algebras
We decided to try to compute H^{2}_{`}(H)for the Hopf algebrasOk(G)
I **Proposition.**For anyfinite groupG,

H^{2}`(Ok(G))∼=H^{2}`(G)

where H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

I **Proof.**The Hopf algebraOk(G)is dual of the Hopf algebrak[G]

For afinite-dimensionalHopf algebraHwithdualHopf algebraH^{∗}, we
have theidentifications

two-cocycle onH ←→ twist onH^{∗}

lazy two-cocycle onH ←→ invariant twist onH^{∗}
cohomologically trivial two-cocycle onH ←→ trivial twist onH^{∗}

## The second lazy cohomology group

I For thesecondlazy cohomology group H^{2}_{`}(H)onlyfew computations
(for non-cocommutative Hopf algebras) had been done

**Our original motivation.**Compute H^{2}`(H)formoreHopf algebras
We decided to try to compute H^{2}_{`}(H)for the Hopf algebrasOk(G)
I **Proposition.**For anyfinite groupG,

H^{2}`(Ok(G))∼=H^{2}`(G)

where H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

I **Proof.**The Hopf algebraOk(G)is dual of the Hopf algebrak[G]

For afinite-dimensionalHopf algebraHwithdualHopf algebraH^{∗}, we
have theidentifications

two-cocycle onH ←→ twist onH^{∗}

lazy two-cocycle onH ←→ invariant twist onH^{∗}
cohomologically trivial two-cocycle onH ←→ trivial twist onH^{∗}

## The second lazy cohomology group

^{2}_{`}(H)onlyfew computations
(for non-cocommutative Hopf algebras) had been done

**Our original motivation.**Compute H^{2}`(H)formoreHopf algebras
We decided to try to compute H^{2}_{`}(H)for the Hopf algebrasOk(G)
I **Proposition.**For anyfinite groupG,

H^{2}`(Ok(G))∼=H^{2}`(G)

where H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

I **Proof.**The Hopf algebraOk(G)is dual of the Hopf algebrak[G]

For afinite-dimensionalHopf algebraHwithdualHopf algebraH^{∗}, we
have theidentifications

two-cocycle onH ←→ twist onH^{∗}

^{∗}
cohomologically trivial two-cocycle onH ←→ trivial twist onH^{∗}

## The second lazy cohomology group

^{2}_{`}(H)onlyfew computations
(for non-cocommutative Hopf algebras) had been done

**Our original motivation.**Compute H^{2}`(H)formoreHopf algebras
We decided to try to compute H^{2}_{`}(H)for the Hopf algebrasOk(G)
I **Proposition.**For anyfinite groupG,

H^{2}`(Ok(G))∼=H^{2}`(G)

where H^{2}_{`}(G) ={invariant twists onk[G]}/{trivial twists}

I **Proof.**The Hopf algebraOk(G)is dual of the Hopf algebrak[G]

For afinite-dimensionalHopf algebraHwithdualHopf algebraH^{∗}, we
have theidentifications

two-cocycle onH ←→ twist onH^{∗}

^{∗}
cohomologically trivial two-cocycle onH ←→ trivial twist onH^{∗}

## Plan

### Drinfeld twists and cohomology

### The main theorem

### On the proof

### Examples

### Rationality issues

### References

## First ingredient: the set B(G)

I For anyfinite groupG

letB(G)be the set ofpairs(A,b)where (i)Ais anabelian normal subgroupofGand

(ii)b:Ab×Ab→k^{×}is ak^{×}-valuedG-invariant non-degenerate
alternatingbiadditive form on thecharacter groupbA=Hom(A,k^{×})

I The setB(G)isfiniteand can be “computed in finite time”

It isnon-emptysince it always contains the trivial pair({1},b≡1)

I There is a further restriction on the groupsAsuch that(A,b)∈ B(G):

Such a group is what is called agroup of central type

In particular, it is asquare group:A∼=A^{0}×A^{0}for some subgroupA^{0}
I **Examples.**B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(F^{q})

## First ingredient: the set B(G)

I For anyfinite groupG

letB(G)be the set ofpairs(A,b)where (i)Ais anabelian normal subgroupofGand

(ii)b:Ab×Ab→k^{×}is ak^{×}-valuedG-invariant non-degenerate
alternatingbiadditive form on thecharacter groupbA=Hom(A,k^{×})

I The setB(G)isfiniteand can be “computed in finite time”

It isnon-emptysince it always contains the trivial pair({1},b≡1)

I There is a further restriction on the groupsAsuch that(A,b)∈ B(G):

Such a group is what is called agroup of central type

In particular, it is asquare group:A∼=A^{0}×A^{0}for some subgroupA^{0}
I **Examples.**B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(F^{q})

## First ingredient: the set B(G)

I For anyfinite groupG

letB(G)be the set ofpairs(A,b)where (i)Ais anabelian normal subgroupofGand

(ii)b:Ab×Ab→k^{×}is ak^{×}-valuedG-invariant non-degenerate
alternatingbiadditive form on thecharacter groupbA=Hom(A,k^{×})

I The setB(G)isfiniteand can be “computed in finite time”

It isnon-emptysince it always contains the trivial pair({1},b≡1)

I There is a further restriction on the groupsAsuch that(A,b)∈ B(G):

Such a group is what is called agroup of central type

In particular, it is asquare group:A∼=A^{0}×A^{0}for some subgroupA^{0}
I **Examples.**B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(F^{q})

## First ingredient: the set B(G)

I For anyfinite groupG

letB(G)be the set ofpairs(A,b)where (i)Ais anabelian normal subgroupofGand

^{×}is ak^{×}-valuedG-invariant non-degenerate
alternatingbiadditive form on thecharacter groupbA=Hom(A,k^{×})

I The setB(G)isfiniteand can be “computed in finite time”

It isnon-emptysince it always contains the trivial pair({1},b≡1)

I There is a further restriction on the groupsAsuch that(A,b)∈ B(G):

Such a group is what is called agroup of central type

^{0}×A^{0}for some subgroupA^{0}
I **Examples.**B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(F^{q})

## First ingredient: the set B(G)

I For anyfinite groupG

letB(G)be the set ofpairs(A,b)where (i)Ais anabelian normal subgroupofGand

^{×}is ak^{×}-valuedG-invariant non-degenerate
alternatingbiadditive form on thecharacter groupbA=Hom(A,k^{×})

I The setB(G)isfiniteand can be “computed in finite time”

It isnon-emptysince it always contains the trivial pair({1},b≡1)

I There is a further restriction on the groupsAsuch that(A,b)∈ B(G):

Such a group is what is called agroup of central type

^{0}×A^{0}for some subgroupA^{0}
I **Examples.**B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(F^{q})

## First ingredient: the set B(G)

I For anyfinite groupG

letB(G)be the set ofpairs(A,b)where (i)Ais anabelian normal subgroupofGand

^{×}is ak^{×}-valuedG-invariant non-degenerate
alternatingbiadditive form on thecharacter groupbA=Hom(A,k^{×})

I The setB(G)isfiniteand can be “computed in finite time”

It isnon-emptysince it always contains the trivial pair({1},b≡1)

I There is a further restriction on the groupsAsuch that(A,b)∈ B(G):

Such a group is what is called agroup of central type

^{0}×A^{0}for some subgroupA^{0}
I **Examples.**B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(F^{q})

## Second ingredient: class-preserving automorphisms

•Let Intk(G)be the group ofautomorphismsofGinduced by the conjugation
ad(a) :g 7→aga^{−1} (g∈G)

by some invertible elementaof the group algebrak[G]:

Intk(G) =Aut(G) \

ad(k[G]^{×})

•**Remarks.**(i) If ad(a)∈Intk(G), then the unitabelongs to thenormalizerN
ofGink[G]^{×}

(ii) By character theory, ifkis of characteristic prime to|G|and is big enough (e.g., algebraically closed), then Intk(G)consists of all automorphisms preserving each conjugacy classofG

•The group Intk(G)contains the group Inn(G)ofinner automorphismsas a normal subgroup and we may consider the quotient group

Intk(G)/Inn(G)

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero
I **Theorem**(with P. Guillot)There is a (set-theoretic) map

Θ :H^{2}`(G)→ B(G)
whosefibersare in bijection withIntk(G)/Inn(G)and
which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

(b) We have H^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I **Theorem**(with P. Guillot)There is a (set-theoretic) map
Θ :H^{2}`(G)→ B(G)

whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

(b) We have H^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I **Theorem**(with P. Guillot)There is a (set-theoretic) map
Θ :H^{2}`(G)→ B(G)

whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

(b) We have H^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I **Theorem**(with P. Guillot)There is a (set-theoretic) map
Θ :H^{2}`(G)→ B(G)

whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I **Theorem**(with P. Guillot)There is a (set-theoretic) map
Θ :H^{2}`(G)→ B(G)

whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I **Theorem**(with P. Guillot)There is a (set-theoretic) map
Θ :H^{2}`(G)→ B(G)

whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I **Theorem**(with P. Guillot)There is a (set-theoretic) map
Θ :H^{2}`(G)→ B(G)

whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I **Theorem**(with P. Guillot)There is a (set-theoretic) map
Θ :H^{2}`(G)→ B(G)

whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I **Consequences**(a) The group H^{2}_{`}(G)isfinite

^{2}`(G) =1 ifB(G)is trivial and Intk(G)/Inn(G) =1
There are many groups with Intk(G)/Inn(G) =1 and havingno normal
abelian square subgroups:

e.g.,Gsimple,G=Sn, andG=SLn(F^{q})
For such groupsall invariant twists are trivial

## Plan

### Drinfeld twists and cohomology

### The main theorem

### On the proof

### Examples

### Rationality issues

### References

## The abelian case

I

I Before explaining the construction of the mapΘ :H^{2}_{`}(G)→ B(G)for an
arbitrary group, we compute the lazy cohomology of abelian groups

I LetAbe anabelian groupandAb=Hom(A,k^{×})its group ofcharacters

The followingisomorphismshold:

H^{2}`(A) ∼= H^{2}`(Ok(A)) (by definition)

∼= H^{2}`(k[bA]) (discrete Fourier transform)

∼= H^{2}(bA,k^{×})

∼= Hom(H2(bA,Z),k^{×}) (universal coefficient theorem)

∼= Hom(Λ^{2}_{Z}(bA),k^{×})

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism
H^{2}_{`}(A)−→ {alternating bicharacters on^{∼}^{=} bA}

## The abelian case

I Before explaining the construction of the mapΘ :H^{2}_{`}(G)→ B(G)for an
arbitrary group, we compute the lazy cohomology of abelian groups

I LetAbe anabelian groupandAb=Hom(A,k^{×})its group ofcharacters

The followingisomorphismshold:

H^{2}`(A) ∼= H^{2}`(Ok(A)) (by definition)

∼= H^{2}`(k[bA]) (discrete Fourier transform)

∼= H^{2}(bA,k^{×})

∼= Hom(H2(bA,Z),k^{×}) (universal coefficient theorem)

∼= Hom(Λ^{2}_{Z}(bA),k^{×})

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism
H^{2}_{`}(A)−→ {alternating bicharacters on^{∼}^{=} bA}

## The abelian case

I Before explaining the construction of the mapΘ :H^{2}_{`}(G)→ B(G)for an
arbitrary group, we compute the lazy cohomology of abelian groups

I LetAbe anabelian groupandAb=Hom(A,k^{×})its group ofcharacters

The followingisomorphismshold:

H^{2}`(A) ∼= H^{2}`(Ok(A)) (by definition)

∼= H^{2}`(k[bA]) (discrete Fourier transform)

∼= H^{2}(bA,k^{×})

∼= Hom(H2(bA,Z),k^{×}) (universal coefficient theorem)

∼= Hom(Λ^{2}_{Z}(bA),k^{×})

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism
H^{2}_{`}(A)−→ {alternating bicharacters on^{∼}^{=} bA}

## The abelian case

^{2}_{`}(G)→ B(G)for an
arbitrary group, we compute the lazy cohomology of abelian groups

I LetAbe anabelian groupandAb=Hom(A,k^{×})its group ofcharacters

The followingisomorphismshold:

H^{2}`(A) ∼= H^{2}`(Ok(A)) (by definition)

∼= H^{2}`(k[bA]) (discrete Fourier transform)

∼= H^{2}(bA,k^{×})

∼= Hom(H2(bA,Z),k^{×}) (universal coefficient theorem)

∼= Hom(Λ^{2}_{Z}(bA),k^{×})

= {alternating bicharacters onbA}

^{2}_{`}(A)−→ {alternating bicharacters on^{∼}^{=} bA}

## The abelian case

^{2}_{`}(G)→ B(G)for an
arbitrary group, we compute the lazy cohomology of abelian groups

I LetAbe anabelian groupandAb=Hom(A,k^{×})its group ofcharacters

The followingisomorphismshold:

H^{2}`(A) ∼= H^{2}`(Ok(A)) (by definition)

∼= H^{2}`(k[bA]) (discrete Fourier transform)

∼= H^{2}(bA,k^{×})

∼= Hom(H2(bA,Z),k^{×}) (universal coefficient theorem)

∼= Hom(Λ^{2}_{Z}(bA),k^{×})

= {alternating bicharacters onbA}

^{2}_{`}(A)−→ {alternating bicharacters on^{∼}^{=} bA}

## Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H^{2}_{`}(A)we
attach analternating bicharacterbF :bA×bA→k^{×}as follows:

I Each characterρ∈Abdefines anidempotentof the algebrak[A]by

eρ= 1

|A|

X

g∈A

ρ(g)g

The set{eρ}_{ρ∈b}_{A}is a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H^{2}_{`}(A)
ConsiderRF =F21F^{−1}and express it in the basis{eρ⊗eσ}_{ρ,σ∈b}_{A}

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

ThenbF is abicharacteronA; it isb alternatingbecause(RF)^{−1}= (RF)21

## Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H^{2}_{`}(A)we
attach analternating bicharacterbF :bA×bA→k^{×}as follows:

I Each characterρ∈Abdefines anidempotentof the algebrak[A]by eρ= 1

|A|

X

g∈A

ρ(g)g

The set{eρ}_{ρ∈b}_{A}is a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H^{2}_{`}(A)
ConsiderRF =F21F^{−1}and express it in the basis{eρ⊗eσ}_{ρ,σ∈b}_{A}

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

ThenbF is abicharacteronA; it isb alternatingbecause(RF)^{−1}= (RF)21

## Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H^{2}_{`}(A)we
attach analternating bicharacterbF :bA×bA→k^{×}as follows:

I Each characterρ∈Abdefines anidempotentof the algebrak[A]by eρ= 1

|A|

X

g∈A

ρ(g)g

The set{eρ}_{ρ∈b}_{A}is a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H^{2}_{`}(A)
ConsiderRF =F21F^{−1}and express it in the basis{eρ⊗eσ}_{ρ,σ∈b}_{A}

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

ThenbF is abicharacteronA; it isb alternatingbecause(RF)^{−1}= (RF)21

## Explicit isomorphism

^{2}_{`}(A)we
attach analternating bicharacterbF :bA×bA→k^{×}as follows:

I Each characterρ∈Abdefines anidempotentof the algebrak[A]by eρ= 1

|A|

X

g∈A

ρ(g)g

The set{eρ}_{ρ∈b}_{A}is a basis ofk[A]

^{2}_{`}(A)
ConsiderRF =F21F^{−1}and express it in the basis{eρ⊗eσ}_{ρ,σ∈b}_{A}

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

ThenbF is abicharacteronA; it isb alternatingbecause(RF)^{−1}= (RF)21

## Explicit isomorphism

^{2}_{`}(A)we
attach analternating bicharacterbF :bA×bA→k^{×}as follows:

I Each characterρ∈Abdefines anidempotentof the algebrak[A]by eρ= 1

|A|

X

g∈A

ρ(g)g

The set{eρ}_{ρ∈b}_{A}is a basis ofk[A]

^{2}_{`}(A)
ConsiderRF =F21F^{−1}and express it in the basis{eρ⊗eσ}_{ρ,σ∈b}_{A}

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

ThenbF is abicharacteronA; it isb alternatingbecause(RF)^{−1}= (RF)21

## Explicit isomorphism

^{2}_{`}(A)we
attach analternating bicharacterbF :bA×bA→k^{×}as follows:

I Each characterρ∈Abdefines anidempotentof the algebrak[A]by eρ= 1

|A|

X

g∈A

ρ(g)g

The set{eρ}_{ρ∈b}_{A}is a basis ofk[A]

^{2}_{`}(A)
ConsiderRF =F21F^{−1}and express it in the basis{eρ⊗eσ}_{ρ,σ∈b}_{A}

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

ThenbF is abicharacteronA; it isb alternatingbecause(RF)^{−1}= (RF)21

## The universal R-matrix attached to a twist

I Let us now explain how to construct the map
Θ :H^{2}`(G)→ B(G)
for anarbitrary finite groupG

Start with an element of H^{2}`(G)and

represent it by aninvariant twistF ∈k[G]⊗k[G]

I As in the abelian case, consider

RF =F21F^{−1}∈k[G]⊗k[G]

It is auniversalR-matrixfor the Hopf algebrak[G], i.e., an invertible element such that

∆(a)RF =RF∆(a) for alla∈k[G]

and

(∆⊗idH)(RF) = (RF)13(RF)23 and (idH⊗∆)(RF) = (RF)13(RF)12

## The universal R-matrix attached to a twist

I Let us now explain how to construct the map
Θ :H^{2}`(G)→ B(G)
for anarbitrary finite groupG

Start with an element of H^{2}`(G)and

represent it by aninvariant twistF ∈k[G]⊗k[G]

I As in the abelian case, consider

RF =F21F^{−1}∈k[G]⊗k[G]

It is auniversalR-matrixfor the Hopf algebrak[G], i.e., an invertible element such that

∆(a)RF =RF∆(a) for alla∈k[G]

and

(∆⊗idH)(RF) = (RF)13(RF)23 and (idH⊗∆)(RF) = (RF)13(RF)12

## The universal R-matrix attached to a twist

I Let us now explain how to construct the map
Θ :H^{2}`(G)→ B(G)
for anarbitrary finite groupG

Start with an element of H^{2}`(G)and

represent it by aninvariant twistF ∈k[G]⊗k[G]

I As in the abelian case, consider

RF =F21F^{−1}∈k[G]⊗k[G]

It is auniversalR-matrixfor the Hopf algebrak[G], i.e., an invertible element such that

∆(a)RF =RF∆(a) for alla∈k[G]

and

(∆⊗idH)(RF) = (RF)13(RF)23 and (idH⊗∆)(RF) = (RF)13(RF)12

## The universal R-matrix attached to a twist

I Let us now explain how to construct the map
Θ :H^{2}`(G)→ B(G)
for anarbitrary finite groupG

Start with an element of H^{2}`(G)and

represent it by aninvariant twistF ∈k[G]⊗k[G]

I As in the abelian case, consider

RF =F21F^{−1}∈k[G]⊗k[G]

It is auniversalR-matrixfor the Hopf algebrak[G], i.e., an invertible element such that

∆(a)RF =RF∆(a) for alla∈k[G]

and

(∆⊗idH)(RF) = (RF)13(RF)23 and (idH⊗∆)(RF) = (RF)13(RF)12

## The map Θ

I By work of Radford, there is aminimalHopf subalgebraH^{0}⊂k[G]such
that

RF ∈H^{0}⊗H^{0}

Moreover, sincek[G]is cocommutative,H^{0}isbicommutative

•One deduces thatH^{0}=k[A]whereAis anabelian subgroupofG.

SinceFis invariant, so isRF, andAisnormalinG

I By duality, the universalR-matrixRF ∈k[A]⊗k[A]corresponds to a
G-invariant alternating bicharacterbF :Ab×bA→k^{×}

•The bicharacterbF isnon-degenerateby the minimality ofA; one sets Θ(F) = (A,bF)∈ B(G)

•It is easy to check thatΘ(F)depends only on the class ofFin H^{2}_{`}(G)

## The map Θ

I By work of Radford, there is aminimalHopf subalgebraH^{0}⊂k[G]such
that

RF ∈H^{0}⊗H^{0}

Moreover, sincek[G]is cocommutative,H^{0}isbicommutative

•One deduces thatH^{0}=k[A]whereAis anabelian subgroupofG.

SinceFis invariant, so isRF, andAisnormalinG

I By duality, the universalR-matrixRF ∈k[A]⊗k[A]corresponds to a
G-invariant alternating bicharacterbF :Ab×bA→k^{×}

•The bicharacterbF isnon-degenerateby the minimality ofA; one sets Θ(F) = (A,bF)∈ B(G)

•It is easy to check thatΘ(F)depends only on the class ofFin H^{2}_{`}(G)

## The map Θ

I By work of Radford, there is aminimalHopf subalgebraH^{0}⊂k[G]such
that

RF ∈H^{0}⊗H^{0}

Moreover, sincek[G]is cocommutative,H^{0}isbicommutative

•One deduces thatH^{0}=k[A]whereAis anabelian subgroupofG.

SinceFis invariant, so isRF, andAisnormalinG

I By duality, the universalR-matrixRF ∈k[A]⊗k[A]corresponds to a
G-invariant alternating bicharacterbF :Ab×bA→k^{×}

•The bicharacterbF isnon-degenerateby the minimality ofA; one sets Θ(F) = (A,bF)∈ B(G)

•It is easy to check thatΘ(F)depends only on the class ofFin H^{2}_{`}(G)

## The map Θ

I By work of Radford, there is aminimalHopf subalgebraH^{0}⊂k[G]such
that

RF ∈H^{0}⊗H^{0}

Moreover, sincek[G]is cocommutative,H^{0}isbicommutative

•One deduces thatH^{0}=k[A]whereAis anabelian subgroupofG.

SinceFis invariant, so isRF, andAisnormalinG

^{×}

•The bicharacterbF isnon-degenerateby the minimality ofA; one sets Θ(F) = (A,bF)∈ B(G)

•It is easy to check thatΘ(F)depends only on the class ofFin H^{2}_{`}(G)

## The map Θ

I By work of Radford, there is aminimalHopf subalgebraH^{0}⊂k[G]such
that

RF ∈H^{0}⊗H^{0}

Moreover, sincek[G]is cocommutative,H^{0}isbicommutative

•One deduces thatH^{0}=k[A]whereAis anabelian subgroupofG.

SinceFis invariant, so isRF, andAisnormalinG

^{×}

•The bicharacterbF isnon-degenerateby the minimality ofA; one sets Θ(F) = (A,bF)∈ B(G)

•It is easy to check thatΘ(F)depends only on the class ofFin H^{2}_{`}(G)

## The map Θ

I By work of Radford, there is aminimalHopf subalgebraH^{0}⊂k[G]such
that

RF ∈H^{0}⊗H^{0}

Moreover, sincek[G]is cocommutative,H^{0}isbicommutative

•One deduces thatH^{0}=k[A]whereAis anabelian subgroupofG.

SinceFis invariant, so isRF, andAisnormalinG

^{×}

•The bicharacterbF isnon-degenerateby the minimality ofA; one sets Θ(F) = (A,bF)∈ B(G)

•It is easy to check thatΘ(F)depends only on the class ofFin H^{2}_{`}(G)

## The map Θ

I By work of Radford, there is aminimalHopf subalgebraH^{0}⊂k[G]such
that

RF ∈H^{0}⊗H^{0}

Moreover, sincek[G]is cocommutative,H^{0}isbicommutative

•One deduces thatH^{0}=k[A]whereAis anabelian subgroupofG.

SinceFis invariant, so isRF, andAisnormalinG

^{×}

•The bicharacterbF isnon-degenerateby the minimality ofA; one sets Θ(F) = (A,bF)∈ B(G)

•It is easy to check thatΘ(F)depends only on the class ofFin H^{2}_{`}(G)

## On the proof of the main theorem

I **Fibers of**Θ:To determine them, we use the following fact:

(Etingof and Gelaki, 2000)IfF is a twist such thatRF =1⊗1, equivalentlyFissymmetric:F=F21, then

F= (a⊗a) ∆(a^{−1})
for some invertible elementa∈k[G]

**Observation.**The elementF = (a⊗a) ∆(a^{−1})is aninvarianttwist if
and only ifabelongs to thenormalizerN, which is equivalent to ad(a)
belonging to Intk(G)

I Etingof and Gelaki’s result follows from classicalTannakian theory:

(Deligne and Milne, 1982)Any exact and fully faithful symmetric tensor functor from the category ofk[G]-modules to the category ofk-vector spaces isisomorphic to the forgetful functor

A symmetric twist gives rise to a symmetric tensor functor to which Etingof and Gelaki apply this result

## On the proof of the main theorem

I **Fibers of**Θ:To determine them, we use the following fact:

(Etingof and Gelaki, 2000)IfF is a twist such thatRF =1⊗1, equivalentlyFissymmetric:F=F21, then

F= (a⊗a) ∆(a^{−1})
for some invertible elementa∈k[G]

**Observation.**The elementF = (a⊗a) ∆(a^{−1})is aninvarianttwist if
and only ifabelongs to thenormalizerN, which is equivalent to ad(a)
belonging to Intk(G)

I Etingof and Gelaki’s result follows from classicalTannakian theory:

(Deligne and Milne, 1982)Any exact and fully faithful symmetric tensor functor from the category ofk[G]-modules to the category ofk-vector spaces isisomorphic to the forgetful functor

A symmetric twist gives rise to a symmetric tensor functor to which Etingof and Gelaki apply this result

## On the proof of the main theorem

I **Fibers of**Θ:To determine them, we use the following fact:

(Etingof and Gelaki, 2000)IfF is a twist such thatRF =1⊗1, equivalentlyFissymmetric:F=F21, then

F= (a⊗a) ∆(a^{−1})
for some invertible elementa∈k[G]

**Observation.**The elementF = (a⊗a) ∆(a^{−1})is aninvarianttwist if
and only ifabelongs to thenormalizerN, which is equivalent to ad(a)
belonging to Intk(G)

I Etingof and Gelaki’s result follows from classicalTannakian theory:

(Deligne and Milne, 1982)Any exact and fully faithful symmetric tensor functor from the category ofk[G]-modules to the category ofk-vector spaces isisomorphic to the forgetful functor

A symmetric twist gives rise to a symmetric tensor functor to which Etingof and Gelaki apply this result

## On the proof of the main theorem

I **Fibers of**Θ:To determine them, we use the following fact:

(Etingof and Gelaki, 2000)IfF is a twist such thatRF =1⊗1, equivalentlyFissymmetric:F=F21, then

F= (a⊗a) ∆(a^{−1})
for some invertible elementa∈k[G]

**Observation.**The elementF = (a⊗a) ∆(a^{−1})is aninvarianttwist if
and only ifabelongs to thenormalizerN, which is equivalent to ad(a)
belonging to Intk(G)

I Etingof and Gelaki’s result follows from classicalTannakian theory: