Invariant Drinfeld twists on group algebras

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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

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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 H2`(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 computeH2`(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

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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 H2`(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 computeH2`(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

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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 H2`(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 computeH2`(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

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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 H2`(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 computeH2`(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

(6)

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 H2`(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 computeH2`(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

(7)

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 H2`(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 computeH2`(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

(8)

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 H2`(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 computeH2`(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

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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 H2`(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 computeH2`(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

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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 H2`(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 computeH2`(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

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Plan

Drinfeld twists and cohomology

The main theorem

On the proof

Examples

Rationality issues

References

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Plan

Drinfeld twists and cohomology

The main theorem

On the proof

Examples

Rationality issues

References

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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. . .

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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. . .

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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}. . .

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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}. . .

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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}. . .

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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}. . .

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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}. . .

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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}. . .

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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

H2`(G) ={invariant twists onk[G]}/{trivial twists}

Our aim.Determinethe group H2`(G)for anyfinitegroupG

Whyconsider H2`(G)?

I Becauseit is a first step infinding all invariant twistsonk[G]and. . . . . . the group H2`(G)is isomorphic to thesecond lazy cohomology group of the Hopf algebraOk(G)ofk-valued functions onG:

H2`(G)∼=H2`(Ok(G))

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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

H2`(G) ={invariant twists onk[G]}/{trivial twists}

Our aim.Determinethe group H2`(G)for anyfinitegroupG

Whyconsider H2`(G)?

I Becauseit is a first step infinding all invariant twistsonk[G]and. . . . . . the group H2`(G)is isomorphic to thesecond lazy cohomology group of the Hopf algebraOk(G)ofk-valued functions onG:

H2`(G)∼=H2`(Ok(G))

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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

H2`(G) ={invariant twists onk[G]}/{trivial twists}

Our aim.Determinethe group H2`(G)for anyfinitegroupG

Whyconsider H2`(G)?

I Becauseit is a first step infinding all invariant twistsonk[G]and. . . . . . the group H2`(G)is isomorphic to thesecond lazy cohomology group of the Hopf algebraOk(G)ofk-valued functions onG:

H2`(G)∼=H2`(Ok(G))

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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

H2`(G) ={invariant twists onk[G]}/{trivial twists}

Our aim.Determinethe group H2`(G)for anyfinitegroupG

Whyconsider H2`(G)?

I Becauseit is a first step infinding all invariant twistsonk[G]and. . . . . . the group H2`(G)is isomorphic to thesecond lazy cohomology group of the Hopf algebraOk(G)ofk-valued functions onG:

H2`(G)∼=H2`(Ok(G))

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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

H2`(G) ={invariant twists onk[G]}/{trivial twists}

Our aim.Determinethe group H2`(G)for anyfinitegroupG

Whyconsider H2`(G)?

I Becauseit is a first step infinding all invariant twistsonk[G]and. . . . . . the group H2`(G)is isomorphic to thesecond lazy cohomology group of the Hopf algebraOk(G)ofk-valued functions onG:

H2`(G)∼=H2`(Ok(G))

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Lazy cohomology

I In the last decade Schauenburg, Bichon, Carnovaleet al. defined cohomology groups H1`(H)and H2`(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

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

I Thefirstlazy cohomology group for ageneralHopf algebra is given by H1`(H) ={µ∈Alg(H,k)|µ∗idH =idH∗µ}

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Lazy cohomology

I In the last decade Schauenburg, Bichon, Carnovaleet al. defined cohomology groups H1`(H)and H2`(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

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

I Thefirstlazy cohomology group for ageneralHopf algebra is given by H1`(H) ={µ∈Alg(H,k)|µ∗idH =idH∗µ}

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Lazy cohomology

I In the last decade Schauenburg, Bichon, Carnovaleet al. defined cohomology groups H1`(H)and H2`(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

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

I Thefirstlazy cohomology group for ageneralHopf algebra is given by H1`(H) ={µ∈Alg(H,k)|µ∗idH =idH∗µ}

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The second lazy cohomology group

I For thesecondlazy cohomology group H2`(H)onlyfew computations (for non-cocommutative Hopf algebras) had been done

Our original motivation.Compute H2`(H)formoreHopf algebras We decided to try to compute H2`(H)for the Hopf algebrasOk(G) I Proposition.For anyfinite groupG,

H2`(Ok(G))∼=H2`(G)

where H2`(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

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The second lazy cohomology group

I For thesecondlazy cohomology group H2`(H)onlyfew computations (for non-cocommutative Hopf algebras) had been done

Our original motivation.Compute H2`(H)formoreHopf algebras We decided to try to compute H2`(H)for the Hopf algebrasOk(G) I Proposition.For anyfinite groupG,

H2`(Ok(G))∼=H2`(G)

where H2`(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

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The second lazy cohomology group

I For thesecondlazy cohomology group H2`(H)onlyfew computations (for non-cocommutative Hopf algebras) had been done

Our original motivation.Compute H2`(H)formoreHopf algebras We decided to try to compute H2`(H)for the Hopf algebrasOk(G) I Proposition.For anyfinite groupG,

H2`(Ok(G))∼=H2`(G)

where H2`(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

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The second lazy cohomology group

I For thesecondlazy cohomology group H2`(H)onlyfew computations (for non-cocommutative Hopf algebras) had been done

Our original motivation.Compute H2`(H)formoreHopf algebras We decided to try to compute H2`(H)for the Hopf algebrasOk(G) I Proposition.For anyfinite groupG,

H2`(Ok(G))∼=H2`(G)

where H2`(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

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The second lazy cohomology group

I For thesecondlazy cohomology group H2`(H)onlyfew computations (for non-cocommutative Hopf algebras) had been done

Our original motivation.Compute H2`(H)formoreHopf algebras We decided to try to compute H2`(H)for the Hopf algebrasOk(G) I Proposition.For anyfinite groupG,

H2`(Ok(G))∼=H2`(G)

where H2`(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

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Plan

Drinfeld twists and cohomology

The main theorem

On the proof

Examples

Rationality issues

References

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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∼=A0×A0for some subgroupA0 I Examples.B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(Fq)

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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∼=A0×A0for some subgroupA0 I Examples.B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(Fq)

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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∼=A0×A0for some subgroupA0 I Examples.B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(Fq)

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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∼=A0×A0for some subgroupA0 I Examples.B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(Fq)

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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∼=A0×A0for some subgroupA0 I Examples.B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(Fq)

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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∼=A0×A0for some subgroupA0 I Examples.B(G)istrivialifGis simple, orG=Sn(symmetric groups),

orG=GLn(Fq)

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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)

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The main theorem

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

Θ :H2`(G)→ B(G) whosefibersare in bijection withIntk(G)/Inn(G)and which issurjectiveif|G|is odd

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

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The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I Theorem(with P. Guillot)There is a (set-theoretic) map Θ :H2`(G)→ B(G)

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

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

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The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I Theorem(with P. Guillot)There is a (set-theoretic) map Θ :H2`(G)→ B(G)

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

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

(45)

The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I Theorem(with P. Guillot)There is a (set-theoretic) map Θ :H2`(G)→ B(G)

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

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

(46)

The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I Theorem(with P. Guillot)There is a (set-theoretic) map Θ :H2`(G)→ B(G)

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

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

(47)

The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I Theorem(with P. Guillot)There is a (set-theoretic) map Θ :H2`(G)→ B(G)

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

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

(48)

The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I Theorem(with P. Guillot)There is a (set-theoretic) map Θ :H2`(G)→ B(G)

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

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

(49)

The main theorem

Assume that the ground fieldkisalgebraically closedofcharacteristic zero I

I Theorem(with P. Guillot)There is a (set-theoretic) map Θ :H2`(G)→ B(G)

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

I Consequences(a) The group H2`(G)isfinite

(b) We have H2`(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(Fq) For such groupsall invariant twists are trivial

(50)

Plan

Drinfeld twists and cohomology

The main theorem

On the proof

Examples

Rationality issues

References

(51)

The abelian case

I

I Before explaining the construction of the mapΘ :H2`(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:

H2`(A) ∼= H2`(Ok(A)) (by definition)

∼= H2`(k[bA]) (discrete Fourier transform)

∼= H2(bA,k×)

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

∼= Hom(Λ2Z(bA),k×)

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism H2`(A)−→ {alternating bicharacters on= bA}

(52)

The abelian case

I Before explaining the construction of the mapΘ :H2`(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:

H2`(A) ∼= H2`(Ok(A)) (by definition)

∼= H2`(k[bA]) (discrete Fourier transform)

∼= H2(bA,k×)

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

∼= Hom(Λ2Z(bA),k×)

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism H2`(A)−→ {alternating bicharacters on= bA}

(53)

The abelian case

I Before explaining the construction of the mapΘ :H2`(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:

H2`(A) ∼= H2`(Ok(A)) (by definition)

∼= H2`(k[bA]) (discrete Fourier transform)

∼= H2(bA,k×)

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

∼= Hom(Λ2Z(bA),k×)

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism H2`(A)−→ {alternating bicharacters on= bA}

(54)

The abelian case

I Before explaining the construction of the mapΘ :H2`(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:

H2`(A) ∼= H2`(Ok(A)) (by definition)

∼= H2`(k[bA]) (discrete Fourier transform)

∼= H2(bA,k×)

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

∼= Hom(Λ2Z(bA),k×)

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism H2`(A)−→ {alternating bicharacters on= bA}

(55)

The abelian case

I Before explaining the construction of the mapΘ :H2`(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:

H2`(A) ∼= H2`(Ok(A)) (by definition)

∼= H2`(k[bA]) (discrete Fourier transform)

∼= H2(bA,k×)

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

∼= Hom(Λ2Z(bA),k×)

= {alternating bicharacters onbA}

I Let us give adirect expressionfor the composite isomorphism H2`(A)−→ {alternating bicharacters on= bA}

(56)

Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H2`(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∈bAis a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H2`(A) ConsiderRF =F21F−1and express it in the basis{eρ⊗eσ}ρ,σ∈bA

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

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

(57)

Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H2`(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∈bAis a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H2`(A) ConsiderRF =F21F−1and express it in the basis{eρ⊗eσ}ρ,σ∈bA

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

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

(58)

Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H2`(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∈bAis a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H2`(A) ConsiderRF =F21F−1and express it in the basis{eρ⊗eσ}ρ,σ∈bA

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

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

(59)

Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H2`(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∈bAis a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H2`(A) ConsiderRF =F21F−1and express it in the basis{eρ⊗eσ}ρ,σ∈bA

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

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

(60)

Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H2`(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∈bAis a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H2`(A) ConsiderRF =F21F−1and express it in the basis{eρ⊗eσ}ρ,σ∈bA

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

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

(61)

Explicit isomorphism

I To anytwistF ∈k[A]⊗k[A]representing an element of H2`(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∈bAis a basis ofk[A]

I LetF∈k[A]⊗k[A]be atwistrepresenting an element of H2`(A) ConsiderRF =F21F−1and express it in the basis{eρ⊗eσ}ρ,σ∈bA

RF =X

ρ∈bA

bF(ρ, σ)eρ⊗eσ∈k[A]⊗k[A]

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

(62)

The universal R-matrix attached to a twist

I Let us now explain how to construct the map Θ :H2`(G)→ B(G) for anarbitrary finite groupG

Start with an element of H2`(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

(63)

The universal R-matrix attached to a twist

I Let us now explain how to construct the map Θ :H2`(G)→ B(G) for anarbitrary finite groupG

Start with an element of H2`(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

(64)

The universal R-matrix attached to a twist

I Let us now explain how to construct the map Θ :H2`(G)→ B(G) for anarbitrary finite groupG

Start with an element of H2`(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

(65)

The universal R-matrix attached to a twist

I Let us now explain how to construct the map Θ :H2`(G)→ B(G) for anarbitrary finite groupG

Start with an element of H2`(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

(66)

The map Θ

I By work of Radford, there is aminimalHopf subalgebraH0⊂k[G]such that

RF ∈H0⊗H0

Moreover, sincek[G]is cocommutative,H0isbicommutative

•One deduces thatH0=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 H2`(G)

(67)

The map Θ

I By work of Radford, there is aminimalHopf subalgebraH0⊂k[G]such that

RF ∈H0⊗H0

Moreover, sincek[G]is cocommutative,H0isbicommutative

•One deduces thatH0=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 H2`(G)

(68)

The map Θ

I By work of Radford, there is aminimalHopf subalgebraH0⊂k[G]such that

RF ∈H0⊗H0

Moreover, sincek[G]is cocommutative,H0isbicommutative

•One deduces thatH0=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 H2`(G)

(69)

The map Θ

I By work of Radford, there is aminimalHopf subalgebraH0⊂k[G]such that

RF ∈H0⊗H0

Moreover, sincek[G]is cocommutative,H0isbicommutative

•One deduces thatH0=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 H2`(G)

(70)

The map Θ

I By work of Radford, there is aminimalHopf subalgebraH0⊂k[G]such that

RF ∈H0⊗H0

Moreover, sincek[G]is cocommutative,H0isbicommutative

•One deduces thatH0=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 H2`(G)

(71)

The map Θ

I By work of Radford, there is aminimalHopf subalgebraH0⊂k[G]such that

RF ∈H0⊗H0

Moreover, sincek[G]is cocommutative,H0isbicommutative

•One deduces thatH0=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 H2`(G)

(72)

The map Θ

I By work of Radford, there is aminimalHopf subalgebraH0⊂k[G]such that

RF ∈H0⊗H0

Moreover, sincek[G]is cocommutative,H0isbicommutative

•One deduces thatH0=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 H2`(G)

(73)

On the proof of the main theorem

I Fibers ofΘ: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

(74)

On the proof of the main theorem

I Fibers ofΘ: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

(75)

On the proof of the main theorem

I Fibers ofΘ: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

(76)

On the proof of the main theorem

I Fibers ofΘ: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

(77)

Plan

Drinfeld twists and cohomology

The main theorem

On the proof

Examples

Rationality issues

References

Figure

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