Invariant Drinfeld twists on group algebras à lire en Document, Kassel

Invariant Drinfeld twists on group algebras

Christian Kassel

InstitutdeRechercheMath´ematiqueAvance´e 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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH2`(H)whenH=Ok(G)

IThe 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

II 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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH2`(H)whenH=Ok(G)

IThe 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

II 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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH`2(H)whenH=Ok(G)

IThe 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

II 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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH`2(H)whenH=Ok(G)

IThe 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

II 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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH`2(H)whenH=Ok(G)

IThe 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

II 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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH2`(H)whenH=Ok(G)

IThe 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

II 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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH2`(H)whenH=Ok(G)

IThe answer involves theabelian normal subgroups of central typeofG as well as the group ofclass-preserving outer automorphismsofG

I

The proof uses tools fromquantum group theory, mainlyR-matrices

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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH2`(H)whenH=Ok(G)

IThe answer involves theabelian normal subgroups of central typeofG as well as the group ofclass-preserving outer automorphismsofG

I

The proof uses tools fromquantum group theory, mainlyR-matrices

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

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

We reformulate the problem in terms ofinvariant Drinfeld twists and obtain amethod to computeH2`(H)whenH=Ok(G)

IThe answer involves theabelian normal subgroups of central typeofG as well as the group ofclass-preserving outer automorphismsofG

I

The proof uses tools fromquantum group theory, mainlyR-matrices

I shall illustrate all this with severalexamples