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Non commutative torsors

De
79 pages
Non-commutative torsors Christian Kassel Institut de Recherche Mathematique Avancee CNRS - Universite de Strasbourg Strasbourg, France Oslo University 2 September 2011

  • proof using

  • quantum group

  • drinfeld twists

  • main results

  • universite de strasbourg

  • commutative torsors

  • group algebras


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Non-commutative torsors
Christian Kassel
Institut de Recherche Mathe´ matique Avance´ e CNRS - Universite´ de Strasbourg Strasbourg, France
Oslo University 2 September 2011
Introduction
Report onjoint workwithPierre Guillot(Strasbourg) about the classification ofnon-commutative torsorsover afinite groupG
Reference:P. Guillot, C. Kassel, Cohomology of invariant Drinfeld twists on group algebras, Internat. Math. Res. Notices 2010 no. 10, 1894–1939; arXiv:0903.2807
The idea of considering non-commutativeG-torsors comes fromquantum group theoryas well as the proofs of the main results
We obtain anew (quantum) invariantfor finite groups
Introduction
Report onjoint workwithPierre Guillot(Strasbourg) about the classification ofnon-commutative torsorsover afinite groupG
Reference:P. Guillot, C. Kassel, Cohomology of invariant Drinfeld twists on group algebras, Internat. Math. Res. Notices 2010 no. 10, 1894–1939; arXiv:0903.2807
The idea of considering non-commutativeG-torsors comes fromquantum group theoryas well as the proofs of the main results
We obtain anew (quantum) invariantfor finite groups
Plan
I. Torsors and bitorsors
II. Non-commutative torsors
III. Main results
IV. A proof using quantum group
theory
Plan
I. Torsors and bitorsors
II. Non-commutative torsors
III. Main results
IV. A proof using quantum group
theory
ClassicalG-torsors
Recall:LetGbe analgebraic groupdefined over a fieldk. A rightG-torsor is a rightG-varietyT(overk) such that the map
is an isomorphism
T×G−→T×T (t,g)7(t,tg)
This means that for anyt,t0Tthere is a uniquegGsuch thatt0=tg
¯ Ifk=kisalgebraically closed, then any torsor is isomorphic toT=G withGacting by right translations
ClassicalG-torsors
Recall:LetGbe analgebraic groupdefined over a fieldk. A rightG-torsor is a rightG-varietyT(overk) such that the map
is an isomorphism
T×G−→T×T (t,g)7(t,tg)
This means that for anyt,t0Tthere is a uniquegGsuch thatt0=tg
¯ Ifk=kisalgebraically closed, then any torsor is isomorphic toT=G withGacting by right translations