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2012.112: Cocycle twists and extensions of braided doubles

2012.112: Y Bazlov and A Berenstein (2012) Cocycle twists and extensions of braided doubles.

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Abstract

It is well known that central extensions of a group G correspond to 2-cocycles on G. Cocycles can be used to construct extensions of G-graded algebras via a version of the Drinfeld twist introduced by Majid. We show how to define the second cohomology group of an abstract monoidal category C, generalising the Schur multiplier of a finite group and the lazy cohomology of a Hopf algebra, recently studied by Schauenburg, Bichon, Carnovale and others. A braiding on C leads to analogues of Nichols algebras in C, and we explain how the recent work on twists of Nichols algebras by Andruskiewitsch, Fantino, Garcia and Vendramin fits in our context.

In the second part of the paper we propose an approach to twisting the multiplication in braided doubles, which are a class of algebras with triangular decomposition over G. Braided doubles are not G-graded, but may be embedded in a double of a Nichols algebra, where a twist is carried out. This is a source of new algebras with triangular decomposition. As an example, we show how to twist the rational Cherednik algebra of the symmetric group by the cocycle arising from the Schur covering group, obtaining the spin Cherednik algebra introduced by Wang.

Item Type:MIMS Preprint
Additional Information:

60 pages

Uncontrolled Keywords:Cocycles, Schur multiplier, Cherednik algebras, deformations, Drinfeld twists, Hopf algebras
Subjects:MSC 2000 > 16 Associative rings and algebras
MIMS number:2012.112
Deposited By:Dr Yuri Bazlov
Deposited On:24 November 2012

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