Cocycle Twists and Extensions of Braided Doubles

TL;DR

This paper explores the use of 2-cocycles in monoidal categories to construct and twist braided doubles and Nichols algebras, leading to new algebraic structures such as the spin Cherednik algebra.

Contribution

It extends cocycle twisting techniques to monoidal categories and braided doubles, introducing methods to generate novel algebras with triangular decomposition.

Findings

Defined 2-cocycles in monoidal categories following Panaite, Staic, and Van Oystaeyen

Connected twists of Nichols algebras to recent research by Andruskiewitsch et al.

Constructed the spin Cherednik algebra via cocycle twisting of the rational Cherednik algebra.

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 2-cocycles can be defined for an abstract monoidal category C, following Panaite, Staic and Van Oystaeyen. 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 this context. Furthermore, 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 may be carried out if careful choices are made. 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.