Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group

TL;DR

This paper introduces braided matrix structures for quantum groups, demonstrating their properties and applications to the Sklyanin algebra and quantum Lorentz group, revealing new algebraic insights in braided categories.

Contribution

It constructs braided group versions of quantum groups with a matrix braided-coproduct and shows their application to the Sklyanin algebra and quantum Lorentz group.

Findings

Degenerate Sklyanin algebra is isomorphic to braided matrices BM_q(2).

Quantum double D(sl_2) is a semidirect product of two copies of U_q(sl_2).

Braided structures provide new perspectives on quantum group doubles.

Abstract

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct $\und\Delta L=L\und\tens L$ where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double $D(\usl)$ (also known as the `quantum Lorentz group') is the semidirect product as an algebra of two copies of $\usl$, and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.