Heisenberg and Drinfeld doubles of Uq(gl(1|1)) and Uq(osp(1|2)) super-algebras

TL;DR

This paper investigates the structures of Heisenberg and Drinfeld doubles of certain superalgebras, establishing new isomorphisms and extending known results to graded and quantum cases relevant for Chern-Simons theory.

Contribution

It proves isomorphisms between Heisenberg doubles and handle algebras, extending these to graded and quantum settings, filling gaps in the literature.

Findings

Established isomorphism between Heisenberg doubles and handle algebras.

Extended isomorphisms to graded Heisenberg doubles and handle algebras.

Generalized Drinfeld double and loop algebra isomorphism to the graded context.

Abstract

We study the Heisenberg double and the Drinfeld double of the Borel half of $Uq (gl(1|1))$ and of the $Uq (gl(1|1))$ when q is a root of unity. We also study the Borel half of Uq (osp(1|2)) for both cases when qis a root of unity and when it is not. We prove the isomorphism between the Heisenberg doubles and the handle algebras, which is missing in the literature, and extend the isomorphism to the graded Heisenberg doubles and the handle algebras in the context of the Z2-graded generalisation of Alekseev-Schomerus combinatorial quantisation of Chern-Simons theory [1, 2], as well as illustrate it on the example of the Heisenberg double of the $Uq (gl(1|1))$ Hopf algebra for q being a root of unity. In addition, we generalise an isomorphism between the Drinfeld double and the loop algebra from the Alekseev-Schomerus combinatorial quantisation to the graded setting.