Graded discrete Heisenberg and Drinfeld doubles (2411.01681v3)

Abstract: We study the Heisenberg double and the Drinfeld double of the Z2-graded Hopf algebras. To present the constructions, we consider in detail the Borel half of Uq(sl(2)) and two super Hopf algebra examples: the Borel half of Uq(osp(1|2)) and the Borel half of Uq(gl(1|1)) for q being a root of unity. 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 full Uq(gl(1|1)) Hopf algebra. In addition, we generalise an isomorphism between the Drinfeld double and the loop algebra from the combinatorial quantisation to the graded setting.