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Graded discrete Heisenberg and Drinfeld doubles

Nezhla Aghaei, M. K. Pawelkiewicz

TL;DR

The paper develops a complete $\Z_2$-graded theory of the Heisenberg and Drinfeld doubles for Hopf algebras, connecting them to graded handle and loop algebras in Alekseev–Schomerus combinatorial quantisation. It provides both basis-free and basis-dependent formulations, introduces the graded canonical element $S$ with a pentagon relation, and proves an isomorphism between graded Heisenberg doubles and graded handle algebras, as well as graded embeddings of Drinfeld doubles into tensor squares of Heisenberg doubles. The authors work through explicit graded constructions for the Borel halves (and the full case) of $U_q(sl(2))$, $U_q(osp(1|2))$, and $U_q(gl(1|1))$ (including root-of-unity cases), deriving detailed (anti-)commutation relations, basis elements, and universal $R$-matrices. They further generalise the Drinfeld double–loop algebra correspondence to the graded setting, thereby extending combinatorial Chern–Simons quantisation to $\Z_2$-graded, potentially non-semisimple gauge groups and foreshadow applications to graded lattice models and non-compact theories.

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.

Graded discrete Heisenberg and Drinfeld doubles

TL;DR

The paper develops a complete -graded theory of the Heisenberg and Drinfeld doubles for Hopf algebras, connecting them to graded handle and loop algebras in Alekseev–Schomerus combinatorial quantisation. It provides both basis-free and basis-dependent formulations, introduces the graded canonical element with a pentagon relation, and proves an isomorphism between graded Heisenberg doubles and graded handle algebras, as well as graded embeddings of Drinfeld doubles into tensor squares of Heisenberg doubles. The authors work through explicit graded constructions for the Borel halves (and the full case) of , , and (including root-of-unity cases), deriving detailed (anti-)commutation relations, basis elements, and universal -matrices. They further generalise the Drinfeld double–loop algebra correspondence to the graded setting, thereby extending combinatorial Chern–Simons quantisation to -graded, potentially non-semisimple gauge groups and foreshadow applications to graded lattice models and non-compact theories.

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.

Paper Structure

This paper contains 13 sections, 21 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Heisenberg doubles
  3. Heisenberg double of the Borel half of $U_q(sl(2))$
  4. Heisenberg double of the Borel half of $U_q(osp(1|2))$
  5. Heisenberg double of the Borel half of $U_q(gl(1|1))$
  6. Heisenberg double of $U_q(gl(1|1))$
  7. Drinfeld doubles
  8. Drinfeld double of the Borel half of $U_q(sl(2))$
  9. Drinfeld double of the Borel half of $U_q(osp(1|2))$
  10. Drinfeld double of the Borel half of $U_q(gl(1|1))$
  11. Outlook
  12. Assorted proofs of Section 2
  13. Assorted proofs of Section 3

Key Result

Proposition 2.1

The canonical element $S$ defined above satisfies the graded pentagon relation where we use a notation for which $S_{12} = S \otimes (1\otimes1)$, $S_{23} = (1\otimes1)\otimes S$ and $S_{13} = \sigma_{12} [(1\otimes1) \otimes S]$, where $\sigma$ is an exchange operator.

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • Proposition 2.3
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 37 more