Quantum Heisenberg groups and Sklyanin algebras

TL;DR

This paper introduces novel quantizations of the Heisenberg group and constructs Sklyanin algebras through new multiplications in the algebra of functions, preserving key representations.

Contribution

It presents new quantizations of the Heisenberg group and establishes a connection to Sklyanin algebras via modified multiplications.

Findings

Preservation of Stone--von Neumann representation coefficients

Introduction of new multiplication in theta function algebra

Construction of Sklyanin algebras from these quantizations

Abstract

We define new quantizations of the Heisenberg group by introducing new quantizations in the universal enveloping algebra of its Lie algebra. Matrix coefficients of the Stone--von Neumann representation are preserved by these new multiplications on the algebra of functions on the Heisenberg group. Some of the new quantizations provide also a new multiplication in the algebra of theta functions; we obtain in this way Sklyanin algebras.