Multiparameter quantum general linear supergroup

TL;DR

This paper develops new quantum algebraic structures for the general linear supergroup, introducing uniparametric and multiparametric quantisations with explicit presentations and PBW theorems, bridging formal and polynomial frameworks.

Contribution

It constructs explicit uniparametric and multiparametric quantum supergroup algebras, including duals and polynomial versions, with detailed presentations and PBW-like theorems, extending prior work.

Findings

Explicit constructions of quantum formal series Hopf superalgebras.

Development of multiparametric quantisations via 2-cocycle deformations.

Establishment of PBW-like theorems for both formal and polynomial cases.

Abstract

We introduce uniparametric and multiparametric quantisations of the general linear supergroup, in the form of "quantised function algebras", both in a formal setting - yielding "quantum formal series Hopf superalgebras", a` la Drinfeld - and in a polynomial one - closer to Manin's point of view. In the uniparametric setting, we start from quantised universal enveloping superalgebras over gl(n) - endowed with a super-structure - as in [Ya1] and [Zha]: through a direct approach, we construct their linear dual, thus finding the quantum formal series Hopf superalgebras mentioned above, which are described in detail via an explicit presentation. Starting from the latter, then, we perform a deformation by a well-chosen 2-cocycle, thus getting a multiparametric quantisation, described again by an explicit presentation: this is, in turn, the dual to the multiparametric quantised universal enveloping algebra over gl(n) from [GGP]. We also provide some "polynomial versions" of these quantisations, both for the uniparametric and the multiparametric case. In particular, we compare the latter to Manin's quantum function algebras from [Ma]. Finally, both for the uniparametric and the multiparametric setting, we provide suitable PBW-like theorems, in "formal" and in "polynomial" versions alike.