The gl(1|1) Lie superbialgebras

TL;DR

This paper classifies all gl(1|1) Lie superbialgebras, explores their classical r-matrices and super Poisson structures, and constructs related integrable systems and quantum deformations, advancing understanding of superalgebra structures.

Contribution

It provides a complete classification of gl(1|1) Lie superbialgebras, their classical r-matrices, and associated structures, including new integrable systems and quantum deformations.

Findings

Classified all gl(1|1) Lie superbialgebras.

Determined super Poisson structures on GL(1|1).

Constructed a new integrable system on a superspace.

Abstract

By direct calculations of matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation, and using the automorphism supergroup of the gl(1|1) Lie superalgebra, we determine and classify all gl(1|1) Lie superbialgebras. Then, by calculating their classical r-matrices, the gl(1j1) coboundary Lie superbialgebras and their types (triangular, quasi-triangular or factorizable) are determined, furthermore in this way super Poisson structures on the GL(1|1) Lie supergroup are obtained. Also, we classify Drinfeld superdoubles based on the gl(1|1) as a theorem. Afterwards, as a physical application of the coboundary Lie superbialgebras, we construct a new integrable system on the homogeneous superspace OSp(1|2)/U(1). Finally, we make use of the Lyakhovsky and Mudrov formalism in order to build up the deformed gl(1|1) Lie superalgebra related to all gl(1|1) coboundary Lie superbialgebras. For one case, the quantization at the supergroup level is also provided, including its quantum R-matrix.