A class of quadratic deformations of Lie superalgebras

TL;DR

This paper introduces quadratic Lie superalgebras, a new class of deformations of Lie superalgebras, exploring their structure, representations, and specific examples, including a generalization of gl_2(n/1).

Contribution

It defines quadratic Lie superalgebras, proves the PBW theorem for them, and develops their representation theory, including Kac modules and atypical modules, with detailed examples.

Findings

Established the PBW basis theorem for quadratic Lie superalgebras.

Constructed Kac modules and analyzed atypical modules.

Provided concrete examples of low-dimensional quadratic Lie superalgebras.

Abstract

We study certain Z_2-graded, finite-dimensional polynomial algebras of degree 2 which are a special class of deformations of Lie superalgebras, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalised Jacobi relations in the context of the Koszul property, and give a proof of the PBW basis theorem. We give several concrete examples of quadratic Lie superalgebras for low dimensional cases, and discuss aspects of their structure constants for the `type I' class. We derive the equivalent of the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate in detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.