PBW degenerations of Lie superalgebras and their typical representations

TL;DR

This paper extends PBW degenerations to Lie superalgebras, constructing new monomial bases for various types, parametrized by lattice points, and establishing foundational steps for further development in the field.

Contribution

It introduces PBW degenerations for basic classical Lie superalgebras and constructs new monomial bases with desirable properties, a novel extension of existing theory.

Findings

Constructed monomial bases for all type I, osp(1,2n), and exceptional Lie superalgebras.

Bases are parametrized by lattice points in convex polytopes with the integer decomposition property.

First step towards extending PBW degeneration framework to Lie superalgebras.

Abstract

We introduce the PBW degeneration for basic classical Lie superalgebras and construct for all type I, $\mathfrak{osp}(1,2n)$ and exceptional Lie superalgebras new monomial bases. These bases are parametrized by lattice points in convex lattice polytopes, sharing useful properties such as the integer decomposition property. This paper is the first step towards extending the framework of PBW degenerations to the Lie superalgebra setting.