Complexity of simple modules over the Lie superalgebra $\mathfrak{osp}(k|2)$

TL;DR

This paper computes the complexity and $z$-complexity of simple modules over the Lie superalgebra $rak{osp}(k|2)$, providing geometric interpretations through support and associated varieties.

Contribution

It introduces explicit calculations of module complexities for $rak{osp}(k|2)$ and links these to geometric support and associated varieties, advancing understanding of supermodule structure.

Findings

Calculated complexities for simple modules over $rak{osp}(k|2)$

Established geometric interpretations via support and associated varieties

Enhanced understanding of module growth and structure in Lie superalgebras

Abstract

The complexity of a module is the rate of growth of the minimal projective resolution of the module while the $z$-complexity is the rate of growth of the number of indecomposable summands at each step in the resolution. Let $\mathfrak{g}=\mathfrak{osp}(k|2)$ ($k>2$) be the type II orthosymplectic Lie superalgebra of types $B$ or $D$. In this paper, we compute the complexity and the $z$-complexity of the simple finite-dimensional $\mathfrak{g}$-supermodules. We then give geometric interpretations using support and associated varieties for these complexities.