Complexity of modules over classical Lie superalgebras

TL;DR

This paper extends the computation of module complexity from Type I Lie superalgebras to Type II, providing geometric interpretations and calculating new invariants for various classical Lie superalgebras.

Contribution

It computes the complexity and z-complexity of modules over Type I and Type II classical Lie superalgebras, offering geometric insights and extending previous results.

Findings

Complexity of modules over $rak{osp}(2|2n)$ matches geometric interpretations.

Z-complexity is interpreted geometrically via a detecting subsuperalgebra.

Computed complexity and z-complexity for simple modules over several Type II Lie superalgebras.

Abstract

The complexity of the simple and the Kac modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$ of type $A$ was computed by Boe, Kujawa, and Nakano in 2012. A natural continuation to their work is computing the complexity of the same family of modules over the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2|2n)$ of type $C$. The two Lie superalgebras are both of Type I which will result in similar computations. In fact, our geometric interpretation of the complexity agrees with theirs. We also compute a categorical invariant, z-complexity, introduced in Boe et al., and we interpret this invariant geometrically in terms of a specific detecting subsuperalgebra. In addition, we compute the complexity and the z-complexity of the simple modules over the Type II Lie superalgebras $\mathfrak{osp}(3|2)$, $D(2,1;\alpha)$, $G(3)$, and $F(4)$.