On First and Second Cohomology Groups for BBW Parabolics for Classical Lie Superalgebras

TL;DR

This paper computes the first and second cohomology groups of certain subalgebras in classical Lie superalgebras using spectral sequences, providing explicit formulas and tables for these cohomologies.

Contribution

It introduces explicit calculations of cohomology groups for BBW parabolics in classical Lie superalgebras, utilizing collapsing spectral sequences.

Findings

Hochschild-Serre spectral sequence collapses for all classical families

Explicit formulas for first and second cohomology groups

Tables with weight decompositions and dimensions

Abstract

Let ${\mathfrak g}$ be a classical simple Lie superalgebra. In this paper, the author studies the cohomology groups for the subalgebra $\mathfrak{n}^{+}$ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition ${\mathfrak g}={\mathfrak n}^{-}\oplus {\mathfrak f} \oplus {\mathfrak n}^{+}$ where $\mathfrak f$ is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. It is shown that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of classical simple Lie superalgebras. This enables the author to explicitly compute the first and second cohomologies for ${\mathfrak n}^{+}$. The paper concludes with tables listing the weight space decompositions and dimension formulas for these cohomology groups.