Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms

TL;DR

This paper investigates the cohomology of Lie superalgebras, especially focusing on superforms, pseudoforms, and integral forms, revealing larger-than-expected cohomology spaces and new classes through spectral sequence techniques.

Contribution

It extends classical cohomology theorems to include pseudoforms and integral forms, and demonstrates the algebraic Poincaré duality for Lie superalgebras, with explicit cohomology representatives.

Findings

Existence of non-empty cohomology spaces among pseudoforms.

Extension of classical theorems to pseudoforms and integral forms.

Proof of algebraic Poincaré duality for f4b2(2|2).

Abstract

We study the cohomology of Lie superalgebras for the full complex of forms: superforms, pseudoforms and integral forms. We use the technique of spectral sequences to abstractly compute the Chevalley-Eilenberg cohomology. We first focus on the superalgebra $\mathfrak{osp}(2|2)$ and show that there exist non-empty cohomology spaces among pseudoforms related to sub-superalgebras. We then extend some classical theorems by Koszul, as to include pseudoforms and integral forms. Further, we conjecture that the algebraic Poincar\'e duality extends to Lie superalgebras, as long as all the complexes of forms are taken into account and we prove that this holds true for $\mathfrak{osp}(2|2)$. We finally construct the cohomology representatives explicitly by using a distributional realisation of pseudoforms and integral forms. On one hand, these results show that the cohomology of Lie superalgebras is actually larger than expected, whereas one restricts to superforms only; on the other hand, we show the emergence of completely new cohomology classes represented by pseudoforms. These classes realise as integral form classes of sub-superstructures.