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Gelfand-Fuchs cohomology for affine superspaces A^(m,n)

Slava Pimenov

Abstract

We study the cohomology of Lie superalgebra of vector fields on affine super-spaces $\mathbb{A}^{m,n}$ with trivial coefficients. In this paper we extend the methodology developed in the previous paper (arXiv:2210.16585) to perform the calculation for arbitrary dimensions $(m, n)$ with $m \geq n$. This combined with previously known results for other dimensions $(m, n)$ fully settles the local problem of Gelfand-Fuchs cohomology with trivial coefficients for super-manifolds. As part of the proof we also compute the cohomology of Lie superalgebras $\mathfrak{gl}(m,n)$ with coefficients given by Schur functors of the standard representation.

Gelfand-Fuchs cohomology for affine superspaces A^(m,n)

Abstract

We study the cohomology of Lie superalgebra of vector fields on affine super-spaces with trivial coefficients. In this paper we extend the methodology developed in the previous paper (arXiv:2210.16585) to perform the calculation for arbitrary dimensions with . This combined with previously known results for other dimensions fully settles the local problem of Gelfand-Fuchs cohomology with trivial coefficients for super-manifolds. As part of the proof we also compute the cohomology of Lie superalgebras with coefficients given by Schur functors of the standard representation.
Paper Structure (4 sections, 7 theorems, 54 equations)

This paper contains 4 sections, 7 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Notations and recollections
  3. Cohomology of $\mathfrak{gl}(m,n)$
  4. Cohomology of $\mathcal{V}_{m,n}$

Key Result

Theorem 1.1

Let $\mathfrak{g} = \mathfrak{gl}(m,1)$, $V$ the standard representation of $\mathfrak{g}$ and $\lambda \in \mathcal{H}_{m,1}$, then where generators $e_i$ are of cohomological degree $i$ and $e'_1$ is of degree $1$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 3.1