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Second cohomology space of $\frak {sl}(2)$ acting on the space of $n$-ary differential operators on $\mathbb{R}$

Mabrouk Ben Ammar

Abstract

We consider the spaces $\mathcal{F}_μ$ of polynomial $μ$-densities on the line as $\mathfrak{sl}(2)$-modules and then we compute the cohomological spaces $\mathrm{H}^2_\mathrm{diff}(\mathfrak{sl}(2), \mathcal{D}_{\barλ,μ})$, where $μ\in \mathbb{R}$, $\barλ=(λ_1,\dots,λ_n) \in\mathbb{R}^n$ and $\mathcal{D}_{\barλ,μ}$ is the space of $n$-ary differential operators from $\mathcal{F}_{λ_1}\otimes\cdots\otimes \mathcal{F}_{λ_n}$ to $\mathcal{F}_μ$.

Second cohomology space of $\frak {sl}(2)$ acting on the space of $n$-ary differential operators on $\mathbb{R}$

Abstract

We consider the spaces of polynomial -densities on the line as -modules and then we compute the cohomological spaces , where , and is the space of -ary differential operators from to .
Paper Structure (6 sections, 13 theorems, 123 equations)

This paper contains 6 sections, 13 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Cohomology
  3. The spaces $\mathrm{H}^2_\mathrm{diff}(\mathfrak{sl}(2),\mathcal{D}_{\underline{\lambda},\mu})$
  4. Singular cases
  5. If $t_1\leq m$.
  6. If $t_1>m$

Key Result

Lemma 3.1

Any 2-cocycle $f\in\mathrm{Z}^2_\mathrm{diff}(\mathfrak{sl}(2),\mathcal{D}_{\underline{\lambda},\mu})$ has the following general form where $A_\alpha,\,B_\alpha$ and $C_\alpha$ are, a priori, polynomial functions satisfying the following cocycle condition:

Theorems & Definitions (13)

  • Lemma 3.1
  • Proposition 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 3 more