Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Duality properties for induced and coinduced representations in positive characteristic

Sophie Chemla

Abstract

Let $k$ be a field of positive characteristic $p>2$. We prove a duality property concerning the kernel of coinduced representations of Lie superalgebras. This property was already proved by M. Duflo for Lie algebras in any characteristic under more restrictive finiteness conditions. It was then generalized to Lie superalgebras in characteristic 0 in previous works of the author. In a second part of the article, we study the links between coinduced representations and induced representations in the case of restricted Lie superalgebras.

Duality properties for induced and coinduced representations in positive characteristic

Abstract

Let be a field of positive characteristic . We prove a duality property concerning the kernel of coinduced representations of Lie superalgebras. This property was already proved by M. Duflo for Lie algebras in any characteristic under more restrictive finiteness conditions. It was then generalized to Lie superalgebras in characteristic 0 in previous works of the author. In a second part of the article, we study the links between coinduced representations and induced representations in the case of restricted Lie superalgebras.
Paper Structure (9 sections, 14 theorems, 73 equations)

This paper contains 9 sections, 14 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Generalities on Lie-Rinehart superalgebras
  4. Algebraic structure on the ${\rm Coind}_{\mathfrak h}^{\mathfrak g} (k)$
  5. The symmetric superalgebra
  6. The superalgebra ${\rm Coind}_{\mathfrak h}^{\mathfrak g}(k)$
  7. Restricted Lie superalgebra
  8. Applications
  9. Declarations

Key Result

Proposition 2.1

Assume that $M$ is a free $A$-module with finite dimension $dim M=(n,m)$. Set $M^*=Hom_A(M,A)$. If $(e_1, \dots , e_{n+m})\in M_{\overline{0}}^n \times M_{\overline{1}}^m$ is a basis of the $A$-module $M$, denote by $d$ left multiplication by ${ \sum_{i=1}^{n+m}(-1)^{\mid e_i \mid +1}\Pi e_i \otimes has no cohomology except in degree $n$. The $A$-module $H^{n}(J(M))$ is free of dimension (1,0) or

Theorems & Definitions (26)

  • Proposition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • Definition 3.1
  • Definition 3.3
  • Remark 3.4
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.5
  • ...and 16 more