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Classification of degenerate Verma modules over $E(4,4)$

Nicoletta Cantarini, Fabrizio Caselli, Victor Kac

Abstract

In this paper we classify degenerate Verma modules over the linearly compact Lie superalgebra $E(4,4)$. This completes the description of Verma modules over the exceptional linearly compact Lie superalgebras. As in the other cases all degenerate modules and morphisms between them give rise to infinite bilateral complexes which may be viewed as a generalization of de Rham complexes.

Classification of degenerate Verma modules over $E(4,4)$

Abstract

In this paper we classify degenerate Verma modules over the linearly compact Lie superalgebra . This completes the description of Verma modules over the exceptional linearly compact Lie superalgebras. As in the other cases all degenerate modules and morphisms between them give rise to infinite bilateral complexes which may be viewed as a generalization of de Rham complexes.
Paper Structure (10 sections, 39 theorems, 168 equations, 8 figures, 1 table)

This paper contains 10 sections, 39 theorems, 168 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Lie superalgebra $E(4,4)$
  3. Irreducible $\hat{\mathfrak{p}}(4)$-modules
  4. Singular vectors of $W(4)$ and $S(4)$
  5. $E(4,4)$-Verma modules and their singular vectors
  6. Singular vectors of degree 1
  7. Singular vectors of degree 2
  8. Singular vectors of degree 3
  9. Singular vectors of degree 4 and 5
  10. Exceptional de Rham complexes for E(4,4)

Key Result

Proposition 3.1

If $t\neq 0$ the Kac module $K_t(a,b,c)$ has a unique irreducible quotient unless $a=c=0$ and $b\geq 1$. Moreover, for all $b\geq 2$, $K_t(0,b,0)$ is the direct sum of two irreducible submodules and $K_t(0,1,0)$ is the direct sum of two submodules one of which is irreducible.

Figures (8)

  • Figure 1: Morphisms between Kac modules
  • Figure 2: All non-zero morphisms between Verma modules for $W(4)$.
  • Figure 3: All non-zero morphisms between Verma modules for $S(4)$. External morphisms have degree 1, and the "diagonal" morphism has degree 2.
  • Figure 4: Nonzero morphisms of degree 1 between Verma modules.
  • Figure 5: Elements $y_{i,jk}$ and $z_{ijk}$ and their weights
  • ...and 3 more figures

Theorems & Definitions (84)

  • Proposition 3.1
  • Example 3.2
  • Example 3.3: The standard representation of $\hat{\mathfrak{p}}(4)$
  • Example 3.4
  • Example 3.5
  • Remark 3.6
  • Definition 3.7
  • Proposition 3.8
  • proof
  • Proposition 3.9
  • ...and 74 more