Homogeneous irreducible supermanifolds and graded Lie superalgebras

TL;DR

This paper classifies complex Lie superalgebras with specific depth one gradings, showing they are semisimple with a particular socle structure, and links these to isotropy irreducible homogeneous supermanifolds.

Contribution

It characterizes all complex Lie superalgebras admitting transitive nonlinear irreducible depth one gradings and describes their structure and associated supermanifolds.

Findings

Such Lie superalgebras are semisimple with socle $rak{s} ensor igwedge(C^n)$.

The gradings correspond to isotropy irreducible homogeneous supermanifolds.

Provides a classification of these gradings and their geometric realizations.

Abstract

A depth one grading $\mathfrak{g}= \mathfrak{g}^{-1}\oplus \mathfrak{g}^0 \oplus \mathfrak{g}^1 \oplus \cdots \oplus \mathfrak{g}^{\ell}$ of a finite dimensional Lie superalgebra $\mathfrak{g}$ is called nonlinear irreducible if the isotropy representation $\mathrm{ad}_{\mathfrak{g}^0}|_{\mathfrak{g}^{-1}}$ is irreducible and $\mathfrak{g}^1 \neq (0)$. An example is the full prolongation of an irreducible linear Lie superalgebra $\mathfrak{g}^0 \subset \mathfrak{gl}(\mathfrak{g}^{-1})$ of finite type with non-trivial first prolongation. We prove that a complex Lie superalgebra $\mathfrak{g}$ which admits a depth one transitive nonlinear irreducible grading is a semisimple Lie superalgebra with the socle $\mathfrak{s}\otimes \Lambda(\mathbb{C}^n)$, where $\mathfrak{s}$ is a simple Lie superalgebra, and we describe such gradings. The graded Lie superalgebra $\mathfrak{g}$ defines an isotropy irreducible homogeneous supermanifold $M=G/G_0$ where $G$, $G_0$ are Lie supergroups respectively associated with the Lie superalgebras $\mathfrak{g}$ and $\mathfrak{g}_0 := \bigoplus_{p\geq 0} \mathfrak{g}^p$.