Homogeneous irreducible supermanifolds and graded Lie superalgebras

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 ${\mathfrak{g}}^1\ne (0)$. 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 ${\mathfrak{g}}_0:={⨁}_{p\ge 0}{\mathfrak{g}}^p$.

28 pages, 8 Tables (v2: acknowledgments updated, final version to be published in IMRN)

References in corpus (2)

• The analogs of Riemann and Penrose tensors on supermanifolds
• Irreducible complex skew-Berger algebras