Superizations of Cahen-Wallach symmetric spaces and spin representations of the Heisenberg algebra

TL;DR

This paper explores the superizations of Cahen-Wallach symmetric spaces by extending spin representations of the Heisenberg algebra, constructing associated Lie supergroups, and analyzing their algebraic structures across different dimensions.

Contribution

It provides a detailed construction of superizations of Cahen-Wallach spaces, focusing on extending spin representations of the Heisenberg algebra within a supergeometric framework.

Findings

Classifies spin representations of the Heisenberg algebra.

Describes extensions of these representations to Cahen-Wallach Lie algebras.

Constructs examples of superizations for various dimensions.

Abstract

Let M_0=G_0/H be a (n+1)-dimensional Cahen-Wallach Lorentzian symmetric space associated with a symmetric decomposition g_0=h+m and let S(M_0) be the spin bundle defined by the spin representation r:H->GL_R(S) of the stabilizer H. This article studies the superizations of M_0, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to Lambda(S^*(M_0)). Here G is a Lie supergroup which is the superization of the Lie group G_0 associated with a certain extension of the Lie algebra g_0 to a Lie superalgebra g=g_0+g_1=g_0+S, via the Kostant construction. The construction of the superization g consists of two steps: extending the spin representation r:h->gl_R(S) to a representation r:g_0->gl_R(S) and constructing appropriate r(g_0)-equivariant bilinear maps on S. Since the Heisenberg algebra heis is a codimension one ideal of the Cahen-Wallach Lie algebra g_0, first we describe spin representations of gheis and then determine their extensions to g_0. There are two large classes of spin representations of gheis and g_0: the zero charge and the non-zero charge ones. The description strongly depends on the dimension n+1 (mod 8). Some general results about superizations g=g_0+g_1 are stated and examples are constructed.