Superizations of Cahen-wallach Symmetric Spaces and Spin Representations of the Heisenberg Algebra

Let M0 = G0/H be a (n+1)-dimensional Cahen-Wallach Lorentzian symmetric space associated with a symmetric decomposition g0 = h + m and let S(M0) be the spin bundle defined by the spin representation ρ : H → GLR(S) of the stabilizer H . This article studies the superizations of M0, i.e. its extensions to a homogeneous supermanifold M = G/H whose sheaf of superfunctions is isomorphic to Λ(S∗(M0)). Here G is a Lie supergroup which is the superization of the Lie group G0 associated with a certain extension of the Lie algebra g0 to a Lie superalgebra g = g0 + g1 = g0 + S, via the Kostant construction. The construction of the superization g consists of two steps: extending the spin representation ρ : h → glR(S) to a representation ρ : g0 → glR(S) and constructing appropriate ρ(g0)equivariant bilinear maps on S. Since the Heisenberg algebra heis is a codimension one ideal of the Cahen-Wallach Lie algebra g0, first we describe spin representations of heis and then determine their extensions to g0. There are two large classes of spin representations of heis and g0: 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 = g0 + g1 are stated and examples are constructed.