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(mod8). Some general results about superizations g=g0¯+g1¯ are stated and examples are constructed.