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 = gover(0, -) + gover(1, -) = g0 + S, via the Kostant construction. The construction of the superization g consists of two steps: extending the spin representation ρ : h → g lR (S) to a representation ρ : g0 → g lR (S) and constructing appropriate ρ (g0)-equivariant bilinear maps on S. Since the Heisenberg algebra h e i s is a codimension one ideal of the Cahen-Wallach Lie algebra g0, first we describe spin representations of h e i s and then determine their extensions to g0. There are two large classes of spin representations of h e i s 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 = gover(0, -) + gover(1, -) are stated and examples are constructed. © 2009 Elsevier B.V. All rights reserved.