Cahen-Wallach对称空间的超化与Heisenberg代数的旋量表示
Superizations of Cahen-Wallach symmetric spaces and spin representations of the Heisenberg algebra
摘要：
设M_0=G_0/H是一个n+1维Cahen-Wallach Lorentz对称空间，其对称分解为g_0=h+m，S（M_0）是由稳定子H的旋量表示r：H->GL_R（S）定义的自旋丛。本文研究M_0的超化，即它到齐次超流形M=G/H的扩张，其超函数层同构于Lambda（S^*（M_0））.这里G是一个李超群，它是李群G_0的超化，并通过Kostant构造将李代数g_0扩张为李超代数g=g_0+g_1=g_0+S。超化g的构造包括两个步骤：将S上的旋量表示r：h->gl_R（S）推广为表示r：g_0->gl_R（S）和构造适当的r（g_0）-等变双线性映射。由于Heisenberg代数Heis是Cahen-Wallach李代数g_0的余维1理想，我们首先描述Heis的旋量表示，然后确定它们对g_0的扩张. gheis和g_0的旋量表示有两大类：零荷表示和非零荷表示。描述强烈地依赖于维数n+1（mod 8）。给出了关于超化g=g_0+g_1的一些一般性结果，并构造了一些例子.
英文摘要：
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.
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