Lowest weight representations of super Schrödinger algebras in one dimensional space

Abstract

Lowest weight modules, in particular, Verma modules over the N = 1, 2 super Schrödinger algebras in (1 + 1) dimensional spacetime are investigated. The reducibility of the Verma modules is analyzed via explicitly constructed singular vectors. The classification of the irreducible lowest weight modules is given for both massive and massless representations. A vector field realization of the N = 1, 2 super Schrödinger algebras is also presented.

Lowest weight representations of super Schrödinger algebras in low dimensional spacetime

“…All irreducible modules are infinite dimensional. Finite dimensional irreducible modules appear when the mass is set equal to zero [36]. One may introduce a bilinear form analogous to the Shapovalov form [38] of the semisimple Lie algebra to the Verma modules.…”

“…Here we give only the outline of the calculation. Detailed proof for (1 + 1) dimensional algebras is found in [36] and for (2 + 1) dimension will be published separately [37].…”

“…To obtain the invariant equations we need a vector field representation of the algebra. Vector field representations for s(1/1) and s(1/2) are found in [36]. Another link of the present work to physical problem is a a supersymmetric extension of the group theoretical approach to nonrelativistic holography discussed in [43].…”

“…After giving the Verma modules in section 3 we give all the singular vectors and list of irreducible lowest weight modules without detailed calculation. The detailed proof is found in [36,37]. The employed procedure in the references is the same one as [32,33].…”

Simple Harish-Chandra supermodules over the super Schrödinger algebra

“…It is known that the representation theory of super Schrödinger algebras play fundamental roles in classifying non-relativistic superconformal field theories. The N = 1 super Schrödinger algebra in (1 + 1)-dimensional spacetime is defined to be Lie superalgebra: S = S(1 | 1) = S0 ⊕ S1, where the even part S0 = span C {e, f, h, p, q, z} is the Schrödinger algebra: Aizawa [1,2] investigated the Verma modules of super Schrödinger algebras in low dimensional spacetime. And other researchers in [21] studied the simple weight modules over S. However, it seems that the supermodules (A left S-supermodules is a supervector space V which is a left S-module in the usual sense such that S θ V τ ⊆ V θ+τ for θ, τ ∈ Z/2Z ) for this Lie superalgebra are not studied well.…”