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Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled
by two parameters and . The aim of the present work is to investigate
the lowest weight representations of CGA with for any integer value of
. First we focus on the reducibility of the Verma modules. We give a
formula for the Shapovalov determinant and it follows that the Verma module is
irreducible if and the lowest weight is nonvanishing. We prove that
the Verma modules contain many singular vectors, i.e., they are reducible when
. Using the singular vectors, hierarchies of partial differential
equations defined on the group manifold are derived. The differential equations
are invariant under the kinematical transformation generated by CGA. Finally we
construct irreducible lowest weight modules obtained from the reducible Verma
modules
generalizations of superconformal Galilei algebras and their representations
We introduce two classes of novel color superalgebras of grading. This is done by realizing members of each in the
universal enveloping algebra of the supersymmetric extension of
the conformal Galilei algebra. This allows us to upgrade any representation of
the super conformal Galilei algebras to a representation of the graded algebra. As an example, boson-fermion Fock space
representation of one class is given. We also provide a vector field
realization of members of the other class by using a generalization of the
Grassmann calculus to graded setting.Comment: 17 pages, no figur
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