Z 2 × Z 2 generalizations of N=1 superconformal Galilei algebras and their representations

Abstract

We introduce two classes of novel color superalgebras of Z 2 × Z 2 grading. This is done by realizing members of each in the universal enveloping algebra of the N = 1 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 Z 2 × Z 2 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 Z 2 × Z 2 graded setting.

Double-graded quantum superplane

“…To our knowledge, the double-graded quantum superplane is the first such example to be defined and studied. We must remark that there has been some renewed interest in Z n 2 -gradings in physics, see for example [1,2,3,9,37,38]. It is not clear if these 'higher gradings' play a fundamental rôle in physics in the same way as Z 2 -gradings do.…”

Superfield approach to higher derivative $$ \mathcal{N} $$ = 1 superconformal mechanics

“…N = 1, N = 2, and N = 4 supersymmetric extensions of the l-conformal Galilei algebra for l > 1 have been constructed in Refs. [70]- [75]. In contrast to N = 2 and N = 4 cases, there is a unique N = 1 superextension.…”

A classification of lowest weight irreducible modules over $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$

“…In order to explore symmetry, we need representation theories of the algebra generating the symmetry. However, the present knowledge of representations of colour Lie algebras is far from completion since only the limited number of works has been done so far [29][30][31][32][33][34][35][36]. Motivated by this situation, in the present work, we study the lowest weight Verma modules over a Z 2 2 -graded extension of osp(1|2) (Z 2 2 -osp(1|2) in short) and present a classification of its irreducible lowest weight modules.…”