Generalization of Superalgebras to Color Superalgebras and Their Representations

Abstract

For a given Lie superalgebra, two ways of constructing color superalgebras are presented. One of them is based on the color superalgebraic nature of the Clifford algebras. The method is applicable to any Lie superalgebras and results in color superalgebra of Z ⊗N 2 grading. The other is discussed with an example, a superalgebra of boson and fermion operators. By treating the boson operators as "second" fermionic sector we obtain a color superalgebra of Z2 ⊗ Z2 grading. A vector field representation of the color superalgebra obtaind from the boson-fermion system is also presented.Mathematics Subject Classification (2010). Primary 17B75; Secondary 15A66.

${\mathcal N}$ -extension of double-graded supersymmetric and superconformal quantum mechanics

“…While in Proposition 4.2 tensor product of s and Clifford algebra gives a Z 2 2 -graded superalgebra. This is an example of the way obtaining Z 2 2 -graded superalgebras discussed in the early works [3][4][5] and in the recent study [27].…”

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

“…This is a suitable approach because it is also applicable to higher N SCGA and other Lie (super)algebras. For instance, in [14] a color superalgebra is constructed by realizing it in the enveloping algebra of N = 2 super Schrödinger algebras and in [16], a Heisenberg algebra of multi-mode boson operators is used to construct a color superalgebra. Furthermore, one may apply this procedure to construction of infinite dimensional color superalgebras [44].…”

Z₂³-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics

“…with the energy 1 2 (1 − 2β) where C i is a constant. Thus the ground state is four-fold degenerate and belongs to either parity odd or even subspaces of H .…”

“…It is observed in [2,3] that the Z n 2 -graded SQM is constructed by a combination of the standard SQM and Clifford algebras. In fact, it is known that a tensor product of a Clifford algebra and a standard Lie superalgebra realizes a Z n 2 -graded Lie superalgebra [1,25]. Such realization is not unique since for a given Lie superalgebra there exist some distinct ways of tensoring Clifford algebras.…”