Generalization of Superalgebras to Color Superalgebras and Their Representations

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 ${\mathbb{Z}}_2^{\otimes N}$ 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 ${\mathbb{Z}}_2\otimes {\mathbb{Z}}_2$ grading. A vector field representation of the color superalgebra obtaind from the boson-fermion system is also presented.