Generalized Grassmann variables for quantum kit (k-level) systems and Barut-Girardello coherent states for su(r+1) algebras

This paper concerns the construction of $su(r+1)$ Barut--Girardello coherent states in term of generalized Grassmann variables. We first introduce a generalized Weyl-Heisenberg algebra ${\cal A}(r)$ ($r \geq 1$) generated by $r$ pairs of creation and annihilation operators. This algebra provides a useful framework to describe qubit and qukit ($k$-level) systems. It includes the usual Weyl-Heisenberg and $su(2)$ algebras. We investigate the corresponding Fock representation space. The generalized Grassmann variables are introduced as variables spanning the Fock--Bargmann space associated with the algebra ${\cal A}(r)$. The Barut--Girardello coherent states for $su(r+1)$ algebras are explicitly derived and their over--completion properties are discussed.