Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

Abstract

Ifq is ap th root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU q (sl 2), whose coproduct yields the truncated tensor product of physical representations ofU q (sl 2), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (“quasi-quantum group algebras”). We describe a space of “functions on the quasi quantum plane,” i.e. of polynomials in noncommuting complex coordinate functionsz a , on which multiplication operatorsZ a and the elements of can act, so thatz a will transform according to some representation τf of can be made into a quasi-associative graded algebra on which elements of act as generalized derivations. In the special case of the truncatedU q (sl 2) algebra we show that the subspaces of monomials inz a ofn th degree vanish forn≥p−1, and that carries the 2J+ 1 dimensional irreducible representation of ifn=2J, J=0,1/2, ..., 1/2(p−2). Assuming that the representation τf of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on. This implies construction of an exterior differentiation, a graded algebra of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under gauge transformations.