Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus |
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Authors: | Gerhard Mack Volker Schomerus |
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Institution: | (1) II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, W-2000 Hamburg 50, FRG |
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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 f 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 fornp–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 f 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. |
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Keywords: | |
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