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

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## Abstract

If

*q*is a*p*^{th}root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations of*U*_{ q }(*sl*_{2}), whose coproduct yields the truncated tensor product of physical representations of*U*_{ q }(*sl*_{2}), and whose*R*-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 functions*z*_{ a }, on which multiplication operators*Z*_{ a }and the elements of can act, so that*z*_{ a }will transform according to some representation τ^{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 truncated*U*_{ q }(*sl*_{2}) algebra we show that the subspaces of monomials in*z*_{ a }of*n*^{th}degree vanish for*n≥p−1*, and that carries the 2*J*+ 1 dimensional irreducible representation of if*n=2J, J*=0,1/2, ..., 1/2(*p*−2). Assuming that the representation τ^{f}of the quasi-quantum group algebra gives rise to an*R*-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.### Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Differential Geometry## Preview

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### References

- 1.Manin, Yu.I.: Quantum groups and Non-commutative geometry. Proc. Int. Congr. Math., Berkeley
**1**, 798 (1986)Google Scholar - 2.Wess, J., Zumino, B.: Covariant differential calculus on the quantum hyperplane. Preprint CERN TH 5697/90, April 1990Google Scholar
- 3.Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys.
**122**, 125 (1989)Google Scholar - 4.Drinfel'd, V.G.: Quantum groups. Proc. ICM 798 (1987)Google Scholar
- 5.Drinfel'd, V.G.: Quasi-Hopf algebras and Knizhnik Zamolodchikov equations. In: Problems of modern quantum field theory, Proceedings Alushta 1989, Research reports in Physics. Berlin, Heidelberg, New York: Springer 1989Google Scholar
- 6.Mack, G., Schomerus, V.: In preparationGoogle Scholar
- 7.Mack, G., Schomerus, V.: Quasi-Hopf Quantum Symmetry in Quantum Theory. Nucl. Phys. B
**370**, 185 (1992)Google Scholar - 8.Mack, G., Schomerus, V.: Quasi-quantum group symmetry and local braid relations in the conformal Ising model. Phys. Lett. B
**267**, 207 (1991)Google Scholar - 9.Sudbury, A.: Non-commuting coordinates and differential operators. In: Quantum groups, Curtright, T., et al. (eds.) Singapore: World Scientific 1991Google Scholar
- 10.Reshetikhin, N.Yu., Takhtajan, L.A., Faddeev, L.D.: Algebra Analysis
**1**, 178 (1989)Google Scholar - 11.Connes, A.: Noncommutative differential geometry. Publ. Math. I.H.E.S.
**62**(1985)Google Scholar

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© Springer-Verlag 1992