arXiv · q-alg/9708022
We introduce anyonic Lie algebras in terms of structure constants. We provide the simplest examples and formulate some open problems.
S. Majid. 1997-08-21. Anyonic Lie Algebras. https://doi.org/10.1023/a%3A1022877616496
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In the paper "On some unsolved problems in quantum group theory", V.Drinfeld formulated the problem of the existence of a universal quantization for Lie bialgebras. When the paper "Tensor structures arising from affine Lie algebras, III", by Kazhdan and Lusztig, appeared, Drinfeld asked whether its methods could be useful for the problem of universal quantization of Lie bialgebras. In this paper we use these methods to construct the universal quantization, which gives a positive answer to Drinfeld's question. We also show the existence of universal quantization of classical r-matrices, unitary r-matrices, and quasitriangular Lie bialgebras, which answers the corresponding questions of Drinfeld.
We present a conjecture on the irreducibility of the tensor products of fundamental representations of quantized affine algebras. This conjecture implies in particular that the irreducibility of the tensor products of fundamental representations is completely described by the poles of the R-matrices. The conjecture is proved in certain cases.
We study some properties of the Askey-Wilson polynomials (AWP) when q is a primitive N-th root of unity. For general four-parameter AWP, zeros of the N-th polynomial and the orthogonality measure are found explicitly. Special subclasses of the AWP, e.g., the continuous q-Jacobi and big q-Jacobi polynomials, are considered in detail. A set of discrete weight functions positive on a real interval is described. Some new trigonometric identities related to the AWP are obtained. Normalization conditions of some polynomials are expressed in terms of the Gauss sums.