1998/01/09 by Pavel Etingof, David Kazhdan, Etingof, Pavel +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.math/9801043
22 pages, amstex. This is the 4-th part of the Quantization series, starting from q-alg/9506005; in the revised version, some errors in formulas in Chapter 5 have been corrected
openalex publication_date 1998/01/09 · arxiv created 1998/08/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is a continuation of "Quantization of Lie bialgebras, III" (q-alg/9610030, revised version). In QLB-III, we introduced the Hopf algebra F(R)_\z associated to a quantum R-matrix R(z) with a spectral parameter, and a set of points \z=(z1,...,zn). This algebra is generated by entries of a matrix power series Ti(u), i=1,...,n,subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GLN[[t]]. In this paper we consider the quotient F0(R)_\z of F(R)_\z by the relations \qdetR(Ti)=1, where \qdetR is the quantum determinant associated to R (for rational, trigonometric, or elliptic R-matrices). This is also a Hopf algebra, which is a quantization of the group SLN[[t]]. This paper was inspired by the pioneering paper of I.Frenkel and Reshetikhin. The main goal of this paper is to study the representation theory of the algebra F0(R)_\z and of its quantum double, and show how the consideration of coinvariants of this double (quantum conformal blocks) naturally leads to the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. Our construction for the rational R-matrix is a quantum analogue of the standard derivation of the Knizhnik-Zamolodchikov equations in the Wess-Zumino-Witten model of conformal field theory, and for the elliptic R-matrix is a quantum analogue of the construction of Kuroki and Takebe. Our result is a generalization of the construction of Enriques and Felder, which appeared while this paper was in preparation. Enriques and Felder gave a derivation of the quantum KZ equations from coinvariants in the case of the rational R-matrix and N=2.