Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/9908115

Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples

Pavel Etingof·Olivier Schiffmann
View original source ↗Read source document ↗Export metadata ↓

Abstract

In early eighties, Belavin and Drinfeld showed that nonskewsymmetric classical r-matrices for simple Lie algebras are classified by combinatorial objects which are now called Belavin-Drinfeld triples. Later the second author of the present paper generalized this result to the case of dynamical r-matrices, and showed that they correspond to generalized Belavin-Drinfeld triples. The dynamical r-matrix corresponding a triple is given by a certain explicit formula, which works for any symmetrizable Kac-Moody algebra. In special cases, this formula gives the Felder and the Belavin r-matrix. In 1994, Felder associated with every dynamical r-matrix a system of differential equations called the KZB equations. In the case of the Felder and Belavin r-matrix, solutions of these equations are known to have a representation theoretical presentation, as conformal blocks (i.e. traces of products of intertwining operators between representations of Lie algebras). In this paper, we propose such a presentation for ANY dynamical r-matrix given by the second author's formula. In particular, the paper sheds light on the Belavin-Drinfeld classical r-matrices of 1984. This paper begins a series of papers. In the next paper, joint with T.Schedler, we will give an explicit quantization of the dynamical r-matrices for simple Lie algebras (in particular, Belavin-Drinfeld r-matrices), which has been an open problem for a number of years. In another paper, we plan to generalize the results of the present paper to quantum groups. This should yield quantum KZB equations which involve quantum R-matrices obtained by quantizing the classical r-matrices that appear in this paper.

Explore related subjects

math.QA

Keep this discovery

BibTeXRIS

Pavel Etingof, Olivier Schiffmann. 1999-08-20. Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples. https://arxiv.org/abs/math/9908115

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Diagrammatic Categorification of Verma Modules I: Braiding

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

math.QA↗
arXiv2026

Some finite dimensional representations of shifted quantum affine algebras of type A

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

math.QA↗