Elliptic Quantum Groups

TL;DR

This paper develops the theory of elliptic quantum groups, connecting their representations, vertex operators, and applications to elliptic weight functions, elliptic cohomology, and q-deformations of W-algebras.

Contribution

It provides a unified framework for elliptic quantum groups associated with affine and toroidal algebras, and links their representations to elliptic cohomology and W-algebras.

Findings

Unified description of level-0 and level ≠ 0 representations.

Construction of vertex operators as intertwining operators.

Connection between elliptic weight functions and elliptic stable envelopes.

Abstract

We expose the elliptic quantum groups in the Drinfeld realization associated with both the affine Lie algebra \g and the toroidal algebra \g_tor. There the level-0 and level \not=0 representations appear in a unified way so that one can define the vertex operators as intertwining operators of them. The vertex operators are key for many applications such as a derivation of the elliptic weight functions, integral solutions of the (elliptic) q-KZ equation and a formulation of algebraic analysis of the elliptic solvable lattice models. Identifying the elliptic weight functions with the elliptic stable envelopes we make a correspondence between the level-0 representation of the elliptic quantum group and the equivariant elliptic cohomology. We also emphasize a characterization of the elliptic quantum groups as $q$-deformations of the W-algebras.