Elliptic Quantum Groups
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.11193
- arXiv:
- arXiv:2405.11193
- Bibcode:
- 2024arXiv240511193K
- Keywords:
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- Mathematics - Representation Theory;
- High Energy Physics - Theory;
- Mathematics - Algebraic Geometry;
- Mathematics - Quantum Algebra;
- 17B37
- E-Print:
- 32 pages, to appear in the Encyclopedia of Mathematical Physics 2nd Edition