Elliptic Algebras and Equivariant Elliptic Cohomology

TL;DR

This paper explores the deep connections between elliptic algebras, classical r-matrices, and equivariant elliptic cohomology, providing a new construction linking these mathematical structures through Steinberg varieties.

Contribution

It introduces a novel construction of elliptic algebras associated with Belavin's classical elliptic r-matrix using equivariant elliptic cohomology of Steinberg varieties.

Findings

Established a correspondence between elliptic algebras and classical r-matrices.

Constructed elliptic algebras via equivariant elliptic cohomology.

Linked formal groups with cohomology theories and algebraic structures.

Abstract

In this paper we explain the parallelism in the classification of three different kinds of mathematical objects: (i) Classical r-matrices. (ii) Generalized cohomology theories that have Chern classes for complex vector bundles. (iii) 1-dimensional formal groups. The main point of the paper is a construction of the elliptic algebra associated to Belavin's classical elliptic r-matrix in terms of Equivariant elliptic cohomology of the Steinberg varieties associated to some partial flag manifolds.