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Groups of extended affine Lie type

Saeid Azam, Amir Farahmand Parsa

Abstract

We construct certain Steinberg groups associated to extended affine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended affine Lie algebra. Afterwards, we show that the extended affine Weyl group of the ground Lie algebra can be recovered as a quotient group of two subgroups of the group associated to the underlying algebra similar to Kac-Moody groups.

Groups of extended affine Lie type

Abstract

We construct certain Steinberg groups associated to extended affine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended affine Lie algebra. Afterwards, we show that the extended affine Weyl group of the ground Lie algebra can be recovered as a quotient group of two subgroups of the group associated to the underlying algebra similar to Kac-Moody groups.

Paper Structure

This paper contains 6 sections, 18 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. General setup
  3. Nilpotent pairs
  4. Groups of Extended Affine Lie Type
  5. Groups Associated to EARSs
  6. Groups Associated to EALAs

Key Result

Theorem 2.3

Az00 The Weyl group ${\mathcal{W}}$ is isomorphic to the group $\hat{{\mathcal{W}}}$ defined by generators $\hat{r}_\alpha,$$\alpha\in R^\times$ and relations: where $\hat{c}_{(\alpha_p,\eta_p)}$ is the element in $\hat{{\mathcal{W}}}$ corresponding to $c_{(\alpha_p,\eta_p)}$, under the assignment $w_\alpha\mapsto\hat{w}_\alpha$.

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • proof
  • ...and 22 more