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On Lusztig's canonical bases of simple Lie algebras

Meinolf Geck

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra over~$\mathbb{C}$ with root system~$Φ$. In the simply laced case, Frenkel and Kac found a particularly simple construction of~$\mathfrak{g}$, together with a Chevalley basis and explicitly given structure constants, in terms of a certain multiplicative $2$-cocycle $\varepsilon\colon \mathbb{Z} Φ\times \mathbb{Z}Φ\rightarrow\{\pm 1\}$. We show that Lusztig's canonical basis of~$\mathfrak{g}$ can also be obtained in this way, for a suitable choice of~$\varepsilon$. We also address the problem of explicitly describing the structure constants when $Φ$ is not simply laced.

On Lusztig's canonical bases of simple Lie algebras

Abstract

Let be a simple Lie algebra over~ with root system~. In the simply laced case, Frenkel and Kac found a particularly simple construction of~, together with a Chevalley basis and explicitly given structure constants, in terms of a certain multiplicative -cocycle . We show that Lusztig's canonical basis of~ can also be obtained in this way, for a suitable choice of~. We also address the problem of explicitly describing the structure constants when is not simply laced.
Paper Structure (5 sections, 11 theorems, 92 equations, 3 tables)

This paper contains 5 sections, 11 theorems, 92 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Special Chevalley systems
  3. The construction of Frenkel and Kac
  4. Non simply laced root systems
  5. Computing structure constants for $B_r$ and $C_r$

Key Result

Lemma 2.1

Let ${\mathcal{L}}$ be a Chevalley system, as above. Let $\gamma_1,\gamma_2,\gamma_2\in \Phi$ be such that $\gamma_1+\gamma_2+ \gamma_3=0$. Then $\varepsilon(\gamma_1,\gamma_2)=-\varepsilon(\gamma_2, \gamma_1)$ and

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • proof
  • ...and 31 more