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Groups with $\mathsf{BC}_\ell$-commutator relations

Egor Voronetsky

Abstract

Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system $\mathsf{BC}_\ell$ and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group $G$ has root subgroups indexed by roots of $\mathsf{BC}_\ell$ and satisfying natural conditions, then there is a homomorphism $\mathrm{StU}(R, Δ) \to G$ inducing isomorphisms on the root subgroups, where $\mathrm{StU}(R, Δ)$ is the odd unitary Steinberg group constructed by an odd form ring $(R, Δ)$ with a Peirce decomposition. For groups with root subgroups indexed by $\mathsf A_\ell$ (the already known case) the resulting odd form ring is essentially a generalized matrix ring.

Groups with $\mathsf{BC}_\ell$-commutator relations

Abstract

Isotropic odd unitary groups generalize Chevalley groups of classical types over commutative rings and their twisted forms. Such groups have root subgroups parameterized by a root system and may be constructed by so-called odd form rings with Peirce decompositions. We show the converse: if a group has root subgroups indexed by roots of and satisfying natural conditions, then there is a homomorphism inducing isomorphisms on the root subgroups, where is the odd unitary Steinberg group constructed by an odd form ring with a Peirce decomposition. For groups with root subgroups indexed by (the already known case) the resulting odd form ring is essentially a generalized matrix ring.
Paper Structure (7 sections, 25 theorems, 76 equations, 1 figure)

This paper contains 7 sections, 25 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Odd form groups and odd form rings
  3. Groups with $\mathsf{BC}_\ell$-commutator relations
  4. Action of commutative ring
  5. Construction of the involution algebra
  6. Construction of the odd form parameter
  7. Existence of action of $\mathbb Z$

Key Result

Theorem 1

Let $G$ be a group with root subgroups indexed by a root system of type $\mathsf{BC}_\ell$ for $\ell \geq 4$ and satisfying natural conditions. Then $G$ is a factor-group of the unitary Steinberg group $\mathop{\mathrm{StU}}\nolimits(R, \Delta)$ constructed by an odd form ring with a Peirce decompos

Figures (1)

  • Figure 1: Stereographic projection of $\mathsf B_3$

Theorems & Definitions (48)

  • Theorem
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 38 more