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Locally isotropic Steinberg groups II. Schur multipliers

Egor Voronetsky

TL;DR

This work determines the Schur multipliers of locally isotropic and root‑graded Steinberg groups for root systems of rank at least 3, excluding $H_3$ and $H_4$, and proves that the corresponding Steinberg maps $ ext{St}_G o G$ are central extensions (crossed modules).It develops and exploits a unified framework of unital and non‑unital $ ext Φ$‑rings, colocalization in infinitary pretopoi, and universal central extensions to produce explicit generator–relation descriptions of the multipliers, including the exceptional cases $A_3$, $B_3$, $D_4$, and $F_4$ with concrete presentations.Beyond the simply laced and doubly laced classifications, the paper shows that most Steinberg groups are centrally closed, while carefully analyzing the exceptional instances via root subsystem analysis and Tits indices; finally it establishes the existence of locally isotropic Steinberg groups as genuine groups by assembling local data.Together, these results extend classical Schur multiplier computations for Chevalley groups and root‑graded analogues, providing a broad, structurally unified account applicable to a wide range of reductive groups and their isotropic variants.

Abstract

We compute Schur multipliers of locally isotropic Steinberg groups and of all root graded Steinberg groups with root systems of rank at least $ 3 $ (excluding the types $ \mathsf H_3 $ and $ \mathsf H_4 $). As an application, we show that locally isotropic Steinberg groups are well defined as abstract groups.

Locally isotropic Steinberg groups II. Schur multipliers

TL;DR

This work determines the Schur multipliers of locally isotropic and root‑graded Steinberg groups for root systems of rank at least 3, excluding $H_3$ and $H_4$, and proves that the corresponding Steinberg maps $ ext{St}_G o G$ are central extensions (crossed modules).It develops and exploits a unified framework of unital and non‑unital $ ext Φ$‑rings, colocalization in infinitary pretopoi, and universal central extensions to produce explicit generator–relation descriptions of the multipliers, including the exceptional cases $A_3$, $B_3$, $D_4$, and $F_4$ with concrete presentations.Beyond the simply laced and doubly laced classifications, the paper shows that most Steinberg groups are centrally closed, while carefully analyzing the exceptional instances via root subsystem analysis and Tits indices; finally it establishes the existence of locally isotropic Steinberg groups as genuine groups by assembling local data.Together, these results extend classical Schur multiplier computations for Chevalley groups and root‑graded analogues, providing a broad, structurally unified account applicable to a wide range of reductive groups and their isotropic variants.

Abstract

We compute Schur multipliers of locally isotropic Steinberg groups and of all root graded Steinberg groups with root systems of rank at least (excluding the types and ). As an application, we show that locally isotropic Steinberg groups are well defined as abstract groups.

Paper Structure

This paper contains 17 sections, 34 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Locally isotropic reductive groups and root graded groups
  4. Unital $\Phi$-rings
  5. Colocalization and infinitary pretopoi
  6. Non-unital $\Phi$-rings
  7. Universal central extensions
  8. Some varieties of non-unital $\Phi$-rings
  9. Upper bounds on Schur multipliers
  10. Simply laced cases
  11. The case of $\mathsf B_3$
  12. Other doubly laced cases
  13. Non-triviality of Schur multipliers
  14. Root graded case
  15. Tits indices
  16. ...and 2 more sections

Key Result

Lemma 1

Let $(R, \Delta)$ be a unital $\mathsf B_3$-ring and $x, y \in R$ be two elements. Then the subring $R'$ generated by $x$, $y$, $x^*$, $y^*$, and some nuclear elements is associative. Moreover, if $z \in R$ is another element and $[x, y, z] = 0$, then the subring $R"$ generated by $R'$, $z$, $z^*$ i

Theorems & Definitions (64)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 54 more