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Unipotent normal subgroups of algebraic groups

Damian Sercombe

TL;DR

This work extends the unipotent-radical theory to arbitrary affine $k$-groups by introducing the restricted unipotent radical $\mathrm{Rad}_u(G)$, the largest unipotent normal subgroup, and proves its existence and base-change compatibility via separable extensions. It studies when $\mathrm{Rad}_u(G)$ is trivial (pseudo-$p$-reductive) and connects this to smooth connected pseudo-reductive theory, comparing with the classical $\mathscr{R}_u(G)$ and $p$-reductivity concepts. The paper also characterizes the largest unipotent subgroup in key cases, derives tameness results for groups with no nontrivial unipotent subgroups, and analyzes the relationship between pseudo-$p$-reductivity of Frobenius kernels and pseudo-reductivity of the ambient group using the $i_G$ map into Weil restrictions, with implications for root systems and multiplicative-type quotients. These results provide a framework for understanding reductive and pseudo-reductive phenomena over imperfect fields and highlight subtle differences from the smooth setting.

Abstract

Let $G$ be an affine algebraic group scheme over a field $k$. We show there exists a unipotent normal subgroup of $G$ which contains all other such subgroups; we call it the restricted unipotent radical $\mathrm{Rad}_u(G)$ of $G$. We investigate some properties of $\mathrm{Rad}_u(G)$, and study those $G$ for which $\mathrm{Rad}_u(G)$ is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine $k$-groups.

Unipotent normal subgroups of algebraic groups

TL;DR

This work extends the unipotent-radical theory to arbitrary affine -groups by introducing the restricted unipotent radical , the largest unipotent normal subgroup, and proves its existence and base-change compatibility via separable extensions. It studies when is trivial (pseudo--reductive) and connects this to smooth connected pseudo-reductive theory, comparing with the classical and -reductivity concepts. The paper also characterizes the largest unipotent subgroup in key cases, derives tameness results for groups with no nontrivial unipotent subgroups, and analyzes the relationship between pseudo--reductivity of Frobenius kernels and pseudo-reductivity of the ambient group using the map into Weil restrictions, with implications for root systems and multiplicative-type quotients. These results provide a framework for understanding reductive and pseudo-reductive phenomena over imperfect fields and highlight subtle differences from the smooth setting.

Abstract

Let be an affine algebraic group scheme over a field . We show there exists a unipotent normal subgroup of which contains all other such subgroups; we call it the restricted unipotent radical of . We investigate some properties of , and study those for which is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine -groups.
Paper Structure (4 sections, 13 theorems, 24 equations)

This paper contains 4 sections, 13 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. The restricted unipotent radical
  3. The largest unipotent subgroup
  4. Pseudo-$p$-reductivity

Key Result

Theorem 1.1

Let $k$ be a field. Let $G$ be an affine algebraic $k$-group. There exists a largest unipotent normal subgroup $\mathop{\mathrm{Rad}}\nolimits_u(G)$ of $G$ (that is, a unipotent normal subgroup of $G$ which contains all other unipotent normal subgroups of $G$). The formation of $\mathop{\mathrm{Rad}

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Lemma 2.1
  • ...and 16 more