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Root monoids and active algebraic groups

Ekaterina Nistiuk, Yulia Zaitseva

Abstract

We describe affine monoids whose group of invertible elements is an active semidirect product of a unipotent group and a torus, in terms of comultiplications on the algebra of regular functions. We introduce the notion of a root monoid, which is constructed from a set of Demazure root pairs on an affine toric variety, and study the properties of such monoids.

Root monoids and active algebraic groups

Abstract

We describe affine monoids whose group of invertible elements is an active semidirect product of a unipotent group and a torus, in terms of comultiplications on the algebra of regular functions. We introduce the notion of a root monoid, which is constructed from a set of Demazure root pairs on an affine toric variety, and study the properties of such monoids.
Paper Structure (21 sections, 24 theorems, 67 equations)

This paper contains 21 sections, 24 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Toric varieties
  4. Derivations and Demazure roots
  5. Semidirect products
  6. Definition of the group $G_{\overline{\chi}}$
  7. Cones and homogeneity of coordinate spaces
  8. Torus orbits
  9. Root monoids
  10. Classification of root monoids
  11. Proof of Theorem \ref{['thm:mon']}
  12. On the existence of root monoids
  13. Active monoids
  14. Examples
  15. Affine space
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X = X_\sigma$ be an affine toric variety, where the cone $\sigma$ contains a $k$-dimensional regular face $\tau \subset \sigma$, and let $\{e_1^{(r)}, e_2^{(r)} \mid r = 1,\ldots,k \}$ be a set of Demazure roots compatible with $\tau$. Then the map $\mathbb{K}[X] \rightarrow \mathbb{K}[X] \otim where $\mathbb{K}[X_\sigma] = \bigoplus_{u \in S_\sigma} \mathbb{K} \chi^u$, defines a comultiplica

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Definition 4.1
  • Theorem 4.2
  • Definition 4.3
  • Remark 4.4
  • Proposition 4.5
  • ...and 41 more