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Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras

Alexander Perepechko

TL;DR

The paper addresses realizing any affine algebraic monoid $M$ as the endomorphism monoid of a finite-dimensional (not necessarily associative) algebra. It introduces two constructions, $A(V,S)$ and $D(P,U,S,γ)$, to realize endomorphism monoids up to an isolated zero as normalizers $L(V)_S$ or $L(U)_S$ of subspaces, and then reduces the general problem to representing $M$ as such a normalizer. The main contributions are explicit algebras with ${\rm End}(A(V,S))\cong L(V)_S\sqcup z$ and ${\rm End}(D(P,U,S,γ))\cong L(U)_S\sqcup z$, plus a method to realize any affine monoid $M$ as $L(U)_S$ and hence as ${\rm End}(D(P,U,S,γ))\cong M\sqcup z$. This work extends known results from groups to the broader class of affine monoids by providing concrete finite-dimensional algebra realizations of monoid actions.

Abstract

In this note we prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra.

Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras

TL;DR

The paper addresses realizing any affine algebraic monoid as the endomorphism monoid of a finite-dimensional (not necessarily associative) algebra. It introduces two constructions, and , to realize endomorphism monoids up to an isolated zero as normalizers or of subspaces, and then reduces the general problem to representing as such a normalizer. The main contributions are explicit algebras with and , plus a method to realize any affine monoid as and hence as . This work extends known results from groups to the broader class of affine monoids by providing concrete finite-dimensional algebra realizations of monoid actions.

Abstract

In this note we prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra.

Paper Structure

This paper contains 5 sections, 5 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Some special algebras
  3. Algebra $A(V,S)$
  4. Algebra $D(P,U,S,\mathbf{\gamma})$
  5. Affine monoids as the normalizers of linear subspaces

Key Result

Theorem 1.1

For any affine algebraic monoid $M$ there exists a finite-dimensional algebra $A$ such that $\operatorname{End}(A)\cong M\sqcup\{\mathfrak{z}\}$, where $\{\mathfrak{z}\}$ is an (isolated) component of the monoid $\operatorname{End}(A)$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • ...and 1 more