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Ideals in hom-associative Weyl algebras

Per Bäck, Johan Richter

TL;DR

This work extends the classical ideal theory of Weyl algebras to hom-associative deformations by constructing higher-order hom-associative Weyl algebras $A_n^k$ via the Yau twist with twisting map $\boldsymbol{\alpha}_k$. It proves that, under a non-trivial deformation (i.e., $k_1\cdots k_n\neq 0$), these algebras are simple and all one-sided ideals are principal, aligning with Stafford-type phenomena in a nonassociative, deformed setting. The paper also analyzes when ideals from the associative $A_n$ transfer to $A_n^k$ and demonstrates principality for a broad class of deformed ideals, thereby connecting the hom-deformed structure to canonical generators. Overall, the results illuminate the ideal structure of hom-associative Weyl algebras and extend deformation-theoretic perspectives on ideal generation beyond the associative realm.

Abstract

We introduce hom-associative versions of the higher order Weyl algebras, generalizing the construction of the first hom-associative Weyl algebras. We then show that the higher order hom-associative Weyl algebras are simple, and that all their one-sided ideals are principal.

Ideals in hom-associative Weyl algebras

TL;DR

This work extends the classical ideal theory of Weyl algebras to hom-associative deformations by constructing higher-order hom-associative Weyl algebras via the Yau twist with twisting map . It proves that, under a non-trivial deformation (i.e., ), these algebras are simple and all one-sided ideals are principal, aligning with Stafford-type phenomena in a nonassociative, deformed setting. The paper also analyzes when ideals from the associative transfer to and demonstrates principality for a broad class of deformed ideals, thereby connecting the hom-deformed structure to canonical generators. Overall, the results illuminate the ideal structure of hom-associative Weyl algebras and extend deformation-theoretic perspectives on ideal generation beyond the associative realm.

Abstract

We introduce hom-associative versions of the higher order Weyl algebras, generalizing the construction of the first hom-associative Weyl algebras. We then show that the higher order hom-associative Weyl algebras are simple, and that all their one-sided ideals are principal.
Paper Structure (7 sections, 9 theorems, 2 equations)

This paper contains 7 sections, 9 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hom-associative algebras
  4. The $n$th Weyl algebra
  5. The first hom-associative Weyl algebras
  6. Monomial orderings
  7. Ideals in higher order hom-associative Weyl algebras

Key Result

Proposition 1

Let $A$ be an associative algebra and let $\alpha$ be an algebra endomorphism on $A$. Define a new product $*$ on $A$ by $a*b\colonequals\alpha(ab)$ for any $a,b\in A$. Then $A$ with product $*$, called the Yau twist of $A$, is a hom-associative algebra with twisting map $\alpha$. If $A$ is unital w

Theorems & Definitions (23)

  • Definition 1: Hom-associative algebra
  • Definition 2: Weak unitality
  • Proposition 1: FG09Yau09
  • Definition 3: The first hom-associative Weyl algebra
  • Definition 4: The $n$th hom-associative Weyl algebra
  • Proposition 2
  • proof
  • Example 1
  • Example 2
  • Lemma 1
  • ...and 13 more