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Ore meets homothetic extensions

Tomasz Brzeziński, A. T. M. West

TL;DR

This work addresses when a skew derivation $(\alpha_R,\delta_R)$ on an associative $\mathbb{F}$-algebra $R$ can be extended to an extension $S$ of $R$ by $\mathbb{F}$ or to a homothetic extension $R(\sigma, s)$, preserving the structure via $\alpha_S\iota = \iota\alpha_R$ and $\delta_S\iota = \iota\delta_R$, and shows that such extensions split into two classes distinguished by $\varsigma \in \{0,1\}$. It develops a homothetic framework using double homothetisms and homothetic data $(\sigma,s)$, providing necessary and sufficient conditions for extending $\alpha_R$ (two types) and for extending $\delta_R$ (via parameters $w,e$ and $\mu(\varsigma)$ with $\mu(1)=0$), culminating in the notion of a $(\sigma,s)$-compatible skew derivation. The results yield a (nearly) complete classification of extensions, explicit extension formulas (including inner homothetic derivations), and corollaries linking extensions to zero-multiplication and corona-algebra contexts. A key contribution is the comparison between Ore extensions and homothetic extensions: under suitable compatibility, one gets a monomorphism from $R[x;\alpha_R,\delta_R](\tilde{\sigma},s)$ into $R(\sigma,s)[x;\alpha_S,\delta_S]$ and a commutative diagram with exact rows, clarifying how the two construction pathways relate and when a unique embedding of the Ore extension exists (notably for $\varsigma=1$). The work also discusses an open question for the $\varsigma=0$ case and provides concrete examples illustrating the theory.

Abstract

Sufficient and necessary conditions for an extension of a skew-derivation $(δ_R,α_R)$ of an associative $\mathbb{F}$-algebra $R$ to a skew derivation $(δ_S,α_S)$ on an extension $S$ of $R$ by $\mathbb{F}$ or a {\em homothetic extension $S$ of $R$ by $\mathbb{F}$} are derived. It is then shown that this yields the unique extension of the Ore extension $R[x;α_R,δ_R]$ of $R$ by $\mathbb{F}$ that embeds in the Ore extension $S[x;α_S,δ_S]$ of $S$ by the extended skew-derivation.

Ore meets homothetic extensions

TL;DR

This work addresses when a skew derivation on an associative -algebra can be extended to an extension of by or to a homothetic extension , preserving the structure via and , and shows that such extensions split into two classes distinguished by . It develops a homothetic framework using double homothetisms and homothetic data , providing necessary and sufficient conditions for extending (two types) and for extending (via parameters and with ), culminating in the notion of a -compatible skew derivation. The results yield a (nearly) complete classification of extensions, explicit extension formulas (including inner homothetic derivations), and corollaries linking extensions to zero-multiplication and corona-algebra contexts. A key contribution is the comparison between Ore extensions and homothetic extensions: under suitable compatibility, one gets a monomorphism from into and a commutative diagram with exact rows, clarifying how the two construction pathways relate and when a unique embedding of the Ore extension exists (notably for ). The work also discusses an open question for the case and provides concrete examples illustrating the theory.

Abstract

Sufficient and necessary conditions for an extension of a skew-derivation of an associative -algebra to a skew derivation on an extension of by or a {\em homothetic extension of by } are derived. It is then shown that this yields the unique extension of the Ore extension of by that embeds in the Ore extension of by the extended skew-derivation.
Paper Structure (3 sections, 5 theorems, 80 equations)

This paper contains 3 sections, 5 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Extending skew derivations
  3. Ore vs homothetic extensions

Key Result

Proposition 2.4

Let $R$ be an algebra with a homothetic datum $(\sigma,s)$. Then an algebra endomorphism $\alpha_R$ of $R$ extends to an algebra endomorphism on $R(\sigma,s)$ satisfying the first condition in deriv.ext if and only if there exists $w\in R$, such that for all $a\in R$, where $\varsigma$ is a scalar equal to either 0 or 1.

Theorems & Definitions (22)

  • Definition 2.1: Double homothetism
  • Definition 2.2: Homothetic datum
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Corollary 2.7
  • proof
  • ...and 12 more