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Center of double extension regular algebras of type (14641)

Andrés Rubiano

TL;DR

This work computes centers and selected central subalgebras for a class of double Ore extensions of type $$(14641)$$, using a case-by-case strategy augmented by SageMath computations. By deriving explicit central generators in several instances and demonstrating central subalgebras in others, the authors obtain new concrete examples that contribute to the study of Zariski cancellation in the noncommutative setting. The results provide computable invariants that aid in understanding representations, automorphisms, and rigidity phenomena for these AS-regular algebras, and they yield cancellativity conclusions for a broad swath of the classified families. The paper organizes foundational DE theory, detailed center calculations, and a cancellation application, complemented by an extensive appendix of defining data for the 26 families.

Abstract

In this paper we compute the center and, in several cases, central subalgebras of double Ore extensions of type (14641) under suitable restrictions on the defining parameters. Part of the analysis is supported by computations in SageMath. As an application, we provide new examples related to the Zariski cancellation problem.

Center of double extension regular algebras of type (14641)

TL;DR

This work computes centers and selected central subalgebras for a class of double Ore extensions of type , using a case-by-case strategy augmented by SageMath computations. By deriving explicit central generators in several instances and demonstrating central subalgebras in others, the authors obtain new concrete examples that contribute to the study of Zariski cancellation in the noncommutative setting. The results provide computable invariants that aid in understanding representations, automorphisms, and rigidity phenomena for these AS-regular algebras, and they yield cancellativity conclusions for a broad swath of the classified families. The paper organizes foundational DE theory, detailed center calculations, and a cancellation application, complemented by an extensive appendix of defining data for the 26 families.

Abstract

In this paper we compute the center and, in several cases, central subalgebras of double Ore extensions of type (14641) under suitable restrictions on the defining parameters. Part of the analysis is supported by computations in SageMath. As an application, we provide new examples related to the Zariski cancellation problem.
Paper Structure (5 sections, 12 theorems, 74 equations, 6 tables)

This paper contains 5 sections, 12 theorems, 74 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Double extension regular algebras of type (14641)
  3. Double extensions of Lg
  4. Center and central subalgebras of double extensions
  5. Application to the Zariski cancellation problem

Key Result

Proposition 2.3

Let $R$ be a $\Bbbk$-algebra, $\sigma:R\to M_{2\times 2}(R)$ an algebra homomorphism, $\delta:R\to M_{2\times 1}(R)$ a $\sigma$-derivation, $P=(p_{12},p_{11})\subseteq \Bbbk$, and $\tau=\{\tau_0,\tau_1,\tau_2\}\subseteq R$. Let $B$ be the associative $\Bbbk$-algebra generated by $R,y_1,y_2$ subject

Theorems & Definitions (29)

  • Definition 2.1: ZhangZhang2008; Carvalhoetal2011
  • Remark 2.2
  • Proposition 2.3: ZhangZhang2008; Carvalhoetal2011
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6: ZhangZhang2009
  • Remark 2.7: ZhangZhang2009
  • Proposition 2.8: ZhangZhang2009
  • Proposition 2.9
  • Definition 2.10: ZhangZhang2009
  • ...and 19 more