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On finite dimensional algebras with only trivial derivations(automorphisms) and simple algebras

U. Bekbaev

TL;DR

The paper investigates finite-dimensional algebras over a field $ mathbb{F}$ with only trivial derivations or automorphisms, and the class of simple algebras. It develops inductive construction methods to produce such algebras from lower dimensions, proving openness and, for algebraically closed fields, Zariski-density in the variety of $n$-dimensional algebras; it also shows density of GL$(n, mathbb{F})$-orbits for the constructed simple/algebra-with-trivial-properties families. A key contribution is the complete two-dimensional classification of algebras with these trivial properties, complemented by explicit higher-dimensional constructions and their applications, including to $k$-local derivations/automorphisms. Collectively, the results reveal that algebras with only trivial derivations and automorphisms are abundant in the moduli space, and the presented inductive methods provide practical pathways to generate and analyze such algebras.

Abstract

This paper deals with $n$-dimensional algebras, over any field, which have only trivial derivation (automorphism) and simple algebras. It is shown that the corresponding sets of algebras are not empty and, in algebraically closed field case, they are dense subsets of the variety of $n$-dimensional algebras with respect to the Zariski topology. Moreover, an inductive construction method is offered to create these kind of algebras as well. In two-dimensional case a complete classifications, up to isomorphism, of such algebras are provided.

On finite dimensional algebras with only trivial derivations(automorphisms) and simple algebras

TL;DR

The paper investigates finite-dimensional algebras over a field with only trivial derivations or automorphisms, and the class of simple algebras. It develops inductive construction methods to produce such algebras from lower dimensions, proving openness and, for algebraically closed fields, Zariski-density in the variety of -dimensional algebras; it also shows density of GL-orbits for the constructed simple/algebra-with-trivial-properties families. A key contribution is the complete two-dimensional classification of algebras with these trivial properties, complemented by explicit higher-dimensional constructions and their applications, including to -local derivations/automorphisms. Collectively, the results reveal that algebras with only trivial derivations and automorphisms are abundant in the moduli space, and the presented inductive methods provide practical pathways to generate and analyze such algebras.

Abstract

This paper deals with -dimensional algebras, over any field, which have only trivial derivation (automorphism) and simple algebras. It is shown that the corresponding sets of algebras are not empty and, in algebraically closed field case, they are dense subsets of the variety of -dimensional algebras with respect to the Zariski topology. Moreover, an inductive construction method is offered to create these kind of algebras as well. In two-dimensional case a complete classifications, up to isomorphism, of such algebras are provided.
Paper Structure (7 sections, 19 theorems, 20 equations)

This paper contains 7 sections, 19 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivations
  4. Automotphisms
  5. Simple algebras
  6. Some applications
  7. Appendix section: Two-dimensional algebras

Key Result

Theorem 3.1

The following statements on $\mathcal{A}_{n, triv.der}$ are true 1. $\mathcal{A}_{n, triv.der}\neq \emptyset$. 2. If $\mathbb{F}$ is algebraically closed then $\mathcal{A}_{n, triv.der}$ is an open subset of $\mathcal{A}$, in Zariski topology, and therefore it is dense in $\mathcal{A}$.

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 17 more