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Locally finite varieties of nonassociative algebras

Yuri Bahturin, Alexander Olshanskii

Abstract

We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such as: nilpotence, solvability, simplicity, freeness, projectivity, and injectivity. We are also interested in the numerical estimates of the ratio of the number of algebras with various classical properties to the total number of all algebras of a fixed dimension $n$. Among these properties are having no proper nontrivial subalgebras or no nontrivial automorphisms, etc.

Locally finite varieties of nonassociative algebras

Abstract

We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such as: nilpotence, solvability, simplicity, freeness, projectivity, and injectivity. We are also interested in the numerical estimates of the ratio of the number of algebras with various classical properties to the total number of all algebras of a fixed dimension . Among these properties are having no proper nontrivial subalgebras or no nontrivial automorphisms, etc.
Paper Structure (40 sections, 71 theorems, 84 equations)

This paper contains 40 sections, 71 theorems, 84 equations.

Table of Contents

  1. Introduction, definitions and elementary properties
  2. Topic of the article.
  3. Absolutely free algebras.
  4. Varieties and free algebras.
  5. Two one-dimensional algebras.
  6. Subalgebras of direct powers of simple algebras.
  7. Nilpotent varieties
  8. Equivalent definitions.
  9. Subideals.
  10. Other criteria of nilpotency.
  11. A residual property of finite relatively free nilpotent algebras.
  12. Finite algebras in locally finite varieties
  13. Birkhoff's construction.
  14. Enveloping algebras and varieties.
  15. Minimal representation of subdirect products.
  16. ...and 25 more sections

Key Result

Lemma 1

Given an algebra $A$ over $\mathbf{F}$, for any $i > 1$, we have $A^i \subset A^{i-1}$, $A^{i-1}A\subset A^i$ and $A^{i-1}A\subset A^i$. Thus, $A^{i-1}$ is an ideal of $A$.

Theorems & Definitions (136)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 126 more