Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Radicals of Lie-solvable Novikov algebras

A. S. Panasenko

TL;DR

The paper addresses radical structure in Lie-solvable Novikov algebras, proving that the Baer radical $\mathcal{B}(A)$ equals the right-nilpotent set $\{x\in A:\exists n\, x^n=0\}$ and that the Andrunakievich radical coincides with the largest left-quasiregular ideal. It additionally shows that the left-quasiregular radical provides the Andrunakievich radical, and that these radicals render $A$ locally solvable with $[A,A]$ right-nilpotent; the results reveal deep connections between solvability, nilpotence, and radical theory in this setting. The final section analyzes how these radical properties behave under the Gelfand-Dorfman construction, identifying both preservation results (nil, r-nil, left-quasiregular) and counterexamples where right-quasiregularity or right-nilpotence fail to be preserved. Overall, the work clarifies radical behavior in Lie-solvable Novikov algebras and highlights the nuanced impact of the GD construction on these structures.

Abstract

We prove that in a Lie-solvable Novikov algebra, the Baer radical coincides with the set of all right-nilpotent elements, and the Andrunakievich radical coincides with the largest left-quasiregular ideal. We investigate the stability of some properties of commutative algebras with derivation after applying the Gelfand-Dorfman construction.

Radicals of Lie-solvable Novikov algebras

TL;DR

The paper addresses radical structure in Lie-solvable Novikov algebras, proving that the Baer radical equals the right-nilpotent set and that the Andrunakievich radical coincides with the largest left-quasiregular ideal. It additionally shows that the left-quasiregular radical provides the Andrunakievich radical, and that these radicals render locally solvable with right-nilpotent; the results reveal deep connections between solvability, nilpotence, and radical theory in this setting. The final section analyzes how these radical properties behave under the Gelfand-Dorfman construction, identifying both preservation results (nil, r-nil, left-quasiregular) and counterexamples where right-quasiregularity or right-nilpotence fail to be preserved. Overall, the work clarifies radical behavior in Lie-solvable Novikov algebras and highlights the nuanced impact of the GD construction on these structures.

Abstract

We prove that in a Lie-solvable Novikov algebra, the Baer radical coincides with the set of all right-nilpotent elements, and the Andrunakievich radical coincides with the largest left-quasiregular ideal. We investigate the stability of some properties of commutative algebras with derivation after applying the Gelfand-Dorfman construction.
Paper Structure (4 sections, 12 theorems, 17 equations)

This paper contains 4 sections, 12 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Nil Novikov algebras
  3. Lie-solvable Novikov algebras
  4. Stability of radicals relative to the Gelfand-Dorfman structure

Key Result

Lemma 2.1

Let $A$ be a Novikov algebra, $x\in A$, $(x^n)^2 = (x^{n+1})^2 = 0$ for some $n>0$. Then $x^{2n+2} = 0$.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 13 more