On radicals of Novikov algebras

TL;DR

This paper investigates the structure of radicals in Novikov algebras, establishing key properties of ideals and radicals, and clarifying their behavior in finite-dimensional cases over various fields.

Contribution

It provides new insights into the radicals of Novikov algebras, including their coincidence and non-existence results in finite dimensions over specific fields.

Findings

Nonzero ideals in prime Novikov algebras are non-associative.

Baer and Andrunakievich radicals coincide in finite-dimensional Novikov algebras over certain fields.

Right quasiregular radical does not exist in finite-dimensional Novikov algebras.

Abstract

We show that in a prime nonassociative Novikov algebra every nonzero ideal is non-associative. We prove that Baer (and Andrunakievich) radical and left quasiregular radical coincide in finite dimensional Novikov algebras over a field of characteristic 0 or algebraically closed field of odd characteristic. We show non-existence of right quasiregular radical in finite dimensional Novikov algebras.