Novikov algebras in low dimension: identities, images and codimensions

TL;DR

This paper classifies polynomial identities of two-dimensional Novikov algebras over complex numbers, showing they distinguish non-associative algebras and analyzing their codimension growth and multilinear images.

Contribution

It provides minimal generating sets for identities, linear bases for free algebras, and characterizes algebra isomorphism via identities, advancing understanding of low-dimensional Novikov algebras.

Findings

Polynomial identities distinguish non-associative two-dimensional Novikov algebras.

Codimension sequences grow at most linearly for these algebras.

Multilinear images of these algebras are explicitly described as vector spaces.

Abstract

Polynomial identities of two-dimensional Novikov algebras are studied over the complex field $\mathbb{C}$. We determine minimal generating sets for the T-ideals of the polynomial identities and linear bases for the corresponding relatively free algebras. As a consequence, we establish that polynomial identities separate two-dimensional Novikov algebras, which are not associative. Namely, any two-dimensional Novikov algebras, which are not associative, are isomorphic if and only if they satisfy the same polynomial identities. Moreover, we obtain the codimension sequences of all these algebras. In particular, every two-dimensional Novikov algebra has at most linear growth of its codimension sequence. We explicitly describe multilinear images of every two-dimensional Novikov algebra. In particular, we show that these images are vector spaces.