Novikov algebras in low dimension: identities, images and codimensions
Iritan Ferreira dos Santos, Alexey M. Kuz'min, Artem Lopatin
TL;DR
This work provides a complete description of polynomial identities for all two-dimensional Novikov algebras over $\mathbb{C}$. It gives finite generating sets for the T-ideals and explicit bases for the corresponding relatively free algebras, proving that nonassociative 2D Novikov algebras are uniquely determined by their PI in this dimension. The paper also derives codimension sequences, showing linear growth for all 2D cases, and furnishes explicit descriptions of multilinear images, establishing that these images are vector spaces. Together, these results refine the classification of 2D Novikov algebras, yield isomorphism criteria via identities, and illuminate the interplay between identities, codimensions, and algebraic images in low dimensions.
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.