An algebra with identities $a\circ(b\circ c-c\circ b)=(a\circ b)\circ c-(a\circ c)\circ b$ and $a\circ(b\circ c)=b\circ(a\circ c)$ is called Novikov. We construct free Novikov base in terms of Young diagrams. We show that codimensions exponent for a variety of Novikov algebras exists and is equal 4.
In this paper, we consider Perm algebra with the derivation $d$. The algebra itself is equipped with the new operation $a\succ b = d(a) b$. We construct a linear basis of the free Novikov dialgebra in terms of new operations. Also, we prove that the class of algebras under the new operation form a variety. Finally, we find the defining identities of the variety.
This work is part of the overarching question of whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. The main result of this paper is that the universal enveloping algebra of any Krichever-Novikov algebra is not noetherian, extending a result of Sierra and Walton on the Witt (or classical Krichever-Novikov) algebra. As a subsidiary result, which may be of independent interest, we show that if $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$ of finite codimension, then the noetherianity of $U(\mathfrak{h})$ is equivalent to th…
Distributive lattices of varieties of Novikov algebras
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS（斯特拉斯堡大学与法国国家科学研究中心）；Institute of Mathematics and Mathematical Modeling（数学与数学建模研究所）
Algebras with the polynomial identity (x,y,z)=(x,z,y), where (x,y,z)=x(yz)-(xy)z is the associator, are called right-symmetric. Novikov algebras are right-symmetric algebras satisfying additionally the polynomial identity x(yz)=y(xz). We consider the d-generated free right-symmetric algebra F(R) and the free Novikov algebra F(N) over a filed K of characteristic 0. The general linear group GL(K,d) with its canonical action on the d-dimensional vector space with basis the generators of F(R) and F(N) acts on F(R) and F(N) as a group of linear automorphisms. For a subgroup G of GL(K,d) we study th…
It is shown that the variety of transposed Poisson algebras coincides with the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is commutative. The Gröbner-Shirshov basis for the transposed Poisson operad is calculated up to degree 4. Furthermore, we demonstrate that every transposed Poisson algebra is F-manifold. We verify that the special identities of GD-algebras hold in transposed Poisson algebras. Finally, we propose a conjecture stating that every transposed Poisson algebra is special, i.e., can be embedded into a differential Poisson algebra.
We discuss algorithmic approach to growth of the codimension sequences of varieties of multilinear algebras, or, equivalently, the sequences of the component dimensions of algebraic operads. The (exponentional) generating functions of such sequences are called codimension series of varieties, or generating series of operads. We show that in general there does not exist an algorithm to decide whether the growth exponent of a codimension sequence of a variety defined by given finite sets of operations and identities is equal to a given rational number. In particular, we solve negatively a recent…
A unified approach to the definition of anti-algebras
University of Beira Interior（贝拉内陆大学）；V.I.Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）；National University of Uzbekistan（乌兹别克斯坦国立大学）；Narxoz University（纳尔霍兹大学）
Bicommutative algebras are nonassociative algebras satisfying the polynomial identities of right- and left-commutativity (xy)z=(xz)y and x(yz)=y(xz). We study subvarieties of the variety of all bicommutative algebras over a field of characteristic 0. We prove that if the subvariety satisfies a polynomial identity of degree k, then its codimension sequence is bounded by a polynomial of degree k-1. Since the exponent of the variety of all bicommutative algebras is equal to 2, this gives that it is minimal with respect to the codimension growth. Up to isomorphism there are three one-generated two…
