The article was accepted for publication in the Bulletin of the Iranian Mathematical Society but subsequently rejected four months later on the grounds that it was already available on arXiv.org
This article considers the structure and properties of $δ$-Novikov algebras, a generalization of Novikov algebras characterized by a scalar parameter $δ$. Unlike Novikov algebras, they have simple non-commutative finite-dimensional algebras for $δ=-1$; on the other hand, for every $δ\neq1$, not all of them embed into commutative associative algebras with a $δ$-derivation. We introduce $δ$-Novikov--Poisson algebras, extending several theorems from classical Novikov--Poisson algebras. For $δ\neq1$, the commutator algebras of $δ$-Novikov algebras are metabelian Lie algebras, and every metabelian Lie algebra can be embedded into a suitable $δ$-Novikov algebra with respect to the commutator product. We construct $δ$-Poisson and transposed $δ$-Poisson algebras through $δ$-derivations on commutative associative algebras, and show that for $δ\neq1$ not every transposed $(δ+1)$-Poisson algebra comes from a $δ$-Novikov--Poisson algebra. Finally, we prove that the $δ$-Novikov operad is not Koszul for any value of $δ$, extending the known results for $δ=1$ and $δ=0$, and that a minimal ideal $I$ of a $δ$-Novikov algebra is simple or satisfies $I^2=0$.
We study BiHom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras and Hom-Novikov-Poisson algebras, and find that BiHom-Novikov-Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom-Novikov-Poisson algebras give rise to BiHom-Poisson algebras.
In 1997, X. Xu \cite{Xiaoping Xu Poisson} invented a concept of Novikov-Poisson algebras (we call them Gelfand-Dorfman-Novikov-Poisson (GDN-Poisson) algebras). We construct a linear basis of a free GDN-Poisson algebra. We define a notion of a special GDN-Poisson admissible algebra, based on X. Xu's definition and an S.I. Gelfand's observation (see \cite{Gelfand}). It is a differential algebra with two commutative associative products and some extra identities. We prove that any GDN-Poisson algebra is embeddable into its universal enveloping special GDN-Poisson admissible algebra. Also we prove…
Hom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras, are studied. Hom-Novikov-Poisson algebras are shown to be closed under tensor products and several kinds of twistings. Necessary and sufficient conditions are given under which Hom-Novikov-Poisson algebras give rise to Hom-Poisson algebras.
The purpose of this paper is to study Hom-Novikov algebras and Hom-Novikov-Poisson algebras, both of which were defined by Yau. In the paper, we give several constructions leading us to some interesting examples of Hom-Novikov algebras and Hom-Novikov-Poisson algebras. Also, we introduce the notion of quadratic Hom-Novikov algebras and provide some properties.
We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.
We study prime and semiprime Novikov algebras. We prove that prime nonassociative Novikov algebra has zero nucleus and center. It is well known that an ideal of an alternative (semi)prime algebra is (semi)prime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras. Also, we proved that a minimal ideal of Novikov algebra is either trivial, or a simple algebra.
The algebraic and geometric classification of $δ$-Novikov algebras
Mansoura University（曼苏拉大学）；University of Beira Interior（贝拉内陆大学）；Moscow Center for Fundamental and Applied Mathematics（莫斯科基础与应用数学中心）；Saint Petersburg State University（圣彼得堡国立大学）
The aim of this paper is to introduce Hom-Novikov color algebras and give some constructions of Hom-Novikov color algebras from a given one and a (weak) morphism. Other interesting constructions using averaging operators, centroids, derivations and tensor product are given. We also proved that any Hom-Novikov color algebra is Hom-Lie admissible. Moreover, we introduce Hom-quadratic Hom-Novikov color algebras and provide some properties by twisting. It is also shown that the Hom-Lie color algebra associated to a given quadratic Hom-Novikov color algebra is also quadratic.
We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a $2$-step solvable Lie algebra without any Novikov structure. We use extensions and classical $r$-matrices to construct Novikov structures on certain classes of solvable Lie algebras.
We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification of the Novikov algebras of dimension 3 over R and C, as well as the classification of the 4-dimensional Novikov algebras over C whose associated Lie algebra is nilpotent. In particular this includes a list of all 4-dimensional commutative associative algebras over C.
BiHom-Novikov agebra is a generalized Hom-Novikov algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show that two classes of BiHom-Novikov algebras can be constructed from BiHom-commutative algebras together with derivations and BiHom-Novikov algebras with Rota-Baxter operators, respectively. We show that quadratic BiHom-Novikov algebras are associative algebras and the sub-adjacent BiHom-Lie algebras of BiHom-Novikov algebras are 2-step nilpotent. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov algebras.
We prove the Freiheitssatz for Novikov algebras in characteristic zero. It is also proved that the variety of Novikov algebras is generated by a Novikov algebra on the space of polynomials $k[x]$ in a single variable $x$ over a field $k$ with respect to the multiplication $f\circ g=\partial(f)g$. It follows that the base rank of the variety of Novikov algebras equals 1.
We define an analogue of the Fox derivatives for differential polynomial algebras and give a criterion for differential algebraic dependence of a finite system of elements. In particular, we prove that differential algebraic dependence of a finite set of elements of a differential polynomial algebra over a constructive differential field $k$ of characteristic zero is algorithmically recognizable. Using a representation of free Novikov algebras by differential polynomials we also give a criterion of Novikov dependence of a finite system of elements of free Novikov algebras.
The main feature of color Hom-algebras is that the identities defining the structures are twisted by even linear maps. The purpose of this paper is to introduce and give some constructions of admissible Hom-Novikov-Poisson color Hom-algebras and Hom-Gelfand-Dorfman color Hom-algebras. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. Also, the connections between Hom-Novikov-Poisson color Hom-algebras and Hom-Gelfand-Dorfman color Hom-algebras is proved. Furthermore, we show that the class of admissible Hom-Novikov-Poisson color Hom-algebras is c…
We prove that any Novikov algebra over a field of characteristic $\neq 2$ is Lie-solvable if and only if its commutator ideal $[N,N]$ is right nilpotent. We also construct examples of infinite-dimensional Lie-solvable Novikov algebras $N$ with non nilpotent commutator ideal $[N,N]$.
We establish Gröbner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand-Dorfman-Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand-Dorfman-Novikov algebra which is not free.
In this paper, we construct a kind of infinite-dimensional Novikov algebras and give its realization by hyperbolic sine functions and hyperbolic cosine functions.
We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_0$ is solvable, where $G$ is a finite additive abelean group and the characteristic of $K$ does not divide the order of the group $G$. We also show that any Novikov algebra $N$ with a finite solvable group of automorphisms $G$ is solvable if the algebra of invariants $N^G$ is solvable.
We construct two superalgebras associated to a punctured Riemann surface. One of them is a Lie superalgebra of Krichever-Novikov type, the other one is a Jordan superalgebra. The constructed algebras are related in several ways (algebraic, geometric, representation theoretic). In particular, the Lie superalgebra is the algebra of derivations of the Jordan superalgebra.
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.
An example of a non-triangulable locally nilpotent derivation and an example of a wild exponential automorphism of the free Novikov algebra $N\left\langle x,y,z\right\rangle$ in three variables $x,y,z$ over a field of characteristic zero are constructed.
In this paper, we first study properties of the lower central chains for Novikov algebras. Then we show that for every Lie nilpotent Novikov algebra~$\mathcal{N}$, the ideal of~$\mathcal{N}$ generated by the set~$\{ab - ba\mid a, b\in \mathcal{N}\}$ is nilpotent. We secondly provide necessary and sufficient conditions on the graph $E$ and the field $K$ for which the Leavitt path algebra $L_K(E)$ is Lie solvable. Consequently, we obtain a complete description of Lie nilpotent Leavitt path algebras, and show that the Lie solvability of~$L_K(E)$ and the Lie nilpotency of $[L_K(E),L_K(E)]$ are the…
Institute of Mathematics and Mathematical Modeling（数学与数学建模研究所）；Narxoz University（纳尔霍兹大学）；SICM, Southern University of Science and Technology（南方科技大学信息计算研究中心）；SDU University（南哈萨克斯坦大学）
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…
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…
We classify the orbit closures in the variety Nov_3 of complex, 3-dimensional Novikov algebras and obtain the Hasse diagrams for the closure ordering of the orbits. We provide invariants which are easy to compute and which enable us to decide whether or not one Novikov algebra degenerates to another Novikov algebra.
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 show that Novikov operad is not Koszul.
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.
We study a twisted generalization of Novikov algebras, called Hom-Novikov algebras, in which the two defining identities are twisted by a linear map. It is shown that Hom-Novikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorphisms on complex Novikov algebras of dimensions two or three are computed, and their associated Hom-Novikov algebras are described explicitly. Another class of Hom-Novikov algebras is constructed from Hom-commutative algebras together with a derivation, generalizing a construction due to Dorfman and Gel'fand.…
