Dietrich Burde
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.
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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.
Transposed Novikov-Poisson algebras
本文研究了转置Novikov-泊松代数的定义、结构及其与Novikov代数的关系，展示了其在张量积和导子中的应用，并探讨了其在单代数中的性质。
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.
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
本文研究了δ不属于{0,1}的3维δ-诺维科夫代数的代数与几何分类，证明了3维δ-诺维科夫代数中不存在简单代数。
Jacobi algebras and Jacobi Novikov-Poisson algebras
本文研究了Jacobi Novikov-Poisson代数的定义、分类及其与Jacobi代数的关系，提出了Frobenius Jacobi Novikov-Poisson代数的概念，并给出了相关等价刻画和构造方法。
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 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.
Basis of the free noncommutative Novikov algebra
本文研究由带导子的结合代数推广得到的非交换 Novikov 代数，并为其自由代数构造了单项基。
Novikov algebras and their primitive ideals
本文研究 Novikov 代数的本原理想，通过模极大右理想给出其刻画，证明稠密性型定理，并建立素、单与本原 Novikov 代数间的等价关系。
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.
This paper is devoted to give the complete algebraic and geometric classification of $4$-dimensional nilpotent Novikov algebras over $\mathbb C.$
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.
Extended $\mathcal{O}$-operators, Novikov Yang-Baxter equations and post-Novikov algebras
本文提出扩展 $\mathcal{O}$-算子和 post-Novikov 代数，证明其与权重 $\lambda$ 的 $\mathcal{O}$-算子紧密相关，并引入扩展 Novikov Yang-Baxter 方程，建立四者间的关系。
$δ$-Novikov and $δ$-Novikov--Poisson algebras
本文研究 $δ$-Novikov 代数及其 Poisson 推广，刻画交换子结构、嵌入关系与相关构造，并证明对应 operad 对任意 $δ$ 均非 Koszul。
We give a classification of 5- and 6-dimensional complex one-generated nilpotent Novikov algebras
On radicals of Novikov algebras
本文研究Novikov代数的根理论，证明素非结合Novikov代数的非零理想均非结合，并证明有限维情形下Baer根与左拟正则根重合，同时证明右拟正则根不存在。
On the structure of finite Novikov conformal algebras
本文分类了特征零代数闭域上的有限单Novikov共形代数，包括基域上的当前共形代数和一类单参数Virasoro型共形代数，并证明了半单有限Novikov共形代数是单代数的直和，同时描述了其形变与中心扩张。
The algebraic classification of nilpotent Novikov algebras
本文针对复5维幂零Novikov代数，开展了完整的代数分类研究工作。
Noncommutative Novikov algebras
本文研究非交换Novikov代数，提出从带导子的非交换结合代数构造一类更一般的线性代数，推广了经典Novikov代数的构造方法。
Differential envelopes of Novikov conformal algebras
研究Novikov conformal代数的微分包络性质，证明有限生成代数可嵌入可交换代数，但无限生成代数不成立。
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.
Infinite-dimensional pre-Lie bialgebras via affinization of pre-Novikov bialgebras
本文通过 pre-Novikov 代数的仿射化构造 pre-Lie 代数，并将其推广到双代数情形，证明 pre-Novikov 双代数的仿射化给出无穷维完备 pre-Lie 双代数，同时由 pre-Novikov Yang-Baxter 方程的对称解构造 S-方程的对称完备解。
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.
Free weakly Novikov metabelian algebra of infinite rank
本文在特征不等于2的任意域上，构造了无限秩自由弱Novikov亚交换代数的显式基与乘法表。
On the free metabelian Novikov and metabelian Lie-admissible algebras
本文构造了自由亚交换Novikov代数与自由亚交换Lie-容许代数的线性基，并刻画了二者的对称多项式空间。
Solvability and nilpotency of transposed Novikov-Poisson algebras
本文发展了转置Novikov-Poisson代数的幂零性与可解性理论，建立了dialgebra幂零的等价条件，证明了可解性与右幂零性等价，并建立了与底代数幂零性的联系。
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.
Radicals of Lie-solvable Novikov algebras
研究李可解诺维科夫代数，证明其贝尔根与右幂零元集重合、安德拉诺维奇根与最大左拟正则理想重合，还探讨应用格尔范德 - 多尔夫曼构造后带导子交换代数性质的稳定性。
Novikov algebras in low dimension: identities, images and codimensions
本文研究复数域上二维Novikov代数的多项式恒等式，确定其T-理想极小生成集与相对自由代数基，证明多项式恒等式可区分非结合二维Novikov代数，得出余维数序列至多线性增长并描述多重线性像。
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.
Classification of Novikov-Poisson Algebras and Their Applications
本文完成3维复诺维科夫-泊松代数的同构分类，以此证明除𝔰𝔩₂(ℂ)外的3维复转置泊松代数均可由其得到，且Sartayev猜想在3维情形成立。
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…
The geometric classification of Novikov-Poisson and Gelfand-Dorfman algebras
本文对复三维诺维科夫-泊松代数、盖尔范德-多夫曼代数及诺维科夫代数开展几何分类，明确了各类代数对应簇的维数、不可约分支数量与刚性代数数量。
$δ$-Mock-Novikov Algebras: Structural Theory, Operad and Poisson-Type Constructions
本文引入参数依赖的 $\delta$-mock-Novikov 代数，研究其结构、operad 性质及 Poisson 型构造，证明有限维情形在特征零下幂零并完成低维分类。
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…
The symmetric operation in a free Novikov algebra
本文研究了自由Novikov代数中零拉格朗日量的性质，通过欧拉算子和对称化乘积揭示其结构并描述其在对称群上的模结构。
Free Novikov-Zinbiel algebra
本文研究带有可逆导子的交换结合代数上的运算，构造自由诺维科夫-津比埃尔代数的显式基，证明其多线性恒等式可由定义恒等式导出。
Quasi-Frobenius Novikov algebras and pre-Novikov bialgebras
本文研究拟Frobenius Novikov代数与pre-Novikov双代数，证明拟Frobenius Novikov代数具有自然pre-Novikov结构，引入双重构造及pre-Novikov Yang-Baxter方程，其对称解可产生pre-Novikov双代数。
This paper is to prove that a fermionic Novikov algebra equipped with an invariant non-degenerate symmetric bilinear form is a Novikov algebra.
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.
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…
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.
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…
Anti-pre-Novikov algebras, quasi-triangular and factorizable anti-pre-Novikov bialgebras
本文提出反 pre-Novikov 代数与双代数框架，通过反 O-算子、APN-YBE 及其斜对称解构造余边界双代数，并刻画拟三角与可因子化情形，建立二次 Rota-Baxter 代数与可因子化双代数的一一对应。
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.
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.
Noncommutative Novikov bialgebras and differential antisymmetric infinitesimal bialgebras with weight
该文建立非交换Novikov双代数理论并通过匹配对、Manin三元组等刻画，引入相关方程与算子开展研究，还构造微分ASI双代数并得到其诱导非交换Novikov双代数的条件，推广了Gelfand的经典构造。
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.…
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.
General multi-Novikov algebras, multi-differential algebras and their free constructions
本文提出研究多重Novikov代数和多重微分代数的通用方法，结合泊松代数，扩展了Gelfand构造Novikov代数的方法，并通过有类型装饰根树和非交换多重微分多项式构建自由非交换多重Novikov代数。
An algebraic deformation theory of algebras over the Landweber-Novikov algebra is obtained.
We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent 2-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices and via extensions. In the latter case we lift Novikov structures on an abelian Lie algebra A and a Lie algebra B to certain extensions of B by A. We apply this to prove the existence of affine and Novikov structures on several classes of 2-step solvab…
On Pre-Novikov Algebras and Derived Zinbiel Variety
本文研究代数簇及其二元 operad 的导出构造，给出导出簇代数可嵌入相应微分代数的充分条件，并证明 Zinbiel 导出簇即 pre-Novikov 代数类中存在不可嵌入带导子 Zinbiel 代数的例子。
We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G,G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x,y].z=0. We derive some basic characterizations of such left-symmetric algebras and we highlight their relationships with the so-called Novikov algebras and derivation algebras.
A vanishing criterion for Lie elements in a free Novikov algebra
本文给出自由 Novikov 代数中多重线性元素属于 Lie 子代数的有限消没判据，利用微分实现与微分算子刻画，并应用于维数恢复与实例验证。
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