Narrow Results

As it is known, the defining identities of a free Novikov algebra can be obtained from a commutative algebra with a derivation. In this paper, we consider a class of algebras obtained from the class of associative algebras with a derivation that generalizes Novikov algebras. Such objects are called noncommutative Novikov algebras. We construct a monomial basis for a free noncommutative Novikov algebra.

In this paper, we first introduce a weighted derivation on algebras over an operad ${𝒫}$, and prove that for the free ${𝒫}$-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ($q$-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ($q$-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.

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 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 semiprime algebra is semiprime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras.

We classify all graded compatible left-symmetric algebraic structures on high rank Witt algebras, and classify all non-graded ones satisfying a minor condition. Furthermore, graded compatible left-symmetric algebraic structures on high rank Virasoro algebras are also classified.

We prove that any Novikov algebra over a field of characteristic $\ne 2$ 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 $[N,N]$.

This paper gives some sufficient conditions for determining the simplicity of infinite dimensional Novikov algebras of characteristic 0, and also constructs a class simple Novikov algebras by extending the base field. At last, the deformation theory of Novikov algebras is introduced.

In this paper, we construct two kinds of infinite-dimensional Novikov algebras and give their realizations, respectively. As an application, we describe their sub-adjacent Lie algebras and characterize some of their properties.

Novikov superalgebras are related to the quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play fundamental role in the completely integrable systems. In this note, we divide Novikov superalgebras into two types: N and S. Then we show that the Novikov superalgebras of dimension up to 3 are of type N.

Novikov super-algebras are related to quadratic conformal super-algebras which correspond to Hamiltonian pairs and play fundamental role in completely integrable systems. In this paper, we focus on quadratic Novikov super-algebras, which are Novikov super-algebras with associative non-degenerate super-symmetric bilinear forms. We show that quadratic Novikov super-algebras are associative and the associated Lie-super algebras of quadratic Novikov super-algebras are 2-step nilpotent. Moreover, we give some properties on quadratic Novikov super-algebras and classify the associated Lie-super algebras of quadratic Novikov super-algebras up to dimension 7.