We first prove that a left Novikov algebra N is right nilpotent if and only it solvable. Then we show that, every can be represented as the sum of two solvable subalgebras itself solvable, moreover, are abelian, then whole metabelian. Finally, for n≥2, n-generated non-abelian free (or nilpotent) has wild automorphisms.

We establish Gr\"{o}bner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, PBW type theorem in Shirshov form is given and we provide an algorithm solving the word problem with finite homogeneous relations. also construct subalgebra one generated free algebra which not free.

We first offer a fast method for calculating the Gelfand-Kirillov dimension of finitely presented commutative algebra by investigating certain finite set. Then we establish Gröbner–Shirshov bases theory bicommutative algebras, and show that every generated has basis. As an application, is nonnegative integer.

We construct linear bases of free Gelfand–Dorfman–Novikov (GDN) superalgebras. As applications, we prove a Poincaré–Birkhoff–Witt (PBW) type theorem, that is, every GDN superalgebra can be embedded into its universal enveloping associative differential supercommuative algebra. An Engel theorem is given.

We apply the method of Gröbner–Shirshov bases for replicated algebras developed by Kolesnikov to offer a general approach constructing free products associative trialgebras (or trioids). In particular, open problem Zhuchok on trioids is solved.

We construct an Anick type wild automorphism $\delta$ in a 3-generated free Poisson algebra which induces tame polynomial algebra. also show that is stably tame. Dedicated to the memory of professor V.A.Roman'kov

In this paper, we study automorphisms of finitely generated free metabelian Novikov algebras and show that every tame automorphism a two-generated right nilpotent algebra index 3 is simple reducible. We offer method on recognizing by using the theory Gröbner–Shirshov basis.

We construct a linear basis of free GDN superalgebra over field characteristic $\neq 2$. As applications, we prove PBW theorem, that is, any can be embedded into its universal enveloping commutative associative differential superalgebra. An Engel theorem under some assumptions is given.

By applying a Gröbner-Shirshov basis of the symmetric group $S_{n}$, we give two formulas for Schubert polynomials, either which involves only nonnegative monomials. We also prove some combinatorial properties polynomials. As applications, algorithms to calculate structure constants one depends on Monk's formula.

We first construct a linear basis for free metabelian Poisson algebra generated by an arbitrary well-ordered set. It turns out that such depends on the characteristic of underlying field. Then we elaborate method Gr\"{o}bner--Shirshov bases algebras. Finally, show word problem finitely presented algebras are solvable.

The Gelfand-Kirillov dimension measures the asymptotic rate of growth algebras. For every associative dialgebra D, quotient AD:=D/Id(S), where Id(S) is ideal D generated by set S:={x⊢y−x⊣y∣x,y∈D}, called algebra associated to D. We show that GKdim(D)≤2GKdim(AD). Moreover, we prove no has strictly between 1 and 2.

We provide necessary and sufficient conditions on the graph [Formula: see text] field for which Leavitt path algebra is Lie solvable. Consequently, we obtain a complete description of nilpotent algebras, show that solvability nilpotency are same. Furthermore, compute solvable index algebra.

We apply the method of Gr\"obner-Shirshov bases for replicated algebras developed by Kolesnikov to offer a general approach constructing free products trialgebrs (resp. trioids). In particular, open problem Zhuchok on trioids is solved.

The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is ideal $\mathcal{D}$ generated by set $S:=\{x \vdash y-x\dashv y \mid x,y\in \mathcal{D}\}$, called algebra associated to $\mathcal{D}$. Here we show that Gelfand--Kirillov bounded above twice $\mathcal{A}_\mathcal{D}$. Moreover, prove no has strictly between 1 and 2.

We construct the free products of arbitrary digroups, and thus we solve an open problem Zhuchok.