scholarly journals On the Solvability of ℤ3-Graded Novikov Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 312
Author(s):  
Viktor Zhelyabin ◽  
Ualbai Umirbaev
Keyword(s):  
Finite Order ◽  
Algebraic Systems ◽  
Novikov Algebra ◽  
Novikov Algebras

Symmetries of algebraic systems are called automorphisms. An algebra admits an automorphism of finite order n if and only if it admits a Zn-grading. Let N=N0⊕N1⊕N2 be a Z3-graded Novikov algebra. The main goal of the paper is to prove that over a field of characteristic not equal to 3, the algebra N is solvable if N0 is solvable. We also show that a Z2-graded Novikov algebra N=N0⊕N1 over a field of characteristic not equal to 2 is solvable if N0 is solvable. This implies that for every n of the form n=2k3l, any Zn-graded Novikov algebra N over a field of characteristic not equal to 2,3 is solvable if N0 is solvable.

scholarly journals Gröbner–Shirshov bases method for Gelfand–Dorfman–Novikov algebras

2017 ◽  
Vol 16 (01) ◽  
pp. 1750001 ◽  
Author(s):  
L. A. Bokut ◽  
Yuqun Chen ◽  
Zerui Zhang
Keyword(s):  
Word Problem ◽  
Type Theorem ◽  
Novikov Algebra ◽  
Base Theory ◽  
Novikov Algebras

We establish Gröbner–Shirshov base theory for Gelfand–Dorfman–Novikov algebras over a field of characteristic [Formula: see text]. 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.

scholarly journals On the solvability of graded Novikov algebras

2021 ◽  
pp. 1-14
Author(s):  
Ualbai Umirbaev ◽  
Viktor Zhelyabin
Keyword(s):  
Solvable Group ◽  
Finite Solvable Group ◽  
Group Of Automorphisms ◽  
Novikov Algebra ◽  
Novikov Algebras ◽  
Nilpotent Subalgebra ◽  
The Right ◽  
Text Component

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 [Formula: see text]-graded Novikov algebra [Formula: see text] over a field [Formula: see text] with solvable [Formula: see text]-component [Formula: see text] is solvable, where [Formula: see text] is a finite additive abelean group and the characteristic of [Formula: see text] does not divide the order of the group [Formula: see text]. We also show that any Novikov algebra [Formula: see text] with a finite solvable group of automorphisms [Formula: see text] is solvable if the algebra of invariants [Formula: see text] is solvable.

scholarly journals On the Solvability of Z3-Graded Novikov Algebras

2021 ◽  
Author(s):  
Viktor Zhelyabin ◽  
Ualbai Umirbaev
Keyword(s):  
Novikov Algebra ◽  
Novikov Algebras

Let N = N0+ N1+ N2 be a Z3-graded Novikov algebra. The main goal of the paper is to prove that over a field of characteristic not equal to 3 the algebra N is solvable if N0 is solvable. We also show that a $Z_2$-graded Novikov algebra N=N0+ N2 over a field of characteristic not equal to 2 is solvable if N0 is solvable. This implies that for every n of the form n=2k3l, any Zn-graded Novikov algebra N over a field of characteristic not equal to 2,3 is solvable if N0 is solvable.

CLASSIFICATION OF ORBIT CLOSURES IN THE VARIETY OF THREE-DIMENSIONAL NOVIKOV ALGEBRAS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350081 ◽  
Author(s):  
THOMAS BENEŠ ◽  
DIETRICH BURDE
Keyword(s):  
Three Dimensional ◽  
Hasse Diagrams ◽  
Novikov Algebra ◽  
Orbit Closures ◽  
Novikov Algebras

We classify the orbit closures in the variety [Formula: see text] of complex, three-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.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)

2014 ◽  
Vol 58 (1) ◽  
pp. 13-22
Author(s):  
Roman Wituła ◽  
Edyta Hetmaniok ◽  
Damian Słota
Keyword(s):  
Finite Order ◽  
Convergent Series ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
Composition Relation ◽  
The Continuum

Abstract In the paper we present the selected properties of composition relation of the convergent and divergent permutations connected with commutation. We note that a permutation on ℕ is called the convergent permutation if for each convergent series ∑an of real terms, the p-rearranged series ∑ap(n) is also convergent. All the other permutations on ℕ are called the divergent permutations. We have proven, among others, that, for many permutations p on ℕ, the family of divergent permutations q on ℕ commuting with p possesses cardinality of the continuum. For example, the permutations p on ℕ having finite order possess this property. On the other hand, an example of a convergent permutation which commutes only with some convergent permutations is also presented.

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