Commutative Associative Algebra Research Articles

Quantizations of Transposed Poisson algebras by Novikov deformations

Abstract The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson
 algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding
 transposed Poisson algebra. As a direct
 consequence, we revisit the relationship between transposed
 Poisson algebras and Novikov-Poisson algebras due to the fact that there is a natural Novikov
deformation of the commutative associative algebra in a
Novikov-Poisson algebra. Hence all transposed Poisson algebras of
Novikov-Poisson type, including unital transposed Poisson
algebras, can be quantized. Finally, we classify the
quantizations of $2$-dimensional complex transposed Poisson
algebras in which the Lie brackets are non-abelian up to
equivalence.

Inner isotopes associated with automorphisms of commutative associative algebras

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.

Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras

This article considers arrowhead and diagonal-plus-rank-one matrices in Fn×n where F∈{R,C,H} and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O(n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language.

Generalized derivations of current Lie algebras

Let L be a Lie algebra and let A be an associative commutative algebra with unity, both over the same field F. We consider the following question. Is every generalized derivation (resp. quasiderivation) of L ⊗ A the sum of a derivation and a map from the centroid of L ⊗ A , if the same holds true for L?

Some Results on Zinbiel Algebras and Rota–Baxter Operators

Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way.

Automorphisms of categories of free algebras with unit for classical varieties of non-associative algebras

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry.

A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie Poisson bialgebras

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and [Formula: see text]-operators are studied.

Degenerations of Poisson algebras

We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative algebra, and we apply it to obtain the classification of the [Formula: see text]-dimensional Poisson algebras. In addition, we study the geometric classification, the graph of degenerations and the closures of the orbits of the variety of [Formula: see text]-dimensional Poisson algebras. Finally, we also study the algebraic classification of the Poisson algebras defined on a commutative associative null-filiform or filiform algebra and, to enrich this classification, we study the degenerations between these particular Poisson algebras.

Derivations and Representations of Commutative Algebras Verifying a Polynomial Identity of Degree Five

In this paper we study a class of commutative non associative algebras satisfying a polynomial identity of degree five. We show that under the assumption of the existence of a non-zero idempotent, any commutative algebra verifying such an identity admits a Peirce decomposition. Using this decomposition we proceeded to the study of the derivations and representations of algebras of this class.

THE NOWICKI CONJECTURE FOR BICOMMUTATIVE ALGEBRAS

Let K be a field of characteristic zero, and K[Xn,Yn ] be the commutative associative unitary polynomial algebra of rank 2n generated by the set Xn∪Yn={x1,…,xn,y1,…,yn }. It is well known that the algebra K[Xn,Yn ]^δ of constants of the locally nilpotent linear derivation δ of K[Xn,Yn ] sending yi to xi, and xi to 0, is generated by x1,…,xn and the determinants of the form xi yj-xj yi; that was first conjectured by Nowicki in 1994, and later proved by several authors. Bicommutative algebras are nonassociative noncommutative algebras satisfying the identities (xy)z=(xz)y and x(yz)=y(xz). In this study, we work in the 2n generated free bicommutative algebra as a noncommutative nonassociative analogue of the Nowicki conjecture, and find the generators of the algebra of constants in this algebra.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

Pre-twisted calculus and differential equations

In this paper we introduce the φA-differentiability for functions f:U⊂Rk→Rn, where U is an open set, A is the linear space Rn endowed with a unital associative commutative algebra product, and φ:U⊂Rk→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.

On [formula omitted]-graded vertex algebras associated with cyclic Leibniz algebras with small dimensions

The main goals for this paper are i) to study of the algebraic structure of N-graded vertex algebras VB associated with vertex A-algebroids B when B are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras VB and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex A-algebroids B associated with given cyclic non-Lie left Leibniz algebras B. Next, we use the constructed vertex A-algebroids B to create a family of indecomposable non-simple vertex algebras VB. Finally, we use the algebraic structure of the unital commutative associative algebras A that we found to study relations between certain types of the vertex algebras VB and the vertex operator algebra associated to the rank one Heisenberg algebra.

Noncommutative Novikov algebras

The class of Novikov algebras is a popular object of study among classical nonassociative algebras. A generic example of a Novikov algebra may be constructed from an associative and commutative algebra A with a derivation d: it is enough to consider the operation $$a\,{\circ }\, b = ad(b)$$ , $$a,b\in A$$ , on the same space A. We consider a more general class of linear algebras which may be obtained in the same way from not necessarily commutative associative algebras with a derivation.

σ-monogenic functions in commutative algebras

In finite-dimensional commutative associative algebra, the concept of σ-monogenic function is introduced. Necessary and sufficient conditions for σ-monogeneity have been established. In some low-dimensional algebras, with a special choice of σ, the representation of σ-monogenic functions is obtained using holomorphic functions of a complex variable. We proposed the application of σ-monogenic functions with values in two-dimensional biharmonic algebra to representation of solutions of two-dimensional biharmonic equation.

Cohomology and deformation quantization of Poisson conformal algebras

In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and give some constructions of them. Then we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.

İZİ SIFIR OLAN JENERİK MATRİSLER CEBİRİ İÇİN NOWICKI SANISI

A locally nilpotent linear derivation δ of the commutative polynomial algebra K[Xd ]=K[x1,…,xd ] of rank d is called Weitzenböck. It is well known that the subalgebra K[Xd ]^δ of K[Xd ] consisting of polynomials which are sent to zero by δ is finitely generated. Let the Weitzenböck derivation δ act on K[Xd,Yd ] such that δ(yi )=xi, δ(xi )=0, i=1,…,d. The explicit form of generators of the algebra K[Xd,Yd ]^δ was conjectured by Nowicki in 1994. In this study, we consider the Nowicki conjecture in the algebra W generated by two traceless generic matrices with entries from commutative associative unitary polynomial algebra with six variables, and obtain the free generators of the algebra W^δ of constants in this algebra.

Editorial: Tensor numerical methods and their application in scientific computing and data science

Editorial: Tensor numerical methods and their application in scientific computing and data science

The construction and deformation of Novikov H-pseudoalgebras

In this paper, we are devoted to constructing a large number of Novikov H-pseudoalgebras from commutative associative (H-pseudo)algebras with (H-)derivations, infinitesimal H-pseudobialgebras with H-derivations and Novikov (H-pseudo)algebras with Rota-Baxter (H-)operators. Meanwhile, the relationship between Novikov H-pseudoalgebras and Lie H-pseudoalgebras is established. Furthermore, we develop the 1-parameter formal deformation theory of Novikov H-pseudoalgebras and characterize it in terms of cohomology.

Algebraic Constructions for Novikov–Poisson Algebras

A Novikov–Poisson algebra (A,∘,·) is a vector space with a Novikov algebra structure (A,∘) and a commutative associative algebra structure (A,·) satisfying some compatibility conditions. Give a Novikov–Poisson algebra (A,∘,·) and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space E=A⊕V such that (A,∘,·) is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object GH2(V,A) to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when dim(V)=1 is investigated in detail.