Commutative Associative Algebra Research Articles

Dimension theorem for free ternary partially associative algebras and applications

An n-ary algebra is a vector space provided with a multiplication given by an n-linear map. A 3-ary algebra is said to be partially associative if the product x ⋅ y ⋅ z satisfies ( ( x 1 ⋅ x 2 ⋅ x 3 ) ⋅ x 4 ⋅ x 5 ) + ( x 1 ⋅ ( x 2 ⋅ x 3 ⋅ x 4 ) ⋅ x 5 ) + ( x 1 ⋅ x 2 ⋅ ( x 3 ⋅ x 4 ⋅ x 5 ) ) = 0 . We compute the dimensions of the homogeneous components of degree ⩽7 in the free 3-ary partially associative algebra over a field of characteristic zero and give a system of generators for all the homogeneous components. We show that this system is a basis if and only if none of these generators is zero. As a consequence, we show that the corresponding operad doesnʼt verify the Koszul property. We also prove that any 3-ary partially associative algebra is solvable of derived length at most 2 and any 3-ary partially associative commutative algebra is nilpotent of index at most 4. For arbitrary integer n we describe the operadic cohomology of the n-ary partially associative algebras.

Algebra of differential operators associated with Young diagrams

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers.

A CLASS OF LOCALLY NILPOTENT COMMUTATIVE ALGEBRAS

This paper deals with the variety of commutative non associative algebras satisfying the identity [Formula: see text], γ ∈ K. In [3] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ ≠ -1, then any such algebra is locally nilpotent. Our results require characteristic ≠ 2, 3.

Automorphisms of the endomorphism semigroup of a polynomial algebra

We describe the automorphism group of the endomorphism semigroup End ( K [ x 1 , … , x n ] ) of ring K [ x 1 , … , x n ] of polynomials over an arbitrary field K. A similar result is obtained for automorphism group of the category of finitely generated free commutative–associative algebras of the variety CA commutative algebras. This solves two problems posed by B. Plotkin (2003) [18, Problems 12 and 15]. More precisely, we prove that if φ ∈ Aut End ( K [ x 1 , … , x n ] ) then there exists a semi-linear automorphism s : K [ x 1 , … , x n ] → K [ x 1 , … , x n ] such that φ ( g ) = s ∘ g ∘ s − 1 for any g ∈ End ( K [ x 1 , … , x n ] ) . This extends the result obtained by A. Berzins for an infinite field K.

Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory

We define cut-and-join operator in Hurwitz theory for merging of two branching points of arbitrary type. These operators have two alternative descriptions:(i) they have the GL characters as eigenfunctions and the symmetric-group characters as eigenvalues; (ii) they can be represented as differential operators of the $W$-type (in particular, acting on the time-variables in the Hurwitz-Kontsevich tau-function). The operators have the simplest form if expressed in terms of the matrix Miwa-variables. They form an important commutative associative algebra, a Universal Hurwitz Algebra, generalizing all group algebra centers of particular symmetric groups which are used in description of the Universal Hurwitz numbers of particular orders. This algebra expresses arbitrary Hurwitz numbers as values of a distinguished linear form on the linear space of Young diagrams, evaluated at the product of all diagrams, which characterize particular ramification points of the covering.

Differentiably simple alternative algebras

Differentiably simple alternative nonassociative algebras of characteristic p > 0 are described in terms of differentiably simple associative commutative algebras. Also we look at some properties of differentiably simple alternative nonassociative algebras of characteristic 0.

The classifying algebra for defects

We demonstrate that topological defects in a rational conformal field theory can be described by a classifying algebra for defects – a finite-dimensional semisimple unital commutative associative algebra whose irreducible representations give the defect transmission coefficients. We show in particular that the structure constants of the classifying algebra are traces of operators on spaces of conformal blocks and that the defect transmission coefficients determine the defect partition functions.

Cyclic foam topological field theories

This paper proposes an axiomatic form for cyclic foam topological field theories, that is, topological field theories corresponding to string theories where particles are arbitrary graphs. World surfaces in this case are 2-manifolds with one-dimensional singularities. I prove that cyclic foam topological field theories are in one-to-one correspondence with graph-Cardy–Frobenius algebras that are families ( A , B ⋆ , ϕ ) where A = { A s | s ∈ S } are families of commutative associative Frobenius algebras, B ⋆ = ⨁ σ ∈ Σ B σ is an associative algebra of Frobenius type graduated by graphs, and ϕ = { ϕ σ s : A s → End ( B σ ) | s ∈ S , σ ∈ Σ } is a family of special representations. Examples of cyclic foam topological field theories and graph-Cardy–Frobenius algebras are constructed.

Varieties of linear algebras with colength one

In the case of characteristic zero it is proved that there exist exactly three varieties of linear algebras with the colength equal to one for all degrees. Those are the variety of all associative-commutative algebras, the variety of all metabelian Lie algebras, and the variety of soluble Jordan algebras of the step 2 with the identity x2x ≡ 0.

INVARIANTS OF LIE ALGEBRAS EXTENDED OVER COMMUTATIVE ALGEBRAS WITHOUT UNIT

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac–Moody algebras.

The three-dimensional origin of the classifying algebra

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S 2 × S 1 . Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.

The weak braided Hopf algebra structure of some Cayley–Dickson algebras

It has been shown by Albuquerque and Majid that a class of unital k-algebras, not necessarily associative, obtained through the Cayley–Dickson process can be viewed as commutative associative algebras in some suitable symmetric monoidal categories. In this note we will prove that they are, moreover, commutative and cocommutative weak braided Hopf algebras within these categories. To this end we first define a Cayley–Dickson process for coalgebras. We then see that the k-vector space of complex numbers, of quaternions, of octonions, of sedenions, etc. fit to our theory, hence they are all monoidal coalgebras as well, and therefore weak braided Hopf algebras.

ANTI-COMMUTATIVE ALGEBRAS WITH SKEW-SYMMETRIC IDENTITIES

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If sn is a standard skew-symmetric polynomial of degree n, then any s4-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D2) are s4-Lie. An algebra (U, id ∧ D3 - 2D ∧ D2) is s'5-Lie, where s'5 is a non-standard skew-symmetric polynomial of degree 5.

Problems of classifying associative or Lie algebras over a field of characteristic not two and finite metabelian groups are wild

Let F be a field of characteristic different from 2. It is shown that the problems ofclassifying    (i) local commutative associative algebras over F with zero cube radical,    (ii) Lie algebras over F with central commutator subalgebra of dimension 3, and    (iii) finite p-groups of exponent p with central commutator subgroup of order p3are hopeless since each of them contains     • the problem of classifying symmetric bilinear mappings U × U → V , or     • the problem of classifying skew-symmetric bilinear mappings U × U → V ,in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.

The Second Cohomology of Current Algebras of General Lie Algebras

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

Finite-dimensional modules for the polynomial ring in one variable as a vertex algebra

A commutative associative algebra A over C with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for vertex algebra A and the modules for associative algebra A are not well understood. In this paper, I give the classification of finite-dimensional indecomposable untwisted or twisted modules for the polynomial ring in one variable over C as a vertex algebra.

On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections

We study the properties of real realizations of holomorphic linear connections over associative commutative algebras $$ \mathbb{A} $$ m with unity. The following statements are proved. If a holomorphic linear connection ∇ on M n over $$ \mathbb{A} $$ m (m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ℝ of the Lie algebra of all affine vector fields of the space (M ℝ , ∇ℝ) is no greater than (mn)2 − 2mn + 5, where m = dimℝ $$ \mathbb{A} $$ , $$ n = dim_\mathbb{A} $$ M n , and ∇ℝ is the real realization of the connection ∇. Let ∇ℝ =1 ∇ ×2 ∇ be the real realization of a holomorphic linear connection ∇ over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor 1 R ≠ 0, 2 R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space (M 2 ℝ , ∇ℝ) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( a M n , a ∇) (a = 1, 2).

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras.

The Second Cohomology of Current Algebras of General Lie Algebras

Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\k$ be a Lie algebra, and $\z$ a vector space, considered as a trivial module of the Lie algebra $\g := A \otimes \k$. In this paper we give a description of the cohomology space $H^2(\g,\z)$ in terms of well accessible data associated to $A$ and $\k$. We also discuss the topological situation, where $A$ and $\k$ are locally convex algebras.

Tensor products of algebras and their applications to the construction of Anosov diffeomorphisms

In this paper, we develop algebraic approaches to the construction of Anosov diffeomorphisms on compact manifolds. Two mutually dual constructions are described, which provide numerous new examples of Anosov diffeomorphisms on nilmanifolds. The basis of the constructions is the operation of tensor multiplication of Lie algebras by appropriate finite-dimensional associative-commutative algebras. Several examples illustrating the general method are given.