Finite-dimensional Commutative Algebra Research Articles

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.

Centers of multilinear forms and applications

The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.

On the Lie triple derivations

This article aims to interpret some Lie triple derivations of Tensor algebras , using generalized quaternion algebra over a field F and assuming T as an F-linear associative algebra. The study concludes with complete description of all Lie triple derivations of A ⊗  F L. It is pertinent to mention here that A is assumed to be a finite-dimensional commutative algebra and L is a simple Lie algebra over an algebraically closed field.

The space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism: An approach to the problem of Albert

In this article we explore an approach to the problem of Albert described by U. Umirbaev. We characterize the space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism F:Kn→Kn, where K is a field with characteristic different from two. We obtain some conjectures expressed in the language of polynomial endomorphisms, which are equivalent to the existence of invariant bilinear forms in a finite-dimensional commutative algebra. We give a characterization of the space of invariant bilinear forms in terms of differential forms in the ring K[x1,…,xn]. We also introduce a new kind of algebra, we call them totally symmetric algebras, and we establish the relationship between these algebras and the existence of invariant bilinear forms in any commutative algebra.

Конструктивний опис моногенних функцiй у скiнченновимiрних комутативних алгебрах

Конструктивний опис моногенних функцiй у скiнченновимiрних комутативних алгебрах

Noetherian quotients of the algebra of partial difference polynomials and Gröbner bases of symmetric ideals

In this paper we develop a Gröbner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials. This can be explained in terms of Noetherianity of the corresponding quotient algebra. Among these Noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined. We obtain therefore a consistent Gröbner bases theory for ideals that possess such symmetries.

Jacobi–Jordan algebras

We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions n<7 over algebraically closed fields of characteristic not 2 or 3.

Solvability of a Commutative Algebra which Satisfies (x 2)2 = 0

We studied the solvability of the algebra which satisfies the polynomial identity (x 2)2 = 0. We believe that, if A is a finite dimensional commutative algebra over a field F of characteristic not 2 which satisfies (x 2)2 = 0 for all x ∈ A, then A is solvable. In this article we proved this when dim  F A ≤ 7.

Commutative algebras and representations of the category of finite sets

We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.

Flags in zero dimensional complete intersection algebras and indices of real vector fields

We introduce bilinear forms in a flag in a complete intersection local \(\mathbb {R}\) -algebra of dimension 0, related to the Eisenbud–Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen’s filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the signature function of vector fields tangent to an even dimensional hypersurface singularity, one being topologically defined and the other computable with finite dimensional commutative algebra methods.

Moduli spaces of discrete gravity: I. A few points

Spectral triples describe and generalize Riemannian spin geometries by converting the geometrical information into algebraic data, which consist of an algebra A, a Hilbert space H carrying a representation of A and the Dirac operator D (a selfadjoint operator acting on H). The gravitational action is described by the trace of a suitable function of D. In this paper we examine the (path-integral)-quantization of such a system given by a finite-dimensional commutative algebra. It is then (in concrete examples) possible to construct the moduli space of the theory, i.e. to divide the space of all Dirac operators by the action of the diffeomorphism group, and to construct an invariant measure on this space. We discuss expectation values of various observables and demonstrate some interesting effects such as the effect of coupling the system to fermions (which renders finite quantities in cases, where the bosons alone would give infinite quantities) or the striking effect of spontaneous breaking of spectral invariance.

𝐴-differentiability and 𝐴-analyticity

Let A A be a finite-dimensional commutative algebra over R \mathbb {R} and let C A r ( U ) C_{A}^{r}(U) , C ω ( U , A ) C^{\omega }(U,A) and O A ( U ) \mathcal { O}_{A}(U) be the ring of A A -differentiable functions of class C r , 0 ≤ r ≤ ∞ C^{r},\,0 \leq r \leq \infty , the ring of real analytic mappings with values in A A and the ring of A A -analytic functions, respectively, defined on an open subset U U of A n A^{n} . We prove two basic results concerning A A -differentiability and A A -analyticity: 1 s t 1^{st} ) O A ( U ) = C A ∞ ( U ) ⋂ C ω ( U , A ) \mathcal { O}_{A}(U) = C^{\infty }_{A}(U) \bigcap C^{\omega }(U,A) , 2 n d 2^{nd} ) O A ( U ) = C A ∞ ( U ) \mathcal { O}_{A}(U) = C^{\infty }_{A}(U) if and only if A A is defined over C \mathbb {C} .

Idempotent Elements in a Bernstein Algebra

A finite-dimensional commutative algebra A over a field K is called a Bernstein algebra if there exists a non-trivial homomorphism co: A -> K (baric algebra) such that the identity (x) = CO(X)JC holds in A (see [7]). The origin of Bernstein algebras lies in genetics (see [2,8]). Holgate (in [2]) was the first to translate the problem into the language of non-associative algebras. Information about algebraic properties of Bernstein algebras, as well as their possible genetic interpretations, can be found in [10, Chapter 9B; 11; 12; 3; 1]. The existence of idempotent elements, that is elements e, e # 0, such that e = e, is of interest in the study of the structure of a non-associative algebra. From the biological aspect the existence of such elements is also interesting, because the equilibria of a population which can be described by an algebra correspond to idempotent elements of this algebra. The algebras occurring in applications usually do contain an idempotent. This occurs in Bernstein algebras (see [10]). With respect to an idempotent eeA (whose existence is guaranteed), A splits into the direct sum A = (e) + U+Z, where

Fixed subalgebra of a commutative Frobenius algebra

Let B B be a finite-dimensional commutative algebra generated by a single element, and let A = B ⊗ B A = B \otimes B . We prove that the fixed subalgebra of A A under the involution b 1 ⊗ b 2 ↦ b 2 ⊗ b 1 {b_1} \otimes {b_2} \mapsto {b_2} \otimes {b_1} is Frobenius if and only if either the characteristic of B B is different from 2 or B B is separable.

Linear Fractional Maps and Jordan Algebras

Jordan algebras are characterized from amongst the finite dimensional commutative algebras without the use of identities. This is achieved by investigating the properties of the linear fractional transformations generated by the quasi-inversion and all translations.

Linear fractional maps and Jordan algebras

Jordan algebras are characterized from amongst the finite dimensional commutative algebras without the use of identities. This is achieved by investigating the properties of the linear fractional transformations generated by the quasi-inversion and all translations.