Commutative Finite-dimensional Algebras Research Articles

Weyr structures of matrices and relevance to commutative finite-dimensional algebras

We relate the Weyr structure of a square matrix B to that of the t×t block upper triangular matrix C that has B down the main diagonal and first superdiagonal, and zeros elsewhere. Of special interest is the case t=2 and where C is the nth Sierpinski matrix Bn, which is defined inductively by B0=1 and Bn=[Bn−1Bn−10Bn−1]. This yields an easy derivation of the Weyr structure of Bn as the binomial coefficients arranged in decreasing order. Earlier proofs of the Jordan analogue of this had often relied on deep theorems from such areas as algebraic geometry. The result has interesting consequences for commutative, finite-dimensional algebras.

An Example of a Central Simple Commutative Finite-Dimensional Algebra with an Infinite Basis of Identities

In 1993 I. P. Shestakov came up with the question whether there exists a central simple finite-dimensional algebra over a field of characteristic 0, whose identities are not given by a finite set (Dniester Notebook, Question 3.103). In 2012 I. M. Isaev and the author constructed a desired example, giving an affirmative answer to the question posed. Here research into Shestakov’s question is continued for the case of commutative algebras. A seven-dimensional central simple commutative algebra over a field of characteristics 0 having no finite basis of identities is exemplified.

Entropy distance: New quantum phenomena

We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance, and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology, and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.

On Quadratic Integral Polynomials with only Finitely Many Roots in any Commutative Finite-Dimensional Algebra

The monic quadratic polynomials f with integer coefficients such that each commutative finite-dimensional algebra over a field contains only finitely many roots of f are determined as the polynomials of the form f = X2 + (2m + 1)X + m2 + m, where \({m \in \mathbb{Z}}\).

Commutative Finite-Dimensional Algebras Satisfying x(x(xy)) = 0 are Nilpotent

We investigate the structure of commutative non-associative algebras satisfying the identity x(x(xy)) = 0. Recently, Correa and Hentzel proved that every commutative algebra satisfying above identity over a field of characteristic ≠ 2 is solvable. We prove that every commutative finite-dimensional algebra 𝔄 over a field F of characteristic ≠ 2, 3 which satisfies the identity x(x(xy)) = 0 is nilpotent. Furthermore, we obtain new identities and properties for this class of algebras.

Hyperholomorphic functions on commutative algebras

In this article, we study properties of hyperholomorphic functions on commutative finite-dimensional algebras. The Cauchy–Riemann type conditions for hyperholomorphic functions is investigated. We prove that a hyperholomorphic function on a commutative finite-dimensional algebra can be expanded in a Taylor series. We also present a technique for computing zeros of polynomials in commutative algebras.

On the cohomology of one-relator associative algebras

Let R be a one-relator algebra over a field. We show that R contains two images of a commutative finite-dimensional algebra, E, called the eigenring, and E contains much of the homological information about R. The theory works best in the case where R is graded; here we can describe E explicitly, compute the global dimension of R, and recover J. Backelin's formulas ( C. R. Acad. Sci. Paris Ser. A 287 (1978), 843–846) for the Hilbert and Poincaré series of R.