Congruence-preserving functions on Stone and Kleene algebras

We investigate local polynomial functions on Stone algebras and on Kleene algebras. We find a generating set for the clone of all local polynomial functions. We also represent local polynomial functions on a given algebra by polynomial functions of some canonical extension of this algebra.     

On clones, transformation monoids, and finite Boolean algebras

We prove that, for a finite Boolean algebra, there exist only finitely many clones which consist of polynomial functions of the algebra and contain the monoid of its unary polynomial functions. Received January 7, 2000; accepted in final form December 13, 2000.     

Polynomial Lie Algebras and Growth of Their Finitely Generated Lie Subalgebras

The concept of polynomial Lie algebra of finite rank was introduced by V. M. Buchstaber in his studies of new relationships between hyperelliptic functions and the theory of integrable systems. In this paper we prove the following theorem: the Lie subalgebra generated by the frame of a polynomial Lie algebra of finite rank has at most polynomial growth. In addition, important examples of polynomial Lie algebras of countable rank are considered in the paper. Such Lie algebras arise in the study of certain hyperbolic partial differential equations, as well as in the construction of self-similar infinite-dimensional Lie algebras (such as the Fibonacci algebra).     

Some algebras similar to the 2×2 Jordanian matrix algebra

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on 2×2 matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.     

Affine completeness of Kleene algebras

An algebra is called affine complete if all its compatible (i.e. congruence-preserving) functions are polynomial functions. In this paper we characterize affine complete members in the variety of Kleene algebras. We also characterize local polynomial functions of Kleene algebras and use this result to describe locally affine complete Kleene algebras. Received December 20, 1996; accepted in final form March 24, 1997.     

Maximal commutative subalgebras of functions on spaces dual to Lie algebras

The problem of searching the maximal commutative sets of polynomial functions on the dual space to the semidirect sum of a Lie algebra and a vector space is studied. It is proved that if the first component of the semi-direct sum is a compact algebra, then the set of functions can be described explicitly. This result is applied to some particular Lie algebras.     

Clifford Algebras of Forms of Higher Degrees

The aim of this lecture is to introduce Clifford algebras of polynomial forms of higher degrees. We recall that these algebras are in general of infinite dimension, and we give a basis depending on a given basis of the underlying vector space. We then show that, though they contain large free associative algebras, we may construct finite dimensional representations of these algebras, also called linearizations of the polynomial form. If the polynomial form is, in a certain sense, non degenerate, the dimensions of these representations are multiples of the degree of the form. In the end, we recall some results known for the special case of a binary cubic form with at least one simple zero, when explicit computations can be done: the Clifford algebra is an Azumaya algebra of rank 9 over its center, which is the algebra of functions over a cubic curve depending on the given cubic form.     

Wronskian of derivations

An associative multilinear polynomial depending on 16 variables and being skew-symmetric with respect to 12 of them is presented. This polynomial provides us with a mapping recovering the algebra of regular functions of an irreducible affine variety from any smooth involutive distribution of dimension 2.     

A generalization of Descartes’ rule of signs and fundamental theorem of algebra

Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n ? 1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these properties. The resulting families include, at least, all polynomial functions and sums of exponential functions. As an application of our approach, we consider, among other things, a method for identifying certain type of bases for the Euclidean space.     

Embeddings in Artinian rings and Sylvester rank functions

Using Sylvester rank functions it is shown that a right Noetherian algebra, modulo the torsion ideal determined by elements regular modulo its nil radical, is embeddable in a simple Artinian ring. This is used to show that a right Noetherian algebra which is right Krull homogeneous embeds in a simple Artinian ring, and that a right Noetherian affine algebra satisfying a polynomial identity embeds in a simple Artinian ring. The research of the second author was supported in part by NSF Grant DMS-8317737.     

Types of polynomial completeness of expanded groups

The polynomial functions of an algebra preserve all congruence relations. In addition, if the algebra is finite, they preserve the labelling of the congruence lattice in the sense of Tame Congruence Theory. The question is for which algebras every congruence preserving function, or at least every function that preserves the labelling of the congruence lattice, is a polynomial function. In this paper, we investigate this question for finite algebras that have a group reduct. Presented by K. Kaarli. Received March 12, 2006; accepted in final form October 16, 2008. The second author is supported by Grant No. 144011 of the Ministry of Science of the Republic of Serbia, and the Scholarship “One-Month Visits to Austria for University Graduates” WUS-Austria, from the Austrian Ministry of Education, Science and Culture.     

DG polynomial algebras and their homological properties

In this paper, we introduce and study differential graded(DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A~# is a polynomial algebra K[x_1, x_2,..., x_n] with |xi| = 1 for any i ∈ {1, 2,..., n}. We describe all possible differential structures on DG polynomial algebras, compute their DG automorphism groups, study their isomorphism problems, and show that they are all homologically smooth and Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ?_A≠ 0 and the trivial DG polynomial algebra(A, 0) is Calabi-Yau if and only if n is an odd integer.     

Flag Partial Differential Equations and Representations of Lie Algebras

Flag partial differential equations naturally appear in the problem of decomposing the polynomial algebra (symmetric tensor) over an irreducible module of a Lie algebra into the direct sum of its irreducible submodules. Many important linear partial differential equations in physics and geometry are also of flag type. In this paper, we use the grading technique in algebra to develop the methods of solving such equations. In particular, we find new special functions by which we are able to explicitly give the solutions of the initial value problems of a large family of constant-coefficient linear partial differential equations in terms of their coefficients. As applications to representations of Lie algebras, we find certain explicit irreducible polynomial representations of the Lie algebras $sl(n,\mathbb {F}),\;so(n,\mathbb {F})$ and the simple Lie algebra of type G ₂.     

Representations of skew polynomial algebras

C. De Concini and C. Procesi have proved that in many cases the degree of a skew polynomial algebra is the same as the degree of the corresponding quasi polynomial algebra. We prove a slightly more general result. In fact we show that in case the skew polynomial algebra is a P.I. algebra, then its degree is the degree of the quasi polynomial algebra.

Our argument is then applied to determine the degree of some algebras given by generators and relations.

     

Cauchy transforms of characteristic functions and algebras generated by inner functions

We prove that Cauchy transforms of characteristic functions of subsets of positive measure of the unit circle are equidistributed in the unit disk in the sense that the -closure of the polynomial algebra in these Cauchy transforms coincides with the -closure of the polynomial algebra in a canonical inner function. As a corollary to this result we find conditions describing when the polynomial algebra in two singular inner functions determined by point masses is dense in the Hardy spaces .

     

Polynomial functions over finite monoids

This paper is a completion of [7]; we characterize the monoids (S,.,e) such that the algebra (P(n,S), o_n) of polynomial functions over S is congruence free.     

Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.