An Algebraic Study of Multivariable Integration and Linear Substitution

We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota-Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota-Baxter hierarchy. We conjecture that the operator relations are a noncommutative Groebner basis for the ideal they generate.Comment: 44 pages, 1 tabl

Monoids of IG-type and Maximal Orders

Let G be a finite group that acts on an abelian monoid A. If f: A -> G is a map so that f(a f(a)(b)) = f(a)f(b), for all a, b in A, then the submonoid S = {(a, f(a)) | a in A} of the associated semidirect product of A and G is said to be a monoid of IG-type. If A is a finitely generated free abelian monoid of rank n and G is a subgroup of the symmetric group Sym_n of degree n, then these monoids first appeared in the work of Gateva-Ivanova and Van den Bergh (they are called monoids of I-type) and later in the work of Jespers and Okninski. It turns out that their associated semigroup algebras share many properties with polynomial algebras in finitely many commuting variables. In this paper we first note that finitely generated monoids S of IG-type are epimorphic images of monoids of I-type and their algebras K[S] are Noetherian and satisfy a polynomial identity. In case the group of fractions of S also is torsion-free then it is characterized when K[S] is a maximal order. It turns out that they often are, and hence these algebras again share arithmetical properties with natural classes of commutative algebras. The characterization is in terms of prime ideals of S, in particular G-orbits of minimal prime ideals in A play a crucial role. Hence, we first describe the prime ideals of S. It also is described when the group of fractions is torsion-free.Comment: 21 pages, 0 figure

Primes of height one and a class of Noetherian finitely presented algebras

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type.Comment: 28 pages, no figure

Quadratic algebras of skew type and the underlying semigroups

We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand-Kirillov dimension defined by homogeneous relations satisfy a polynomial identity. The structure of the underlying monoid, defined by the same presentation, is described. This is used to derive information on the prime radical and minimal prime ideals. Some examples are described in detail. Earlier, Etingof, Schedler and Soloviev, Gateva-Ivanova and Van den Bergh, and the authors considered special classes of such algebras in the contexts of noetherian algebras, Grobner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang-Baxter equation.Comment: Work supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), and KBN research grant 2P03A 030 18 (Poland

Groebner-Shirshov basis of the universal enveloping Rota-Baxter algebra of a Lie algebra

Consider the class RBLie of Lie algebras equipped with a Rota---Baxter operator. Then the forgetful functor RBLie --> Lie has a left adjoint one denoted by $U_{RB}(\cdot)$. We prove an "operator" analogue of the Poincare---Birkhoff---Witt theorem for $U_{RB}(L)$, where $L$ is an arbitrary Lie algebra, by means of Gr\"obner---Shirshov bases theory for Lie algebras with an additional operator.Comment: 16 page

Gr\"obner-Shirshov bases and embeddings of algebras

In this paper, by using Gr\"obner-Shirshov bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. We show that in the following classes, each countably generated algebra over a countable field $k$ can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative $\Omega$-algebras, associative $\lambda$-differential algebras. Also we prove that any countably generated module over a free associative algebra $k$ can be embedded into a cyclic $k$-module, where $|X|>1$. We give another proofs of the well known theorems: each countably generated group (resp. associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (resp. associative algebra, semigroup, Lie algebra).Comment: 26 page

Jucys-Murphy elements for Birman-Murakami-Wenzl algebras

The Birman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys--Murphy elements. We show that the set of Jucys--Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.Comment: Proceedings of International Workshop "Supersymmetries and Quantum Symmetries", Dubna, 200

An Algebraic Study of Averaging Operators

A module endomorphism $f$ on an algebra $A$ is called an averaging operator if it satisfies $f(xf(y)) = f(x)f(y)$ for any $x, y\in A$. An algebra $A$ with an averaging operator $f$ is called an averaging algebra. Averaging operators have been studied for over one hundred years. We study averaging operators from an algebraic point of view. In the first part, we construct free averaging algebras on an algebra $A$ and on a set $X$, and free objects for some subcategories of averaging algebras. Then we study properties of these free objects and, as an application, we discuss some decision problems of averaging algebras. In the second part, we show how averaging operators induce Lie algebra structures. We discuss conditions under which a Lie bracket operation is induced by an averaging operator. Then we discuss properties of these induced Lie algebra structures. Finally we apply the results from this discussion in the study of averaging operators.Comment: 75 page

Baxter Algebras and Umbral Calculus

We apply recent constructions of free Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of the umbral calculus that include the classical umbral calculus in a family of $\lambda$-umbral calculi parameterized by $\lambda$ in the base ring.Comment: 22 page

Finiteness spaces and generalized power series

We consider Ribenboim's construction of rings of generalized power series. Ribenboim's construction makes use of a special class of partially ordered monoids and a special class of their subsets. While the restrictions he imposes might seem conceptually unclear, we demonstrate that they are precisely the appropriate conditions to represent such monoids as internal monoids in an appropriate category of Ehrhard's finiteness spaces. Ehrhard introduced finiteness spaces as the objects of a categorical model of classical linear logic, where a set is equipped with a class of subsets to be thought of as finitary. Morphisms are relations preserving the finitary structure. The notion of finitary subset allows for a sharper analysis of computational structure than is available in the relational model. For example, fixed point operators fail to be finitary. In the present work, we take morphisms to be partial functions preserving the finitary structure rather than relations. The resulting category is symmetric monoidal closed, complete and cocomplete. Any pair of an internal monoid in this category and a ring induces a ring of generalized power series by an extension of the Ribenboim construction based on Ehrhard's notion of linearization of a finiteness space. We thus further generalize Ribenboim's constructions. We give several examples of rings which arise from this construction, including the ring of Puiseux series and the ring of formal power series generated by a free monoid