Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.     

Bivariant cyclic theory

In this paper we construct a bivariant version of cyclic cohomology and study its fundamental properties. We prove universal coefficient theorems relating the bivariant theory with cyclic homology and cohomology, we construct products in the bivariant theory, and we analyse the notion of an HC-equivalence.Dedicated to Alexander Grothendieck     

Holomorphic Jackson's theorems in polydiscs

The purpose of this article is to establish Jackson-type inequality in the polydiscs U^N of for holomorphic spaces X, such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely,

where is the deviation of the best approximation of fX by polynomials of degree at most k_j about the jth variable z_j with respect to the X-metric and is the corresponding modulus of continuity.     

A Morita type theorem for a sort of quotient categories

We consider the quotient categories of two categories of modules relative to the Serre classes of modules which are bounded as abelian groups and we prove a Morita type theorem for some equivalences between these quotient categories.     

A Different Version of Flett＇s Mean Value Theorem and an Associated Functional Equation

In this paper we present a mean value theorem derived from Flett‘s mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point. To answer what class of functions have this property, we consider a functional equation associated with this mean value theorem. This equation is then solved in a general setting on abelian groups.     

Polynomial Pell's equation-II

The polynomial Pell's equation is X²−DY²=1, where D is a polynomial with integer coefficients and the solutions X,Y must be polynomials with integer coefficients. Let D=A²+2C be a polynomial in , where . Then for a prime, a necessary and sufficient condition for which the polynomial Pell's equation has a nontrivial solution is obtained. Furthermore, all solutions to the polynomial Pell's equation satisfying the above condition are determined.     

Polynomial detection of matrix subalgebras

The double Capelli polynomial of total degree is

It was proved by Giambruno-Sehgal and Chang that the double Capelli polynomial of total degree is a polynomial identity for . (Here, is a field and is the algebra of matrices over .) Using a strengthened version of this result obtained by Domokos, we show that the double Capelli polynomial of total degree is a polynomial identity for any proper -subalgebra of . Subsequently, we present a similar result for nonsplit inequivalent extensions of full matrix algebras.

     

Sampling sets and sufficient sets for A

We give new characterizations of the subsets S of the unit disc of the complex plane such that the topology of the space A^−∞ of holomorphic functions of polynomial growth on coincides with the topology of space of the restrictions of the functions to the set S. These sets are called weakly sufficient sets for A^−∞. Our approach is based on a study of the so-called (p,q)-sampling sets which generalize the A^−p-sampling sets of Seip. A characterization of (p,q)-sampling and weakly sufficient rotation invariant sets is included. It permits us to obtain new examples and to solve an open question of Khôi and Thomas.     

About И-quantifiers

Gabbay and Pitts observed that the Fraenkel–Mostowski model of set-theory supports useful notions of name-abstraction and fresh-name. In order to understand their work in a more general setting we introduce the notions of -units and -relations in a regular category D. A -relation is given by a functor A # (_-):DD and we show that in the case that D is a topos then A # (_-) has a right adjoint [A](_-) that can be thought of as an object of abstractions. We also explore the existence of a right adjoint to [A](_-) and relate it to the name swapping operations considered as fundamental by Gabbay and Pitts. We present many examples of categories where this notions occur and we relate the results here with Pitts' Nominal Logic.     

Codimension growth and minimal superalgebras

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope of a finite dimensional superalgebra . In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field.

The importance of such algebras is readily proved: is a minimal superalgebra if and only if the ideal of identities of is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties such that and for all proper subvarieties of . This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003).

As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.