Integral representation for powers of elements in locally <Emphasis Type="Italic">m</Emphasis>-convex algebras and applications

This paper proves a new integral representation theorem for powers of elements in locally m-convex algebras. It is shown that the conditions handled in this theorem are satisfied for any sequentially complete locally m-convex Q-algebra. Various applications of this result concerning the spectrum, the spectral radius or the numerical radius are given.      

Nilpotent fusion categories

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger.     

On <Emphasis Type="Italic">n</Emphasis> × <Emphasis Type="Italic">m</Emphasis>-valued Łukasiewicz-Moisil algebras

n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM _n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM _n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM _n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM _n×m -algebra is computed.      

States on sharply dominating effect algebras

We prove that sharply dominating Archimedean atomic lattice effect algebras can be characterized by the property called basic decomposition of elements.As an application we prove the state smearing theorem for these effect algebras.     

Amalgamation in finite dimensional cylindric algebras

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA _n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. February 22, 1999.     

Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.