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For a broad class of Frechet-Lie supergroups we prove that there exists a correspondence between positive definite smooth superfunctions and matrix coefficients of unitary representations. We also give a characterization of linear…
We develop a method to give presentations of quantized function algebras of complex reductive groups. In particular, we give presentations of quantized function algebras of automorphism groups of finite dimensional simple complex Lie…
An analogue of Burnside's Lemma for 2-transitive groups is shown to hold for a class of topological groups. If the group is compact the representation is finite and splits into an irreducible and the constant functions. If both the group…
We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…
We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these…
We continue our study of topological partial *algebras, focusing our attention to the interplay between the various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link…
Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…
We study a positive-definite function associated to a measure-preserving equivalence relation on a standard probability space and use it to measure quantitatively the proximity of subequivalence relations. This is combined with a recent…
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups…
We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the…
We study connections between the ring of symmetric functions and the characters of irreducible finite-dimensional representations of quantum affine algebras. We study two families of representations of the symplectic and orthogonal Lie…
We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates…
A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher…
A strict 2-group is a 2-category with one object in which all morphisms and all 2-morphisms have inverses. 2-Groups have been studied in the context of homotopy theory, higher gauge theory and Topological Quantum Field Theory (TQFT). In the…
A new deformed canonical commutation relation, generalizing various known deformations, is defined together with its structure function of deformation. Then, the related irreducible representations are characterized and classified. Finally,…
In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are…
The aim of the present survey paper is to provide an accessible introduction to a new chapter of representation theory - harmonic analysis for noncommutative groups with infinite-dimensional dual space. I omitted detailed proofs but tried…
In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail the structure of the Hecke algebra corresponding to the commutant of an induced representation and then specialize to the…
In this paper we study the possibility to define irreducible representations of the symmetric groups with the help of finitely many relations. The existence of finite bases is established for the classes of representations corresponding to…
We introduce a notion of positive definiteness for functions $f\!:P\to\mathbb{R}$ defined on meet semilattices $(P,\preceq,\wedge)$ and prove several properties for these functions. In addition, we utilize the $LDL^{\rm T}$ decomposition of…