Related papers: On the representation theory of some noncrossing p…
We develop a theory of crossed products by "actions" of Hecke pairs $(G, \Gamma)$, motivated by applications in non-abelian $C^*$-duality. Our approach gives back the usual crossed product construction whenever $G / \Gamma$ is a group and…
We consider a family of Hecke C*-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke…
Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum groups these categories arise from a…
We find, for each $n\geq2$, the class of $n\times n$ compact quantum groups whose representation theory is similar to that of $SU(2)$: this is the class of "free analogues of $O(n)$" constructed by Van Daele and Wang.
We apply the free product construction to various local algebras in algebraic quantum field theory. If we take the free product of infinitely many identical half-sided modular inclusions with ergodic canonical endomorphism, we obtain a…
We construct generalized regular representations of the wreath product of a compact group with the infinite symmetric group. The characters of these representations are determined by probability measures on families of partitions called the…
A combinatorial approach to free probability theory has been developped by Roland Speicher, based on the notion of noncrossing cumulants, a free analogue of the classical theory of cumulants in probability theory. We review this theory, and…
For a family of unital free *-algebras with a family of states on them, we construct a sequence of noncommutative probability spaces, which are tensor product algebras with tensor product states and which approximate the free product of…
This is mainly a lecture note taken by myself following Weinberg's book, but also contains some corrections to the abuse of mathematical treatment. This article discusses projective unitary representations of Poincare group on the single…
Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also,…
We present a method to construct induced representations of quantum algebras having the structure of bicrossproduct. We apply this procedure to some quantum kinematical algebras in (1+1)--dimensions with this kind of structure: null-plane…
In this paper we study general quantum affinizations $\U_q(\hat{\Glie})$ of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1)…
We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra $H$ associated to the factorization of a finite group into two subgroups. The representations of the quantum double…
We study a physically motivated representation of an algebra of operators in gravitational and non gravitational theories called the covariant representation of an algebra. This is a representation where the symmetries of the operator…
Let U be the quantum group associated to a symmetrizable generalized Cartan matrix. We give a realization of U from the category of the representations of certain product valued quiver.
The structure of representations describing systems of free particles in the theory with the invariance group SO(1,4) is investigated. The property of the particles to be free means as usual that the representation describing a…
We classify certain categories of partitions of finite sets subject to specific rules on the colorization of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one…
The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through…
Given a morphism $\varphi : G \to A \wr B$ from a finitely presented group $G$ to a wreath product $A \wr B$, we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free…
The concept of quantum representation of finite groups (QRFG) has been a fundamental aspect of quantum computing for quite some time, playing a role in every corner, from elementary quantum logic gates to the famous Shor's and Grover's…