Related papers: Easy quantum groups and quantum subgroups of a sem…
We consider several orthogonal quantum groups satisfying the easiness assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr(u^k) with respect to the Haar measure, u being the…
Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite…
We introduce and study a number of invariants of locally compact quantum groups defined by their scaling and modular groups and the spectrum of their modular elements. Focusing mainly on compact quantum groups we consider the question…
We show an organized form of quantum de Finetti theorem for Boolean independence. We define a Boolean analogue of easy quantum groups for the categories of interval partitions, which is a family of sequences of quantum semigroups. We…
We provide a classification of compact quantum groups, which can be obtained by the Woronowicz construction, when the arrays used in the twisted determinant condition are extensions of functions on permutations. General properties of such…
Using the general theory of [10] ( hep-th 9412058 ), quantum Poincar\'e groups (without dilatations) are described and investigated. The description contains a set of numerical parameters which satisfy certain polynomial equations. For most…
We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group $\mathcal Q_4$. Our main tool is a new presentation for the algebra $\rm C(\mathcal Q_4)$, corresponding to an…
Using a suitably noncommutative flat matrix model, it is shown that the quantum permutation group has free orbitals: that is, a monomial in the generators of the algebra of functions can be zero for trivial reasons only. It is shown that…
A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible…
A representation of the group element (also known as ``universal ${\cal T}$-matrix'') which satisfies $\Delta(g) = g\otimes g$, is given in the form $$ g = \left(\prod_{s=1}^{d_B}\phantom.^>\ {\cal…
We consider a constructive modification of quantum-mechanical formalism. Replacement of a general unitary group by unitary representations of finite groups makes it possible to reproduce quantum formalism without loss of its empirical…
A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a…
We study simplicial action of groups on one vertex Kan complexes. We show that every semi-direct product of the fundamental group of an one vertex Kan complex with a finite group can be simplicially realized. We also calculate the…
The main notions of the quantum groups: coproduct, action and coaction, representation and corepresentation are discussed using simplest examples: $GL_q(2)$, $sl_q(2)$, $q$-oscillator algebra ${\cal A}(q)$, and reflection equation algebra.…
Induced representations for quantum groups are defined starting from coisotropic quantum subgroups and their main properties are proved. When the coisotropic quantum subgroup has a suitably defined section such representations can be…
The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet…
We study Karoubian tensor categories which interpolate representation categories of families of so-called easy quantum groups in the same sense in which Deligne's interpolation categories $\mathrm{\underline{Rep}}(S_t)$ interpolate the…
Mathematical core of quantum mechanics is the theory of unitary representations of symmetries of physical systems. We argue that quantum behavior is a natural result of extraction of "observable" information about systems containing…
In this paper, we expose the construction of a possible, simple quantum matrix group (according to Woronowicz), related to elementary formal aspects of the Einstein field equations of General Relativity, and its possible symmetries.
We study actions of compact quantum groups on type I factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz' results concerning Peter-Weyl theory for…