761 research outputs found
Integration over the Pauli quantum group
We prove that the Pauli representation of the quantum permutation algebra
is faithful. This provides the second known model for a free quantum
algebra. We use this model for performing some computations, with the main
result that at the level of laws of diagonal coordinates, the Lebesgue measure
appears between the Dirac mass and the free Poisson law.Comment: 27 page
Unitary Easy Quantum Groups: the free case and the group case
Easy quantum groups have been studied intensively since the time they were
introduced by Banica and Speicher in 2009. They arise as a subclass of
(-algebraic) compact matrix quantum groups in the sense of Woronowicz. Due
to some Tannaka-Krein type result, they are completely determined by the
combinatorics of categories of (set theoretical) partitions. So far, only
orthogonal easy quantum groups have been considered in order to understand
quantum subgroups of the free orthogonal quantum group .
We now give a definition of unitary easy quantum groups using colored
partitions to tackle the problem of finding quantum subgroups of . In
the free case (i.e. restricting to noncrossing partitions), the corresponding
categories of partitions have recently been classified by the authors by purely
combinatorial means. There are ten series showing up each indexed by one or two
discrete parameters, plus two additional quantum groups. We now present the
quantum group picture of it and investigate them in detail. We show how they
can be constructed from other known examples using generalizations of Banica's
free complexification. For doing so, we introduce new kinds of products between
quantum groups.
We also study the notion of easy groups.Comment: 39 page
Compact Kac algebras and commuting squares
We consider commuting squares of finite dimensional von Neumann algebras
having the algebra of complex numbers in the lower left corner. Examples
include the vertex models, the spin models (in the sense of subfactor theory)
and the commuting squares associated to finite dimensional Kac algebras. To any
such commuting square we associate a compact Kac algebra and we compute the
corresponding subfactor and its standard invariant in terms of it.Comment: 14 pages, some minor change
Liberation of orthogonal Lie groups
We show that under suitable assumptions, we have a one-to-one correspondence
between classical groups and free quantum groups, in the compact orthogonal
case. We classify the groups under correspondence, with the result that there
are exactly 6 of them: . We investigate the
representation theory aspects of the correspondence, with the result that for
, this is compatible with the Bercovici-Pata bijection.
Finally, we discuss some more general classification problems in the compact
orthogonal case, notably with the construction of a new quantum group.Comment: 42 page
On polynomial integrals over the orthogonal group
We consider integrals of type , with respect to the Haar measure
on the orthogonal group. We establish several remarkable invariance properties
satisfied by such integrals, by using combinatorial methods. We present as well
a general formula for such integrals, as a sum of products of factorials.Comment: 20 page
Quantum automorphism groups of homogeneous graphs
Associated to a finite graph is its quantum automorphism group . The
main problem is to compute the Poincar\'e series of , meaning the series
whose coefficients are multiplicities of 1 into tensor
powers of the fundamental representation. In this paper we find a duality
between certain quantum groups and planar algebras, which leads to a planar
algebra formulation of the problem. Together with some other results, this
gives for all homogeneous graphs having 8 vertices or less.Comment: 30 page
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