150 research outputs found
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
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
Quantum groups and Fuss-Catalan algebras
The categories of representations of compact quantum groups of automorphisms
of certain inclusions of finite dimensional C*-algebras are shown to be
isomorphic to the categories of Fuss-Catalan diagrams.Comment: 12 page
The planar algebra of a coaction
We study actions of ``compact quantum groups'' on ``finite quantum spaces''.
According to Woronowicz and to general \c^*-algebra philosophy these
correspond to certain coactions . Here is a finite
dimensional \c^*-algebra, and is a certain special type of Hopf
*-algebra. If preserves a positive linear form \phi :A\to\c, a version of
Jones' ``basic construction'' applies. This produces a certain \c^*-algebra
structure on , plus a coaction , for every . The elements satisfying
are called fixed points of . They form a \c^*-algebra . We prove
that under suitable assumptions on the graded union of the algebras
is a spherical \c^*-planar algebra.Comment: 39 page
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