Invariants of knots and 3-manifolds from quantum groupoids

Abstract

We use the categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.

Determination of quantum symmetries for higher ADE systems from the modular T matrix

“…However, T , when evaluated on non-zero entries of the 22 other columns of E 0 , is not constant. We conclude that the set J charactering the ambichiral points of Oc(E 21 ) is a set with two elements: the two extreme vertices of E 21 . The values of the modular exponentT obtained for these two points areT = 21 andT = 39.…”

“…According to A. Ocneanu (unpublished), this object, also called "algebra of double triangles", is a semi-simple weak Hopf algebra (or quantum groupoid) -see [3], [21], for general properties of quantum groupoids. We shall not use it explicitly in our paper and it is enough to say that, as a bialgebra, it possesses two associative algebra structures (say "composition •" and "convolution ⋆"), for which the underlying vector space can be block diagonalized (i.e., decomposed as a sum of matrix algebras) in two different ways.…”

Comments about quantum symmetries of SU(3) graphs

“…This bialgebra is a particular type of weak Hopf algebra (or quantum groupoïd) (see for instance [37,4,5,34,33]). We call it the "Ocneanu quantum groupoïd" associated with the chosen pair.…”

Quantum symmetries of face models and the double triangle algebra

“…The DTA is [21,9] a weak Hopf algebra [2,16,13,14,15] and the algebra of quantum symmetries(AQS) can be obtained as the algebra describing the tensor category of the DTA representations associated with one of its product structures(the same name sometimes denotes the bialgebra itself). The elements of the DTA algebra are certain endomorphisms of the vector space of paths over the corresponding ADE graph.…”