Publication

L-2-Betti Numbers Of Rigid C*-Tensor Categories And Discrete Quantum Groups

Sven Raum
2017
Journal Articles

Abstract

We compute the L-2-Betti numbers of the free C*-tensor categories, which are the representation categories of the universal unitary quantum groups A(u)(F). We show that the L-2-Betti numbers of the dual of a compact quantum group G are equal to the L-2-Betti numbers of the representation category Rep. (G) and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of L-2-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first L-2-Betti number in terms of a generating set of a C*-tensor category.

Official source

https://infoscience.epfl.ch/entities/publication/ce03926e-5e27-48ca-b060-086462fab640

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Sven Raum
2017
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/ce03926e-5e27-48ca-b060-086462fab640

About this result

Related concepts (32)

Monoidal category

In mathematics, a monoidal category (or tensor category) is a equipped with a bifunctor that is associative up to a natural isomorphism, and an I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant s commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.

Tensor algebra

In mathematics, the tensor algebra of a vector space V, denoted T(V) or T^•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V).

Braided monoidal category

In mathematics, a commutativity constraint on a is a choice of isomorphism for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have for all pairs of objects . A braided monoidal category is a monoidal category equipped with a braiding—that is, a commutativity constraint that satisfies axioms including the hexagon identities defined below. The term braided references the fact that the braid group plays an important role in the theory of braided monoidal categories.

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