arXiv subjects

Bas Jordans

Publications and source records attributed to Bas Jordans.

3 recordsLinked to original sources

arXiv2017

Convergence to the boundary for random walks on discrete quantum groups and monoidal categories

We study the problem of convergence to the boundary in the setting of random walks on discrete quantum groups. Convergence to the boundary is established for random walks on $\hat{\textrm{SU}_q(2)}$. Furthermore, we will define the Martin boundary for random walks on C$^*$-tensor categories and give a formulation for convergence to the boundary for such random walks. These categorical definitions are shown to be compatible with the definitions in the quantum group case. This implies that convergence to the boundary for random walks on quantum groups is stable under monoidal equivalence.

math.OA↗

arXiv2014

A classification of $SU(d)$-type C$^*$-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group $SU(d)$. In this paper we consider the C$^*$-analogue of this problem. Given a rigid C$^*$-tensor category $\mathcal{C}$ with fusion ring isomorphic to the fusion ring of the group $SU(d)$, we can extract a constant $q$ from $\mathcal{C}$ such that there exists a $*$-representation of the Hecke algebra $H_n(q)$ into $\mathcal{C}$. The categorical trace on $\mathcal{C}$ induces a Markov trace on $H_n(q)$. Using this Markov trace and a representation of $H_n(q)$ in $\textrm{Rep}\,(SU_{\sqrt{q}}(d))$ we show that $\mathcal{C}$ is equivalent to a twist of the category $\textrm{Rep}\,(SU_{\sqrt{q}}(d))$. Furthermore a sufficient condition on a C$^*$-tensor category $\mathcal{C}$ is given for existence of an embedding of a twist of $\textrm{Rep}\,(SU_{\sqrt{q}}(d))$ in $\mathcal{C}$.

math.OA↗

arXiv2014

Real dimensional spaces in noncommutative geometry

In this paper we will extend the product of spectral triples to a product of semifinite spectral triples. We will prove that finite summability and regularity are preserved under taking products. Connes and Marcolli constructed for each $z\in(0,\infty)$ a type ${\rm II}_\infty$-semifinite spectral triple which can be considered as a geometric space of dimension $z$. A small adaption of their construction yields a type ${\rm I}$-semifinite spectral triple. We will investigate the properties of these semifinite spectral triples. At the same time we will also avoid the need for an infra-red cutoff to compute the dimension spectrum. Using this collection of semifinite spectral triples and the product of semifinite spectral triples one can construct a mathematical tool for dimensional and zeta-function regularisation in quantum field theory.

math-ph↗