Zeta-regularization and the heat-trace on some compact quantum semigroups

TL;DR

This paper extends heat-invariant concepts to non-commutative geometries like quantum groups and algebras, using zeta-regularization to define heat-traces and dimensions where classical methods do not apply.

Contribution

It introduces a method to compute heat-traces on non-commutative spaces via zeta-regularization, including cases lacking classical Brownian motion.

Findings

Heat-traces computed on Toeplitz algebra and quantum groups.

Defined notions of dimension and volume for non-commutative spaces.

Extended heat-invariant analysis to quantum groups like SU_q(2).

Abstract

Heat-invariants are a class of spectral invariants of Laplace-type operators on compact Riemannian manifolds that contain information about the geometry of the manifold, e.g., the metric and connection. Since Brownian motion solves the heat equation, these invariants can be obtained studying Brownian motion on manifolds. In this article, we consider Brownian motion on the Toeplitz algebra, discrete Heisenberg group algebras, and non-commutative tori to define Laplace-type operators and heat-semigroups on these C*-bialgebras. We show that their traces can be $\zeta$-regularized and compute "heat-traces" on these algebras, giving us a notion of dimension and volume. Furthermore, we consider $SU_q(2)$ which does not have a Brownian motion but a class of driftless Gaussians which still recover the dimension of $SU_q(2)$.