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

2018/01/11 by Jason Hancox, Hancox, Jason, Tobias Hartung +1
Mathematics · #46L57 #47B48 #58B32 #58J35 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1801.03982

openalex publication_date 2018/01/11 · openalex created_date 2022/09/15 · openalex updated_date 2026/07/28

Abstract

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

Citations

• On the heat equation and the index theorem
• Mathematica notebook for the explicit functions and functional relations for ``The term a₄ in the heat kernel expansion of noncommutative tori"

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