First-order differential calculi and Laplacians on $q$-deformations of compact semisimple Lie groups

Abstract: In this talk, we suggest a simple definition of Laplacian on a compact quantum group (CQG) associated with a first-order differential calculus (FODC) on it. Applied to the classical differential calculus on a compact Lie group, this definition yields classical Laplacians, as it should. Moreover, on the CQG $ K_q $ arising from the $ q $-deformation of a compact semisimple Lie group $K$, we can find many interesting linear operators that satisfy this definition, which converge to a classical Laplacian on $ K $ as $ q $ tends to 1. In the light of this, we call them $ q $-Laplacians on $ K_q $ and investigate some of their operator theoretic properties. In particlar, we show that the heat semigroups generated by these are not completely positive, suggesting that perhaps on the CQG $ K_q $, stochastic processes that are most relevant to the geometry of it are not quantum Markov processes. This work is based on the preprint arXiv:2410.00720.

operator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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