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

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 Kq 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 Kq 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 Kq, 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.

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