Laplacians on $ q $-deformations of compact semisimple Lie groups

Abstract

The problem of formulating a correct notion of Laplacian on compact quantum groups (CQGs) has long been recognized as both fundamental and nontrivial. Existing constructions typically rely on selecting a specific first-order differential calculus (FODC), but the absence of a canonical choice in the noncommutative setting renders these approaches inherently non-canonical. In this work, we propose a simple set of conditions under which a linear operator on a CQG can be recognized as a Laplacian -- specifically, as the formal modulus square of the differential associated with a bicovariant FODC. A key feature of our framework is its generality: it applies to arbitrary finite-dimensional bicovariant $*$-FODCs on \( K_q \), the \( q \)-deformation of a compact semisimple Lie group \( K \). To each such calculus, we associate a Laplacian defined via the formal modulus square of its differential. Under mild additional assumptions, we demonstrate that these operators converge to classical Laplacians on \( K \) in the classical limit \( q \to 1 \), thereby justifying their interpretation as ``$q$-deformed Laplacians." Furthermore, we prove that the spectra of the $q$-deformed Laplacians are discrete, real, bounded from below, and diverge to infinity, much like those of their classical counterparts. However, in contrast to the classical case, the associated heat semigroups do not define quantum Markov semigroups.