On simple modules and automorphisms of the quantum Galilei group (Q6561444)

The classical Galilean group is the spacetime symmetry group of nonrelativistic quantum mechanics. The quantum Galilei group \({\mathcal F}_q={\mathcal F}_q(G)\) is obtained by deforming the algebra of functions on the one-dimensional Galilei group \(G\) by means of a nontrivial 1-cocycle with values in \(\wedge^2\text{Lie}(G)\) that defines a Lie-Poisson structure. The quantum Galilei algebra \({\mathcal U}_q={\mathcal U}_q({\mathfrak g})\) is the Hopf dual of \({\mathcal F}_q\). Both \({\mathcal F}_q\) and \({\mathcal U}_q\) are non-commutative and non-cocommutativeHopf algebras, and there is a nondegenerated duality pairing between them. The quantum Galilei algebra has been considered as the kinematical symmetry of phonons, and the coproduct rule plays a fundamental physical role as it establishes the way in which single phonons combine. \par The paper under review studies the ring theoretic properties and the representations of \({\mathcal F}_q\) and \({\mathcal U}_q\). The authors classify the primitive ideals and the simple modules over an algebraically closed field of characteristic different from 2. The description of the groups of automorphisms of \({\mathcal F}_q\) and \({\mathcal U}_q\) is obtained in the case of characteristic 0.