Representations of theq-Deformed Lie Algebra of the Group of Motions of the Euclidean Plane

Bounded an unbounded Hilbert space ∗-Representations of theq-deformed Lie algebra of the group of plan motions are studied for different choices of involutions. Integrable (“well-behaved”) representations of the corresponding ∗-algebras are defined and described up to unitary equivalence. In the case of the ∗-algebras with quadratic involutions, analytically defined representations are introduced, and irreducible analytically defined representations are described up to unitary equivalence using dynamical systems. In the case of the ∗-algebras with involutions of the first order all analytically defined representations are shown to be one- dimensional. For these ∗-algebras the problem of unitary classification of all representations defined on some dense invariant domain is shown to be equivalent to the unitary classification of arbitrary families of bounded self-adjoint operators. Integrable (“well-behaved”) representations of these ∗-algebras are defined and described up to unitary equivalence.