SYMMETRIES AND INVARIANTS OF TWISTED QUANTUM ALGEBRAS AND ASSOCIATED POISSON ALGEBRAS
We construct an action of the braid group BN on the twisted quantized enveloping algebra [Formula: see text] where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra [Formula: see text]. We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.