  We construct an action of the braid group B_N on the twisted quantized
enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In
the classical limit q -> 1 we recover the action of B_N 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
re-discovered in the mathematical literature. Furthermore, we construct a
Poisson bracket on the space of polynomials associated with another twisted
quantized enveloping algebra U'_q(sp_{2n}). We use the Casimir elements of both
twisted quantized enveloping algebras to re-produce some well-known and
construct some new polynomial invariants of the corresponding Poisson algebras.
