Symmetries and invariants of twisted quantum algebras and associated Poisson algebras

Details Symmetries and invariants of twisted quantum algebras and associated Poisson algebras Symmetries and invariants of twisted quantum algebras and associated Poisson algebras

2007-01-31

arxiv.org/abs/math/0701902v3

Rev. Math. Phys. 20 (2008), 173-198.

Mathematics

Quantum Algebra

29 pages, more references added

Scientific paper

10.1142/S0129055X08003250

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.

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