Poincare'-Birkhoff-Witt property for bicovariant differential algebras on simple quantum groups

Abstract

We investigate the possibility to construct bicovariant differential calculi on quantum groups SO_q(N) and Sp_q(N) as a quantization of an underlying bicovariant bracket.We show that, opposite to GL(N) and SL(N)-cases, neither of possible graded SO- and Sp- bicovariant brackets (associated with a quasitriangular r-matrices) obey the Jacobi identity when the differential forms are Lie algebra- valued. The absence of a classical Poisson structure gives an indication that differential algebras describing bicovariant differential calculi on quantum orthogonal and symplectic groups are not of Poincar'e- Birkhoff- -Witt type.

12 pages,latex

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Cited by in corpus (4)

• On the algebraic structure of differential calculus on quantum groups
• Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras
• On construction of unitary quantum group differential calculus
• On Poisson-Lie structure on the external algebra of the classical Lie groups