Relating quantum and braided Lie algebras

X. Gomez, S. Majid
2003 Noncommutative Geometry and Quantum Groups   unpublished
We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if g Γ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space k ⊕ g Γ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra U (g Γ ) is a bialgebra in the category of A-comodules. Introduction. Since the appearance of quantum groups in the mid eighties, and in
more » ... ghties, and in particular of the pairs of Hopf algebras O q (G) and U q (g) associated to semi-simple complex Lie algebras, there has been many attempts to define a corresponding notion of quantum Lie algebra [Wor, LS, DG, Br] . It is remarkable that there is still no fully satisfying answer: all these attempts tend to generalize an aspect of usual Lie algebras, but some other properties that one would expect from the generalization seem to be lost. The geometric approach to quantum Lie algebras [Wor] tends to reproduce the relation (given by the Lie functor) between a Lie group G and its Lie algebra g = Lie(G), which is defined as the vector space of left (or right) invariant vector fields on G. To give a "non-commutative" analogue of this, one has to work out a suitable formalism of differential calculus on a Hopf algebra (the leading example would be O q (G)), and this was done in [Wor]. Woronowicz noticed that the (quantum) "tangent space" g Γ of each such calculus Γ comes naturally equipped with maps σ (a braiding) and [ , ] : g Γ ⊗ g Γ → g Γ which satisfy identities that generalize the axioms of usual Lie algebras. Thus, these identities can be chosen as the axioms of abstract quantum Lie algebras. In this construction, the classical geometrical picture is preserved, but some familiar features seem to be lost. First a quantum Lie functor, from the category of Hopf algebras to that of bicovariant differential calculi, doesn't seem to exist (given an arbitrary Hopf algebra A, there is no canonical
doi:10.4064/bc61-0-6 fatcat:u2xesbjssvev3e4tauzs6y3de4