BRAIDED ENVELOPING ALGEBRAS ASSOCIATED TO QUANTUM PARABOLIC SUBALGEBRAS

Associated to each subset J of the nodes I of a Dynkin diagram is a triangular decomposition of the corresponding Lie algebra g into three subalgebras (g(J)) over tilde (generated by e(j), f(j) for j is an element of J and h(i) for i is an element of I), n(D)(-) (generated by f(d), d is an element of D = I\ J) and its dual n(D)(+). We demonstrate a quantum counterpart, generalising work of Majid and Rosso, by exhibiting analogous triangular decompositions of U-q(g) and identifying a graded braided Hopf algebra that quantizes n(D)(-). This algebra has many similar properties to U-q(-)(g), in many cases being a Nichols algebra and therefore completely determined by its associated braiding.