Braided enveloping algebras associated to quantum parabolic subalgebras

Associated to every subset J of nodes of a Dynkin diagram is a standard parabolic subalgebra pJ of the corresponding Lie algebra g, generated by the positive Borel subalgebra of g together with the negative simple generators fj, j ∈ J . Furthermore one has a decomposition of g as a semi-direct product of pJ and the subalgebra n − J generated by the remaining negative simple generators. In the same way, for each J one can construct a quantum parabolic subalgebra of the quantized enveloping algebra Uq(g). In this paper, we show that the quantum analogue of n−J is a graded braided Hopf algebra B in the category of Uq(gJ )-modules, where gJ is the Lie algebra generated by the ej and fj with j ∈ J . We show that Uq(g) has a semi-direct product decomposition into the three subalgebras B, Uq(gJ ) and B ∗ (the graded dual of B), generalising the usual triangular decomposition into negative, Cartan and positive parts. We examine in detail the structure of B, ultimately showing that it is a Nichols algebra. In particular, B is generated in degree one and is completely determined by its associated braiding. Furthermore, we conjecture that the Nichols algebra associated to the q → 1 limit of this braiding is isomorphic to U(n−J ) and hence B is indeed a quantization of this.