Sur les semi-invariants d'une sous-algèbre parabolique d'une algèbre enveloppante quantifiée

Abstract

Let Ŭ _q (g _π) be the simply connected quantized enveloping algebra of the complex semisimple finite dimensional Lie algebrag _π. Letp be a parabolic subalgebra ofg _π and$\mathcal{P}: = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{U} _q \left( {g_\pi } \right)$ the Hopf subalgebra of Ŭ _q (g _π) associated top. LetY $\left( \mathcal{P} \right)$ be the space generated by the semi-invariants of$\mathcal{P}$ under adjoint action. We describe a basis forY $\left( \mathcal{P} \right)$, show that it is a polynomial algebra and describe its rank, which we compute explicitly in a number of cases. Except for the Borel and some special cases a corresponding result is not known for the semi-centre of the enveloping algebra ofp. However we show here that the latter has the same Gelfand-Kirillov dimension asY $\left( \mathcal{P} \right)$. We also describe the multiplication rules of the above basos elements ofY $\left( \mathcal{P} \right)$. These are analogous to “fusion rules” in tensor product decomposition and their derivation obtains from an analysis of theR-matrix.