Cauchon diagrams for quantized enveloping algebras

Details Cauchon diagrams for quantized enveloping algebras Cauchon diagrams for quantized enveloping algebras

2008-07-07

arxiv.org/abs/0807.1012v2

Mathematics

Quantum Algebra

The section 4 was rewritten and split into section 4 and 5 for more clarity. An english version is now available

Scientific paper

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra, $\mathbb{K}$ a commutative field and $q$ a nonzero element of $\mathbb{K}$ which is not a root of unity. To each reduced decomposition of the longest element $w_0$ of the Weyl group $W$ corresponds a PBW basis of the quantised enveloping algebra $\mathcal{U}_q^+(\mathfrak{g})$, and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of $w_0$, this theory constructs a bijection between the set of prime ideals in $\mathcal{U}_q^+(\mathfrak{g})$ that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of $w_0$ corresponds to a good ordering (in the sense of Lusztig \cite{Lu2}) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes \cite{Lu2}: each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitely describe the set of Cauchon diagrams for explicit reduced decomposition of $w_0$ in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of $W$. In \cite{CM}, we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of $w_0$. So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of $w_0$ corresponding to a good ordering.

Affiliated with

Also associated with

No associations

Rating

Cauchon diagrams for quantized enveloping algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Cauchon diagrams for quantized enveloping algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cauchon diagrams for quantized enveloping algebras will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-561299