Vol. 285, No. 2, 2016

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ISSN: 0030-8730
On seaweed subalgebras and meander graphs in type C

Dmitri I. Panyushev and Oksana S. Yakimova

Vol. 285 (2016), No. 2, 485–499
Abstract

In 2000, Dergachev and Kirillov introduced subalgebras of “seaweed type” in gln (or sln) and computed their index using certain graphs. In this article, those graphs are called type-A meander graphs. Then the subalgebras of seaweed type, or just “seaweeds”, were defined by Panyushev (2001) for arbitrary simple Lie algebras. Namely, if p1,p2 g are parabolic subalgebras such that p1 + p2 = g, then q = p1 p2 is a seaweed in g. If p1 and p2 are “adapted” to a fixed triangular decomposition of g, then q is said to be standard. The number of standard seaweeds is finite. A general algebraic formula for the index of seaweeds was proposed by Tauvel and Yu (2004) and then proved by Joseph (2006).

In this paper, elaborating on the “graphical” approach of Dergachev and Kirillov, we introduce the type-C meander graphs, i.e., the graphs associated with the standard seaweed subalgebras of sp2n, and give a formula for the index in terms of these graphs. We also note that the very same graphs can be used in the case of the odd orthogonal Lie algebras.

Recall that q is called Frobenius if the index of q equals 0. We provide several applications of our formula to Frobenius seaweeds in sp2n. In particular, using a natural partition of the set n of standard Frobenius seaweeds, we prove that #n strictly increases for the passage from n to n + 1. The similar monotonicity question is open for the standard Frobenius seaweeds in sln, even for the passage from n to n + 2.

Keywords
index of Lie algebra, Frobenius Lie algebra
Mathematical Subject Classification 2010
Primary: 17B08
Secondary: 17B20
Milestones
Received: 29 December 2015
Revised: 5 May 2016
Accepted: 28 May 2016
Published: 21 November 2016
Authors
Dmitri I. Panyushev
Dobrushin Mathematics Laboratory
Institute for Information Transmission Problems
Russian Academy of Sciences
Bolshoi Karetnyi per. 19
Moscow
127051
Russia
Oksana S. Yakimova
Institut für Mathematik
Friedrich-Schiller-Universität Jena
D-07737 Jena
Germany