Oksana Yakimova

On seaweed subalgebras and meander graphs in type C

In 2000, Dergachev and Kirillov introduced subalgebras of ”seaweed type” in gl n (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”, have been 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 has been 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 sp 2n , and give a formula for the index in terms of these graphs. We also note that the very same graphs can be used in 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 sp 2n . In particular, using a natural partition of the set Fn of standard Frobenius seaweeds, we prove that #Fn 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.

Some Semi-Direct Products with Free Algebras of Symmetric Invariants

Let (mathfrak{g}) be a complex reductive Lie algebra and V the underlying vector space of a finite-dimensional representation of (mathfrak{g}). Then one can consider a new Lie algebra (mathfrak{q} = mathfrak{g}ltimes V), which is a semi-direct product of (mathfrak{g}) and an Abelian ideal V. We outline several results on the algebra (mathbb{C}[mathfrak{q}^{{ast}}]^{mathfrak{q}}) of symmetric invariants of (mathfrak{q}) and describe all semi-direct products related to the defining representation of (mathfrak{s}mathfrak{l}_{n}) with (mathbb{C}[mathfrak{q}^{{ast}}]^{mathfrak{q}}) being a free algebra.

Symmetric invariants related to representations of exceptional simple groups

We classify the finite-dimensional rational representations

Semi-direct products of Lie algebras and covariants

V

Symmetric invariants of ℤ2-contractions and other semi-direct products

of the exceptional algebraic groups

An effective method to compute closure ordering for nilpotent orbits of θ-representations

G

Parabolic contractions of semisimple Lie algebras and their invariants

with

Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

mathfrak g={sf Lie}(G)

Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras

such that the symmetric invariants of the semi-direct product

Z2-contractions of classical Lie algebras and symmetric polynomials

mathfrak gltimes V