Related papers: Indice des sous-alg\`ebres biparaboliques d'une al…
We give some properties on the index of seaweed subalgebras of the complex Lie algebra $\mathfrak{gl}(n)$ which allow to obtain formulas for the index of some interesting classes of this family and to give new families of Frobenius Lie…
The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families…
The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based…
Analogous to the types A, B, and C cases, we address the computation of the index of seaweed subalgebras in the type-D case. Formulas for the algebra's index can be computed by counting the connected components of its associated meander. We…
In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}(n)$ and computed their index using certain graphs. In this article, those graphs are called type-A meander graphs. Then the subalgebras of seaweed…
We give an upper bound for the index of certain Lie algebras, called of seaweed type, introduced by V. Dergachev, A. Kirillov and D. Panyushev. We deduce from this a conjecture of D. Panyushev stated in "Inductive formulas for the index of…
Analogous to the sl(n) case, we address the computation of the index of seaweed subalgebras of sp(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra's index may be computed by counting the…
In this article, we give a simple explicit construction of an affine slice for the coadjoint action of a certain class of biparabolic (also called seaweed) subalgebras of a semisimple Lie algebra over an algebraically closed field of…
We extend the set of known infinite families of Frobenius seaweed Lie subalgebras of $\mathfrak{sl}_{n}$ to include a family which is the first non-trivial general family containing algebras whose associated meanders have an arbitrarily…
Seaweed algebras are a class of Lie algebras that are naturally characterized by a pair of compositions, which in turn are represented visually as planar graphs called meanders. These meanders provide a straightforward method for computing…
In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs. Then seaweed subalgebras $\mathfrak q\subset\mathfrak g$ were defined by Panyushev for any…
If $\mathfrak{g}$ is a Frobenius Lie algebra, then the spectrum of $\mathfrak{g}$ is an algebraic invariant equal to the multiset of eigenvalues corresponding to a particular operator acting on $\mathfrak{g}$. In the case of Frobenius…
The index of a Lie algebra $\mathfrak{g}$ is defined by ind $\mathfrak{g}=$ $\min_{f\in \mathfrak{g}^*}\dim(\ker (B_f))$, where $f$ is an element of the linear dual $\mathfrak{g}^*$ and $B_f(x,y)=f([x,y])$ is the associated skew-symmetric…
In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs, which we call type-${\sf A}$ meander graphs. Then the subalgebras of seaweed type, or just…
This is a continuation of arXiv:0903.0398 [math.RT]. Let g be a simple Lie algebra. In this note, we provide simple formulae for the index of sl(2)-subalgebras in the classical Lie algebras and a new formula for the index of the principal…
The index of a Lie algebra is an important algebraic invariant. In 2000, Vladimir Dergachev and Alexandre Kirillov defined seaweed subalgebras of $\mathfrak{gl}_n$ (or $\mathfrak{sl}_n$) and provided a formula for the index of a seaweed…
We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements.…
A standard seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$ may be parametrized by a pair of compositions of the positive integer $n$. For all $n$ and certain $k(n)$, we provide closed-form formulas and the generating functions for $C(n,k)$…
By using a free monoid of operators on the set of compositions (resp. pairs of compositions), we establish in this paper a bijective correspondence between Frobenius standard parabolic (resp. seaweed) subalgebras and certain elements of…
Seaweed (biparabolic) subalgebras form a large and structurally rich class of subalgebras of simple Lie algebras. We determine their adjoint cohomology. If $\mathfrak{s}$ is an indecomposable seaweed subalgebra of a complex simple Lie…