Combinatorial index formulas for Lie algebras of seaweed type

Abstract

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 focus on a set of distinguished vertices of the meander, called the tail of the meander, and using the tail, we provide comprehensive combinatorial formulas for the index of a seaweed in all the classical types. Using these formulas, we provide all general closed-form index formulas where the index is given by a polynomial greatest common divisor formula in the sizes of the parts that define the seaweed.

Keywords:

• Frobenius Lie algebra
• index
• meander
• seaweed
• special orthogonal Lie algebra

Mathematics Subject Classification 2010:

• 17B08

Notes

1 Frobenius algebras are of special interest in deformation and quantum group theory stemming from their connection with the classical Yang-Baxter equation (see [10] and [11]). More specifically, an index-realizing functional is called regular, and a regular functional F on a Frobenius Lie algebra $\mathfrak{g}$ is called a Frobenius functional; equivalently, $B_F(-,-)$ is non-degenerate. Suppose $B_F(-,-)$ is non-degenerate and let $\left[F\right]$ be the matrix of $B_F(-,-)$ relative to some basis $\{x_1,\dots ,x_n\}$ of $\mathfrak{g}.$ In [1], Belavin and Drinfeld showed that $$\sum _{i,j}{\left[F\right]}_{ij}^{-1}{x}_i\wedge x_j$$ is the infinitesimal of a Universal Deformation Formula (UDF) based on $\mathfrak{g}.$ A UDF based on $\mathfrak{g}$ can be used to deform the universal enveloping algebra of $\mathfrak{g}$ and also the function space on any Lie group which contains $\mathfrak{g}$ in its Lie algebra of derivations.