Combinatorial index formulas for Lie algebras of seaweed type

Seaweed algebras

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 on the connected components of a "meander": a planar graph associated with the algebra. Our index analysis on seaweed algebras requires only basic linear and abstract algebra. Indeed, the main goal of this survey-type article is to introduce a broader audience to seaweed algebras with minimal appeal to specialized language and notation from Lie theory. This s…

Contact Seaweeds

A ($2k+1$)$-$dimensional contact Lie algebra is one which admits a one-form $φ$ such that $φ\wedge (dφ)^k\ne0$. Such algebras have index one, but this is not generally a sufficient condition. Here we show that index-one type-A seaweed algebras are necessarily contact. Examples, together with a method for their explicit construction, are provided.

接触海藻代数 II：C 型（Contact seaweeds II: type C）

Contact seaweeds II: type C

本研究延续海藻代数接触形式构建工作，采用图论与组合方法，将前期关于$A_{n-1}=\mathfrak{sl}(n)$指数为1的海藻子代数均为接触的结论，扩展至$C_{n}=\mathfrak{sp}(2n)$的指数为1海藻子代数情形。

Regular functionals on seaweed Lie algebras

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 Kirillov form. We develop a broad general framework for the explicit construction of regular (index realizing) functionals for seaweed subalgebras of $\mathfrak{gl}(n)$ and the classical Lie algebras: $A_n=\mathfrak{sl}(n+1),$ $B_n=\mathfrak{so}(2n+1)$, and $C_n=\mathfrak{sp}(2n)$. Until now, this problem has remained open in $\mathfrak{gl}(n)$ -- and in all …

The unbroken spectrum of type-A Frobenius seaweeds

If $\mathfrak{g}$ is a Frobenius Lie algebra, then for certain $F\in \mathfrak{g}^*$ the natural map $\mathfrak{g}\longrightarrow \mathfrak{g}^* $ given by $x \longmapsto F[x,-]$ is an isomorphism. The inverse image of $F$ under this isomorphism is called a principal element. We show that if $\mathfrak{g}$ is a Frobenius seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$ then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution. Our proof methods are constructive and combinatorial in nature.

接触海藻代数的分类

Classification of contact seaweeds

本研究针对海藻代数的接触性分类开放问题，证明指数为1的海藻代数是接触的当且仅当它是拟可约的，结合已有拟可约海藻代数分类结果完成接触海藻代数的完整分类，且A型、C型中指数为1即可保证存在接触形式。