Vincent E. Coll Jr. | Searcharxiv

arXiv2016

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.

math.RA↗

arXiv2015

Cohomology of Lie semidirect products and poset algebras

When $\mathfrak h$ is a toral subalgebra of a Lie algebra $\mathfrak g$ over a field $\mathbf k$, and $M$ a $\mathfrak g$-module on which $\mathfrak h$ also acts torally, the Hochschild-Serre filtration of the Chevalley-Eilenberg cochain complex admits a stronger form than for an arbitrary subalgebra. For a semidirect product $\mathfrak g = \mathfrak h \ltimes \mathfrak k$ with $\mathfrak h$ toral one has $H^*(\mathfrak g, M) \cong \bigwedge\mathfrak h^{\vee} \bigotimes H^*(\mathfrak k,M)^{\mathfrak h} = H^*(\mathfrak h, \mathbf k)\bigotimes H^*(\mathfrak k,M)^{\mathfrak h}$, and for a Lie poset algebra $\mathfrak g$, that $H^*(\mathfrak g, \mathfrak g)$, which controls the deformations of $\mathfrak g$, can be computed from the nerve of the underlying poset. The deformation theory of Lie poset algebras, analogous to that of complex analytic manifolds for which it is a small model, is illustrated by examples.

math.RA↗