The Principal Element of a Frobenius Lie Algebra

TL;DR

This paper introduces the concept of the principal element in Frobenius Lie algebras, explores its properties, and demonstrates its behavior in specific cases like ext{sl}_n and certain subalgebras, also discussing stability under deformation.

Contribution

It defines the principal element in Frobenius Lie algebras and analyzes its properties, including semisimplicity and eigenvalues, with applications to specific subalgebras and deformation stability.

Findings

Principal element is often semisimple.

Eigenvalues are integers and F-independent in ext{sl}_n.

Frobenius algebras are stable under deformation.

Abstract

We introduce the notion of the \textit{principal element} of a Frobenius Lie algebra $\f$. The principal element corresponds to a choice of $F\in \f^*$ such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to $\sl_n$, its eigenvalues are integers and are independent of $F$. For certain ``small'' functionals $F$, a simple construction is given which readily yields the principal element. When applied to the first maximal parabolic subalgebra of $\sl_n$, the principal element coincides with semisimple element of the principal three-dimensional subalgebra. We also show that Frobenius algebras are stable under deformation.