Functionals and Graphs of a Frobenius Lie Algebra

An algebra A is Frobenius if there exists a linear functional f on A such that if f(ab)=0 for all a then b=0. Frobenius algebras were named in the late 1930s by Nakayama who discovered the beginnings of remarkably rich duality theories for these objects. Examples include any matrix algebra over any group algebra of a finite group over a field. Current interest in Frobenius algebras is their connection to Hopf algebras and topological quantum field theory. In the 1970s, Ooms extended the notion to the Lie case: a Lie algebra is Frobenius if there is a linear functional f on A such that if f([a,b])=0 for all a then b=0. Unlike the associative case, a simple or semisimple Lie algebra can never be Frobenius.

In this talk the focus will be a survey of Frobenius subalgebras of the Lie algebra of nxn matrices. Many such examples are members of the class of "seaweed Lie algebras." Such algebras are determined by two compositions (unordered partitions) of n. Seaweed Lie algebras were invented by Dergachev and Kirillov in 2000 and are the subject of much current research. Many elementary and fascinating properties of Frobenius seaweed Lie algebras can be combinatorially deduced in terms of a certain directed graph built from the linear functional. I will discuss some of these properties. No knowledge of Lie theory is necessary.