The Principal Element of a Frobenius Lie Algebra

We introduce the notion of the principal element of a Frobenius Lie algebra ${\frak{f}}$ . The principal element corresponds to a choice of ${F \in \frak{f}^{*}}$ 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.