The principal element of a Frobenius Lie algebra

A finite dimensional Lie algebra f is Frobenius if there is a linear Frobenius functional F : f→ C such that the skew bilinear form BF defined by BF (x, y) = F ([x, y]) is non-degenerate. The principal element of f is then the unique element F̂ such that F (x) = F ([F̂ , x]); it depends on the choice of functional. However, if f is a subalgebra of a simple Lie algebra g and not an ideal of any larger subalgebra of g (in particular when f contains a Cartan subalgebra of g) then the principal element is semisimple, and for g = sln the eigenvalues of ad F̂ are shown to be integers which are independent of the choice of Frobenius functional. A basic open question is whether these eigenvalues characterize the algebra. The principal element of the first parabolic subalgebra of sln is shown to be the semisimple element of the principal three-dimensional subalgebra of sln. Deformations of Frobenius Lie algebras remain Frobenius. 1 The principal element For any skew bilinear form B on a vector space V one can define a linear map V → V ∗ sending v ∈ V to the functional sending x to B(v, x) for all x ∈ V . If B is non-degenerate then this is an isomorphism so for every F ∈ V ∗ there is a unique element F̂ ∈ V such that F (x) = B(F̂ , x) for all x ∈ V . A Frobenius functional on a Lie algebra f is an element F ∈ f such that the bilinear form BF (x, y) = F ([x, y]) is non-degenerate. If such an F exists then f is a Frobenius Lie algebra and the element F̂ ∈ f such that F (x) = F ([F̂ , x]) for all x ∈ f will be called the principal element. In this note all Lie algebras will tacitly be complex but much of what is said holds more generally and probably even for most finite characteristics. One reason for the interest in Frobenius Lie algebras f is that if M is the matrix of the non-degenerate form B = BF relative to some basis x1, . . . , xm of f then R = ∑ i,j(M )ijxi ∧ xj is a constant solution to the classical Yang-Baxter equation. A simple Lie algebra itself can never be Frobenius but many subalgebras are. For sln, where we may assume the simple roots numbered from 1 to n−1, basic results of Elashvili [4], [2] assert in particular that the ith maximal parabolic subalgebra (obtained by deleting the ith negative root) is Frobenius if and only