The Principal Element of a Frobenius Lie Algebra

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We introduce the notion of the principal element of a Frobenius Lie algebra 𝔣. The principal element corresponds to a choice of E ∈ such that F[–, –] non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to sln, 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 sln, the principal element coincides with semisimple element of the principal three-dimensional subalgebra. We also show that Frobenius algebras are stable under deformation.

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