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arXiv · 0801.4808

The Principal Element of a Frobenius Lie Algebra

Murray Gerstenhaber·Anthony Giaquinto
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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.

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Murray Gerstenhaber, Anthony Giaquinto. 2009-10-21. The Principal Element of a Frobenius Lie Algebra. https://doi.org/10.1007/s11005-009-0321-8

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