The Principal Element of a Frobenius Lie Algebra

Details The Principal Element of a Frobenius Lie Algebra The Principal Element of a Frobenius Lie Algebra

2008-01-31

arxiv.org/abs/0801.4808v2

Letters in Mathematical Physics, Vol. 88, 2009, 333-341

Mathematics

Representation Theory

10 pages

Scientific paper

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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