Chevalley bases for Lie algebras and the combinatorics of Kac's asymmetry functionRichard Green, CU |
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November 19, 2009 AbstractA Chevalley basis for a Lie algebra over the complex numbers is a basis for the algebra all of whose structure constants are integers. One of the standard constructions of Chevalley bases for simple Lie algebras over the complex numbers involves a certain function $\varepsilon$ taking values in the set {+1, -1}; this function is sometimes called Kac's asymmetry function. Although the definition of this function may appear mysterious at first, I will argue that it has some natural combinatorial interpretations. Along the way, I will present combinatorial constructions of some interesting nonassociative algebras, including the loop algebras associated to simple Lie algebras over the complex numbers. |