A 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 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.
Chevalley bases for Lie algebras and the combinatorics of Kac's asymmetry function
Chevalley bases for Lie algebras and the combinatorics of Kac's asymmetry function