Geodesic
On the combinatorics of Kac's asymmetry function
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 217-235

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We use categories to recast the combinatorial theory of full heaps, which are certain labelled partially ordered sets that we introduced in previous work. This gives rise to a far simpler set of definitions, which we use to outline a combinatorial construction of the so-called loop algebras associated to affine untwisted Kac--Moody algebras. The finite convex subsets of full heaps are equipped with a statistic called parity, and this naturally gives rise to Kac's asymmetry function. The latter is a key ingredient in understanding the (integer) structure constants of simple Lie algebras with respect to certain Chevalley bases, which also arise naturally in the context of heaps.
Classification : 06A07, 17B20, 17B67
Keywords: Lie algebra; Chevalley basis; heap 
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Green, R. M. On the combinatorics of Kac's asymmetry function. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 217-235. http://geodesic.mathdoc.fr/item/CMUC_2010__51_2_a6/