Previous |  Up |  Next


Lie algebra; Chevalley basis; heap
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.
[1] Carter R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley, New York, 1985. MR 0794307 | Zbl 0567.20023
[2] Diekert V., Rozenberg G. (eds.): The Book of Traces. World Scientific, Singapore, 1995. MR 1478992
[3] Green R.M.: Full heaps and representations of affine Kac–Moody algebras. Internat. Electron. J. Algebra 2 (2007), 138–188. MR 2320733 | Zbl 1134.17010
[4] Green R.M.: Full heaps and representations of affine Weyl groups. Internat. Electron. J. Algebra 3 (2008), 1–42. MR 2369402 | Zbl 1184.20037
[5] Green R.M.: Combinatorics of minuscule representations. in preparation.
[6] Kac V.G.: Infinite Dimensional Lie Algebras. third edition, Cambridge University Press, Cambridge, 1990. MR 1104219 | Zbl 0925.17021
[7] Kashiwara M.: On crystal bases of the $q$-analogue of universal enveloping algebras. Duke Math. J. 63 (1991), 465–516. DOI 10.1215/S0012-7094-91-06321-0 | MR 1115118 | Zbl 0739.17005
[8] Littelmann P.: A Littlewood–Richardson type rule for symmetrizable Kac–Moody algebras. Invent. Math. 116 (1994), 329–346. DOI 10.1007/BF01231564 | MR 1253196
[9] McGregor-Dorsey Z.S.: Full heaps over Dynkin diagrams of type $\widetilde{A}$. M.A. thesis, University of Colorado at Boulder, 2008.
[10] Stembridge J.R.: Minuscule elements of Weyl groups. J. Algebra 235 (2001), 722–743. DOI 10.1006/jabr.2000.8488 | MR 1805477 | Zbl 0973.17034
[11] Vavilov N.A.: Can one see the signs of structure constants?. St Petersburg Math. J. 19 (2008), 519–543. DOI 10.1090/S1061-0022-08-01008-X | MR 2381932
[12] Viennot G.X.: Heaps of pieces, I: basic definitions and combinatorial lemmas. in Combinatoire Énumérative (ed. G. Labelle and P. Leroux), Springer, Berlin, 1986, pp. 321–350. MR 0927773 | Zbl 0792.05012
[13] Wildberger N.J.: A combinatorial construction for simply laced Lie algebras. Adv. Appl. Math. 30 (2003), 385–396. DOI 10.1016/S0196-8858(02)00541-9 | MR 1979800 | Zbl 1023.17015
Partner of
EuDML logo