Full heaps and representations of affine Kac--Moody algebras

Abstract

We give a combinatorial construction, not involving a presentation, of almost all untwisted affine Kac--Moody algebras modulo their one-dimensional centres in terms of signed raising and lowering operators on a certain distributive lattice ${\Cal B}$. The lattice ${\Cal B}$ is constructed combinatorially as a set of ideals of a ``full heap'' over the Dynkin diagram, which leads to a kind of categorification of the positive roots for the Kac--Moody algebra. The lattice ${\Cal B}$ is also a crystal in the sense of Kashiwara, and its span affords representations of the associated quantum affine algebra and affine Weyl group. There are analogues of these results for two infinite families of twisted affine Kac--Moody algebras, which we hope to treat more fully elsewhere. By restriction, we obtain combinatorial constructions of the finite dimensional simple Lie algebras over $\complex$, except those of types $E_8$, $F_4$ and $G_2$. The Chevalley basis corresponding to an arbitrary orientation of the Dynkin diagram is then represented explicitly by raising and lowering operators. We also obtain combinatorial constructions of the spin modules for Lie algebras of types $B$ and $D$, which avoid Clifford algebras, and in which the action of Chevalley bases on the canonical bases of the modules may be explicitly calculated.