  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.
