Last updates: 18 February 2012
This section gives a brief treatment of the theory of Chevalley groups. The primary reference
is [St] and the extensions to the Kac-Moody case are found in [Ti].
Let be a Cartan matrix and let
be the real roots of the corresponding
Borcherds-Kac-Moody Lie algebra . Let be the
enveloping algebra of . For each
fix a choice of
in (2.18) (a choice of
). Use the notation
Following [Ti, 3.2], a prenilpotent pair is a pair of roots
such that there exists
This condition guarantees that the Lie subalgebra of
be a prenilpotent pair and let
be as in (2.18).
By [St, Lemma 15] there exist unique integers
Let be a commutative ring. The Steinberg group
for prenilpotent pairs
. In St define
Let be a
-lattice in which is stable under
the -action and such that
with as in (2.2). With
the Tits group
with the relations coming from the third equation of (1.3) and the additional relations
be given by
(see [CC, p48] and [Ti, (3.3)]). By [St, Lemma 37] (see also [Ti, 3.7a])
Thus has a symmetry under the subgroup
If is big enough then is the
normaliser of in [St, Ex(b) p36] and,
by [St, Lemma 27], the homomorphism
Remark 1. [Ti, 3.7b]
If and the first relation of (1.5)
holds in then there is a surjective homomorphism
By [St, Lemma 22], the elements
automatically commute with each
so that .
In many cases is the universal central extension of
(see [Ti, 3.7c] and [St, Theorems 10-12]).
Remark 2. The algebra
in (2.12) is generated by
is integrable if ,
act locally nilpotently so that
are well defined operators on
. The Chevalley group
is the subgroup of
generated by the operators in (1.10).
To do this integrally use a Kostant
-form and choose a lattice in
(see [Ti, 4.3-4] and [St, Ch1]). The
is the group
generated by the
and the additional relations coming from forcing an element to be 1 if it acts by 1 on
This is essentially the Chevalley group
for the case when
is the adjoint representation and so
There are surjective homomorphisms
See [Kac, Exercises 3.16-19] and [Ti, Proposition 1].
Remark 3. [St, Lemma 28] In the setting of Remark 2 let be the subgroup of generated by for Then
and if is big enough
where is a -basis of the -span of the weights of [St, Lemma 35].
Notes and References
These notes are a retyped version of Section 3 of [PRS].
R. Carter and Y. Chen, Automorphisms of affine Kac-Moody groups and related
Chevalley groups over rings, J. Algebra 155 (1) (1993) 44-94.
J. Parkinson, A. Ram and C. Schwer, Combinatorics in affine flag varieties,
J. Algebra 321 (2009) 3469-3493.
R. Steinberg, Lecture Notes on Chevalley groups, Yale University, 1967.
J. Tits, Uniqueness and presentation of Kac-Moody groups over fields,
J. Algebra 105 (2) (1987) 542-573,