1.2. SIMPLE LIE ALGEBRAS AND GROUPS 5
G2
F4
E8
E7
Bn
A
n
Cn
E6
Dn
Figure 1.1. The list of Dynkin diagrams
1.2.3. The Borel (resp., opposite Borel) subalgebra b+ (resp., b−) of g relative
to h is spanned by h and vectors eα, α Φ+ (resp., eα, α Φ−). Let n+ = [b+, b+]
and n− = [b−, b−] be the corresponding maximal nilpotent subalgebras of g. The
connected subgroups that correspond to g, h, b+, b−, n+, n− will be denoted by G,
H, B+, B−, N+, N−.
The weight lattice P consists of elements γ
h∗
such that γ(hi) Z for
i [1,r]. The basis of fundamental weights ω1,...,ωr in P is defined by ωj(hi) =
ωj|αi = δij, i, j [1,r]. Every weight γ P defines a multiplicative character
a
of H as follows: let a = exp(h) for some h h, then = eγ(h). Positive
simple roots, fundamental weights and entries of the Cartan matrix are related via
(1.9) αi =
r
j=1
ajiωj = 2ωi +
r
j=1
j=i
ajiωj.
The Weyl group W is defined as a quotient W = NormGH/H. For each of the
embeddings ρi: SL2 G, i [1,r], defined in (1.7), put
¯i s = ρi
0 −1
1 0
NormG H
and si = ¯iH s W . All si are of order 2, and together they generate W . We
can identify si with the elementary reflection sαi , αi Π, in h0 defined by (1.5).
Consequently, W can be identified with the Coxeter group generated by reflections
s1,...,sr. The Weyl group acts on the weight lattice. For any w W and γ P the
action γ is defined by expanding w into a product of elementary reflections
and applying the reflections one by one to γ.
A reduced decomposition of an element w W is a representation of w as a
product w = si1 · · · sil of the smallest possible length. A reduced decomposition is
not unique, but the number l of reflections in the product depends only on w and
is called the length of w and denoted by l(w). The sequence of indices (i1,...,il)
that corresponds to a given reduced decomposition of w is called a reduced word for
w. The unique element of W of maximal length (also called the longest element of
W ) will be denoted by w0. Braid relations for ¯i s guarantee that the representative
¯ w NormG H can be unambiguously defined for any w W by requiring that
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