its Lie algebra is the vector space of all n × n matrices over k. Let T be the subgroup of diagonal matrices in G. Then the root-space decomposition expresses as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions (i, j). Writing L1,...,Ln for the standard basis for the weight lattice X(T) ≅ Zn, the roots are the elements Li − Lj for all i ≠ j from 1 to n. The roots of a semisimple group form a root system; this is a combinatorial structure which can be completely classified. More generally, the roots of a reductive group form a root datum, a slight variation. The Weyl group of a reductive group G means the quotient group of the normalizer of a maximal torus by the torus, W = NG(T)/T. The Weyl group is in fact a finite group generated by reflections. For example, for the group GL(n) (or SL(n)), the Weyl group is the symmetric group Sn. There are finitely many Borel subgroups containing a given maximal torus, and they are permuted simply transitively by the Weyl group (acting by conjugation). A choice of Borel subgroup determines a set of positive roots Φ+ ⊂ Φ, with the property that Φ is the disjoint union of Φ+ and −Φ+. Explicitly, the Lie algebra of B is the direct sum of the Lie algebra of T and the positive root spaces: For example, if B is the Borel subgroup of upper-triangular matrices in GL(n), then this is the obvious decomposition of the subspace of upper-triangular matrices in . The positive roots are Li − Lj for 1 ≤ i < j ≤ n. A simple root means a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The number r of simple roots is equal to the rank of the commutator subgroup of G, called the semisimple rank of G (which is simply the rank of G if G is semisimple). For example, the simple roots for GL(n) (or SL(n)) are Li − Li+1 for 1 ≤ i ≤ n − 1. Root systems are classified by the corresponding Dynkin diagram, which is a finite graph (with some edges directed or multiple). The set of vertices of the Dynkin diagram is the set of simple roots. In short, the Dynkin diagram