describes the angles between the simple roots and their relative lengths, with respect to a Weyl group-invariant inner product on the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below. For a split reductive group G over a field k, an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra of G, but also a copy of the additive group Ga in G with the given Lie algebra, called a root subgroup Uα. The root subgroup is the unique copy of the additive group in G which is normalized by T and which has the given Lie algebra. The whole group G is generated (as an algebraic group) by T and the root subgroups, while the Borel subgroup B is generated by T and the positive root subgroups. In fact, a split semisimple group G is generated by the root subgroups alone. Parabolic subgroups For a split reductive group G over a field k, the smooth connected subgroups of G that contain a given Borel subgroup B of G are in one-to-one correspondence with the subsets of the set Δ of simple roots (or equivalently, the subsets of the set of vertices of the Dynkin diagram). Let r be the order of Δ, the semisimple rank of G. Every parabolic subgroup of G is conjugate to a subgroup containing B by some element of G(k). As a result, there are exactly 2r conjugacy classes of parabolic subgroups in G over k. Explicitly, the parabolic subgroup corresponding to a given subset S of Δ is the group generated by B together with the root subgroups U−α for α in S. For example, the parabolic subgroups of GL(n) that contain the Borel subgroup B above are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as: By definition, a parabolic subgroup P of a reductive group G over a field k is a smooth k-subgroup such that the quotient variety G/P is proper over k, or equivalently projective over k. Thus the classification of parabolic subgroups amounts to a classification of the projective homogeneous varieties for G (with smooth stabilizer group; that is no restriction for k of characteristic zero). For GL(n), these are the flag varieties, parametrizing sequences of linear subspaces of given dimensions a1,...,ai contained in a fixed vector