reductive groups, up to isomorphism. For a compact Lie group K with complexification G, the inclusion from K into the complex reductive group G(C) is a homotopy equivalence, with respect to the classical topology on G(C). For example, the inclusion from the unitary group U(n) to GL(n,C) is a homotopy equivalence. For a reductive group G over a field of characteristic zero, all finite-dimensional representations of G (as an algebraic group) are completely reducible, that is, they are direct sums of irreducible representations. That is the source of the name "reductive". Note, however, that complete reducibility fails for reductive groups in positive characteristic (apart from tori). In more detail: an affine group scheme G of finite type over a field k is called linearly reductive if its finite-dimensional representations are completely reducible. For k of characteristic zero, G is linearly reductive if and only if the identity component Go of G is reductive. For k of characteristic p>0, however, Masayoshi Nagata showed that G is linearly reductive if and only if Go is of multiplicative type and G/Go has order prime to p. Roots The classification of reductive algebraic groups is in terms of the associated root system, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups. Let G be a split reductive group over a field k, and let T be a split maximal torus in G; so T is isomorphic to (Gm)n for some n, with n called the rank of G. Every representation of T (as an algebraic group) is a direct sum of 1-dimensional representations. A weight for G means an isomorphism class of 1-dimensional representations of T, or equivalently a homomorphism T → Gm. The weights form a group X(T) under tensor product of representations, with X(T) isomorphic to the product of n copies of the integers, Zn. The adjoint representation is the action of G by conjugation on its Lie algebra . A root of G means a nonzero weight that occurs in the action of T ⊂ G on . The subspace of corresponding to each root is 1-dimensional, and the subspace of fixed by T is exactly the Lie algebra of T. Therefore, the Lie algebra of G decomposes into together with 1-dimensional subspaces indexed by the set Φ of roots: For example, when G is the group GL(n),