In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is semisimple, i.e. a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.
Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number field, the classification is well understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field k, or as minor variants of that construction.
Reductive groups have a rich representation theory in various contexts. One can study the representations of a reductive group G over a field k as an algebraic group, which are actions of G on k-vector spaces. Also, one can study the complex representations of the group G(k) when k is a finite field, or the infinite-dimensional unitary representations of a real reductive group, or the automorphic representations of an adelic algebraic group. The structure theory of reductive groups is used in all these areas.
Definitions
A linear algebraic group over a field k is defined as a smooth closed subgroup scheme of GL(n) over k, for some positive integer n. Equivalently, a linear algebraic group over k is a smooth affine group scheme over k.
With the unipotent radical A connected linear algebraic group over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of is trivial. More generally, a connected linear algebraic group over an algebraically closed field is called reductive if the largest smooth connected unipotent normal subgroup of G is trivial.SGA 3 (2011), v. 3, Définition XIX.1.6.1. This normal subgroup is called the unipotent radical and is denoted Ru(G). (Some authors do not require reductive groups to be connected.) A group G over an arbitrary field k is called semisimple or reductive if the base change is semisimple or reductive, where is an algebraic closure of k. (This is equivalent to the definition of reductive groups in the introduction when k is perfect.Milne (2017), Proposition 21.60.) Any torus over k, such as the multiplicative group Gm, is reductive.
With representation theory Over fields of characteristic zero another equivalent definition of a reductive group is a connected group G admitting a faithful semisimple representation which remains semisimple over its algebraic closure kal. page 424.
Simple reductive groups A linear algebraic group G over a field k is called simple (or k-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup of G over k is trivial or equal to G.Conrad (2014), after Proposition 5.1.17. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer n at least 2 and any field k, the group SL(n) over k is simple, and its center is the group scheme μn of n-th roots of unity.
A central isogeny of reductive groups is a surjective homomorphism with kernel a finite central subgroup scheme. Every reductive group over a field admits a central isogeny from the product of a torus and some simple groups. For example, over any field k,
It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect field k, that can be avoided: a linear algebraic group G over k is reductive if and only if every smooth connected unipotent normal k-subgroup of G is trivial. For an arbitrary field, the latter property defines a pseudo-reductive group, which is somewhat more general.
Split-reductive groups A reductive group G over a field k is called split if it contains a split maximal torus T over k; that is, a split torus in G whose base change to is a maximal torus in . It is equivalent to say that T is a split torus in G that is maximal among all k-tori in G.Borel (1991), 18.2(i). These kinds of groups are useful because their classification can be described through combinatorical data called root data.
Examples
GLn and SLn A fundamental example of a reductive group is the general linear group of invertible n × n matrices over a field k, for a natural number n. In particular, the multiplicative group Gm is the group GL(1), and so its group Gm(k) of k-rational points is the group k× of nonzero elements of k under multiplication. Another reductive group is the special linear group SL(n) over a field k, the subgroup of matrices with determinant 1. In fact, SL(n) is a simple algebraic group for n at least 2.
O(n), SO(n), and Sp(n) An important simple group is the symplectic group Sp(2n) over a field k, the subgroup of GL(2n) that preserves a nondegenerate alternating bilinear form on the vector space k2n. Likewise, the orthogonal group O(q) is the subgroup of the general linear group that preserves a nondegenerate quadratic form q on a vector space over a field k. The algebraic group O(q) has two connected components, and its identity component SO(q) is reductive, in fact simple for q of dimension n at least 3. (For k of characteristic 2 and n odd, the group scheme O(q) is in fact connected but not smooth over k. The simple group SO(q) can always be defined as the maximal smooth connected subgroup of O(q) over k.) When k is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group SO(n). For a general field k, different quadratic forms of dimension n can yield non-isomorphic simple groups SO(q) over k, although they all have the same base change to the algebraic closure .
Tori The group and products of it are called the algebraic tori. They are examples of reductive groups since they embed in through the diagonal, and from this representation, their unipotent radical is trivial. For example, embeds in from the map
Non-examples
Any unipotent group is not reductive since its unipotent radical is itself. This includes the additive group . The Borel group of has a non-trivial unipotent radical of upper-triangular matrices with on the diagonal. This is an example of a non-reductive group which is not unipotent.
Associated reductive group Note that the normality of the unipotent radical implies that the quotient group is reductive. For example,
Other characterizations of reductive groups Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex 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.Milne (2017), Theorem 22.42. 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.Milne (2017), Corollary 22.43. 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.Demazure & Gabriel (1970), Théorème IV.3.3.6.
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.Milne (2017), Theorem 12.12. 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.Milne (2017), Theorem 21.11. 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), 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.Milne (2017), Corollary 21.12. 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 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).Milne (2017), Proposition 17.53. 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.