First, define a function from the abstract group GL(n,k) to k to be regular if it can be written as a polynomial in the entries of an n×n matrix A and in 1/det(A), where det is the determinant.
For an arbitrary field k, algebraic varieties over k are defined as a special case of schemes over k. In that language, a linear algebraic group G over a field k is a smooth closed subgroup scheme of GL(n) over k for some natural number n. In particular, G is defined by the vanishing of some set of regular functions on GL(n) over k, and these functions must have the property that for every commutative k-algebra R, G(R) is a subgroup of the abstract group GL(n,R).
Indeed, every group scheme of finite type over a field k of characteristic zero is smooth over k.[2] A group scheme of finite type over any field k is smooth over k if and only if it is geometrically reduced, meaning that the base change
is an algebraic closure of k.[3] Since an affine scheme X is determined by its ring O(X) of regular functions, an affine group scheme G over a field k is determined by the ring O(G) with its structure of a Hopf algebra (coming from the multiplication and inverse maps on G).
This gives an equivalence of categories (reversing arrows) between affine group schemes over k and commutative Hopf algebras over k. For example, the Hopf algebra corresponding to the multiplicative group Gm = GL(1) is the Laurent polynomial ring k[x, x−1], with comultiplication given by For a linear algebraic group G over a field k, the identity component Go (the connected component containing the point 1) is a normal subgroup of finite index.
Also, the normalizer, the center, and the centralizer of a closed subgroup H of a linear algebraic group G are naturally viewed as closed subgroup schemes of G. If they are smooth over k, then they are linear algebraic groups as defined above.
A useful result in this direction is that if the field k is perfect (for example, of characteristic zero), or if G is reductive (as defined below), then G is unirational over k. Therefore, if in addition k is infinite, the group G(k) is Zariski dense in G.[4] For example, under the assumptions mentioned, G is commutative, nilpotent, or solvable if and only if G(k) has the corresponding property.
If k is algebraically closed, a derivation D: O(G) → O(G) over k of the coordinate ring of G is left-invariant if for every x in G(k), where λx: O(G) → O(G) is induced by left multiplication by x.
For an arbitrary field k, left invariance of a derivation is defined as an analogous equality of two linear maps O(G) → O(G) ⊗O(G).
(These properties are in fact independent of the choice of a faithful representation of G.) If the field k is perfect, then the semisimple and unipotent parts of a k-point of G are automatically in G. That is (the Jordan decomposition): every element g of G(k) can be written uniquely as a product g = gssgu in G(k) such that gss is semisimple, gu is unipotent, and gss and gu commute with each other.
For example, the group of diagonal matrices in GL(n) over k is a maximal torus in GL(n), isomorphic to (Gm)n. A basic result of the theory is that any two maximal tori in a group G over an algebraically closed field k are conjugate by some element of G(k).
For an arbitrary field k, a torus T over k means a linear algebraic group over k whose base change
[10] For a linear algebraic group G over a general field k, one cannot expect all maximal tori in G over k to be conjugate by elements of G(k).
[11] As a result, it makes sense to define the k-rank or split rank of a group G over k as the dimension of any maximal split torus in G over k. For any maximal torus T in a linear algebraic group G over a field k, Grothendieck showed that
A basic result of the theory is that any two Borel subgroups of a connected group G over an algebraically closed field k are conjugate by some element of G(k).
For example, the subgroups P ⊂ GL(3) over k that contain the Borel subgroup B of upper-triangular matrices are B itself, the whole group GL(3), and the intermediate subgroups The corresponding projective homogeneous varieties GL(3)/P are (respectively): the flag manifold of all chains of linear subspaces with Vi of dimension i; a point; the projective space P2 of lines (1-dimensional linear subspaces) in A3; and the dual projective space P2 of planes in A3.
Likewise, GL(n) is reductive but not semisimple (because its center Gm is a nontrivial smooth connected solvable normal subgroup).
Remarkably, Claude Chevalley gave a complete classification of the reductive groups over an algebraically closed field: they are determined by root data.
[23] In particular, simple groups over an algebraically closed field k are classified (up to quotients by finite central subgroup schemes) by their Dynkin diagrams.
By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field.
Chevalley showed that the irreducible representations of a split reductive group over a field k are finite-dimensional, and they are indexed by dominant weights.
The Borel–Weil theorem gives a geometric construction of the irreducible representations of a reductive group G in characteristic zero, as spaces of sections of line bundles over the flag manifold G/B.
The representation theory of reductive groups (other than tori) over a field of positive characteristic p is less well understood.
Andersen, Jantzen and Soergel (1994) determined these characters (proving Lusztig's conjecture) when the characteristic p is sufficiently large compared to the Coxeter number of the group.
Part of the theory of group actions is geometric invariant theory, which aims to construct a quotient variety X/G, describing the set of orbits of a linear algebraic group G on X as an algebraic variety.
In the positive direction, the ring of invariants is finitely generated if G is reductive, by Haboush's theorem, proved in characteristic zero by Hilbert and Nagata.
Geometric invariant theory involves further subtleties when a reductive group G acts on a projective variety X.
Even the case of elliptic curves (abelian varieties of dimension 1) is central to number theory, with applications including the proof of Fermat's Last Theorem.
In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes.
For example, over a field of characteristic zero, RepG is a semisimple category if and only if the identity component of G is pro-reductive.