In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .
Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.
One of the first uses for the theory was to define the Chevalley groups.
Examples For a positive integer , the general linear group over a field , consisting of all invertible matrices, is a linear algebraic group over . It contains the subgroups
consisting of matrices of the form, resp., and . The group is an example of a unipotent linear algebraic group, the group is an example of a solvable algebraic group called the Borel subgroup of . It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of is conjugated into . Any unipotent subgroup can be conjugated into .
Another algebraic subgroup of is the special linear group of matrices with determinant 1.
The group is called the multiplicative group, usually denoted by . The group of -points is the multiplicative group of nonzero elements of the field . The additive group , whose -points are isomorphic to the additive group of , can also be expressed as a matrix group, for example as the subgroup in :
These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups). Every representation of the multiplicative group is a direct sum of irreducible representations. (Its irreducible representations all have dimension 1, of the form for an integer .) By contrast, the only irreducible representation of the additive group is the trivial representation. So every representation of (such as the 2-dimensional representation above) is an iterated extension of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.
Definitions For an algebraically closed field k, much of the structure of an algebraic variety X over k is encoded in its set X(k) of k-rational points, which allows an elementary definition of a linear algebraic group. 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. Then a linear algebraic group G over an algebraically closed field k is a subgroup G(k) of the abstract group GL(n,k) for some natural number n such that G(k) is defined by the vanishing of some set of regular functions.
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). (Thus an algebraic group G over k is not just the abstract group G(k), but rather the whole family of groups G(R) for commutative k-algebras R; this is the philosophy of describing a scheme by its functor of points.)
In either language, one has the notion of a homomorphism of linear algebraic groups. For example, when k is algebraically closed, a homomorphism from G ⊂ GL(m) to H ⊂ GL(n) is a homomorphism of abstract groups G(k) → H(k) which is defined by regular functions on G. This makes the linear algebraic groups over k into a category. In particular, this defines what it means for two linear algebraic groups to be isomorphic.
In the language of schemes, a linear algebraic group G over a field k is in particular a group scheme over k, meaning a scheme over k together with a k-point 1 ∈ G(k) and morphisms
over k which satisfy the usual axioms for the multiplication and inverse maps in a group (associativity, identity, inverses). A linear algebraic group is also smooth and of finite type over k, and it is affine (as a scheme). Conversely, every affine group scheme G of finite type over a field k has a faithful representation into GL(n) over k for some n.Milne (2017), Corollary 4.10. An example is the embedding of the additive group Ga into GL(2), as mentioned above. As a result, one can think of linear algebraic groups either as matrix groups or, more abstractly, as smooth affine group schemes over a field. (Some authors use "linear algebraic group" to mean any affine group scheme of finite type over a field.)
For a full understanding of linear algebraic groups, one has to consider more general (non-smooth) group schemes. For example, let k be an algebraically closed field of characteristic p > 0. Then the homomorphism f: Gm → Gm defined by x ↦ xp induces an isomorphism of abstract groups k* → k*, but f is not an isomorphism of algebraic groups (because x1/p is not a regular function). In the language of group schemes, there is a clearer reason why f is not an isomorphism: f is surjective, but it has nontrivial kernel, namely the group scheme μp of pth roots of unity. This issue does not arise in characteristic zero. Indeed, every group scheme of finite type over a field k of characteristic zero is smooth over k.Milne (2017), Corollary 8.39. 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 reduced, where is an algebraic closure of k.Milne (2017), Proposition 1.26(b).
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
Basic notions 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. So there is a group extension
where F is a finite algebraic group. (For k algebraically closed, F can be identified with an abstract finite group.) Because of this, the study of algebraic groups mostly focuses on connected groups.
Various notions from abstract group theory can be extended to linear algebraic groups. It is straightforward to define what it means for a linear algebraic group to be commutative, nilpotent, or solvable, by analogy with the definitions in abstract group theory. For example, a linear algebraic group is solvable if it has a composition series of linear algebraic subgroups such that the quotient groups are commutative. 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.
One may ask to what extent the properties of a connected linear algebraic group G over a field k are determined by the abstract group G(k). 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.Borel (1991), Theorem 18.2 and Corollary 18.4. For example, under the assumptions mentioned, G is commutative, nilpotent, or solvable if and only if G(k) has the corresponding property.
The assumption of connectedness cannot be omitted in these results. For example, let G be the group μ3 ⊂ GL(1) of cube roots of unity over the rational numbers Q. Then G is a linear algebraic group over Q for which G(Q) = 1 is not Zariski dense in G, because is a group of order 3.
Over an algebraically closed field, there is a stronger result about algebraic groups as algebraic varieties: every connected linear algebraic group over an algebraically closed field is a rational variety.Borel (1991), Remark 14.14.
The Lie algebra of an algebraic group The Lie algebra of an algebraic group G can be defined in several equivalent ways: as the tangent space T1(G) at the identity element 1 ∈ G(k), or as the space of left-invariant derivations. 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).Milne (2017), section 10.e. The Lie bracket of two derivations is defined by [D1, D2] =D1D2 − D2D1.
The passage from G to is thus a process of differentiation. For an element x ∈ G(k), the derivative at 1 ∈ G(k) of the conjugation map G → G, g ↦ xgx−1, is an automorphism of , giving the adjoint representation:
Over a field of characteristic zero, a connected subgroup H of a linear algebraic group G is uniquely determined by its Lie algebra .Borel (1991), section 7.1. But not every Lie subalgebra of corresponds to an algebraic subgroup of G, as one sees in the example of the torus G = (Gm)2 over C. In positive characteristic, there can be many different connected subgroups of a group G with the same Lie algebra (again, the torus G = (Gm)2 provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.
Semisimple and unipotent elements
For an algebraically closed field k, a matrix g in GL(n,k) is called semisimple if it is diagonalizable, and unipotent if the matrix g − 1 is nilpotent. Equivalently, g is unipotent if all eigenvalues of g are equal to 1. The Jordan canonical form for matrices implies that every element g of GL(n,k) can be written uniquely as a product g = gssgu such that gss is semisimple, gu is unipotent, and gss and gu commute with each other.