Introduction
Linear algebraic groups are ubiquitous in mathematics. They occur in geometry,
topology, representation theory, number theory and many other areas. One viewpoint is to start from the theory of Lie groups. Remarkably, most Lie groups are
algebraic in the sense that they are defined by solutions of polynomial equations,
and furthermore, multiplication and taking inverses are polynomial maps. This is
the case for the linear group GL n (R), the orthogonal group O n (R), the symplectic
group Sp 2n (R), etc., and similarly for groups over the complex numbers C. We can
quote here Tannaka’s theorem that every compact subgroup of GL n (C) is algebraic.
However, we need to pay attention to the fact that morphisms between Lie groups
of algebraic nature have no reason to be so; the exponential map C → C × is a basic
example. Another viewpoint is to start from finite fields, as Jordan did in his Traité
des substitutions et des équations algébriques for the purpose of developing Galois
theory [102, II.II]. Both viewpoints lead to the theory of linear algebraic groups over
a base field k with algebraic closure k.
An algebraic k-subgroup G of the linear group GL n is given by a finite system
of polynomial equations in the coordinates (a i,j ) of GL n such that for each
(commutative, unital) k-algebra R the subset of R-points G(R) is a subgroup of the
group GL n (R) of invertible R-automorphisms of the R-module R n . It is important
here to distinguish between the algebraic k-subgroup G of GL n and the group G(k)
of k-points of G. A basic tool is the extension of scalars: if F is a field extension
of k, we can define the F -algebraic group G × k F by the same equations used to
define G, but now viewed as equations over F .
An important example is that of the orthogonal algebraic group attached to a
regular quadratic form q : k 2n → k. It is the algebraic k-subgroup of GL 2n given
by the system of quadratic equations q ◦ f = f . For each k-ring R, we have
O q (R) =
f ∈ GL 2n (R) | q ◦ f = q
.
The classification of algebraic groups is a fundamental problem. We are mostly
interested in elementary pieces called almost simple algebraic groups; see Sect. 2.1.2
for the precise definition. Over k, the classification is that of Cartan Killing; it is
xiii
Linear algebraic groups are ubiquitous in mathematics. They occur in geometry,
topology, representation theory, number theory and many other areas. One viewpoint is to start from the theory of Lie groups. Remarkably, most Lie groups are
algebraic in the sense that they are defined by solutions of polynomial equations,
and furthermore, multiplication and taking inverses are polynomial maps. This is
the case for the linear group GL n (R), the orthogonal group O n (R), the symplectic
group Sp 2n (R), etc., and similarly for groups over the complex numbers C. We can
quote here Tannaka’s theorem that every compact subgroup of GL n (C) is algebraic.
However, we need to pay attention to the fact that morphisms between Lie groups
of algebraic nature have no reason to be so; the exponential map C → C × is a basic
example. Another viewpoint is to start from finite fields, as Jordan did in his Traité
des substitutions et des équations algébriques for the purpose of developing Galois
theory [102, II.II]. Both viewpoints lead to the theory of linear algebraic groups over
a base field k with algebraic closure k.
An algebraic k-subgroup G of the linear group GL n is given by a finite system
of polynomial equations in the coordinates (a i,j ) of GL n such that for each
(commutative, unital) k-algebra R the subset of R-points G(R) is a subgroup of the
group GL n (R) of invertible R-automorphisms of the R-module R n . It is important
here to distinguish between the algebraic k-subgroup G of GL n and the group G(k)
of k-points of G. A basic tool is the extension of scalars: if F is a field extension
of k, we can define the F -algebraic group G × k F by the same equations used to
define G, but now viewed as equations over F .
An important example is that of the orthogonal algebraic group attached to a
regular quadratic form q : k 2n → k. It is the algebraic k-subgroup of GL 2n given
by the system of quadratic equations q ◦ f = f . For each k-ring R, we have
O q (R) =
f ∈ GL 2n (R) | q ◦ f = q
.
The classification of algebraic groups is a fundamental problem. We are mostly
interested in elementary pieces called almost simple algebraic groups; see Sect. 2.1.2
for the precise definition. Over k, the classification is that of Cartan Killing; it is
xiii