Algebraic Preliminaries

§ 1. Linear algebraic groups.

Let F be a field of characteristic zero and let V be a vector space of dimension n over F. We denote by End(V) the algebra of all F-linear transformations of V and by GL(V) the group of units in End(V), i. e., the group of all non-singular F-linear transformations of V. When a basis of V is fixed, V is identified with the space of n-tuples Fn, and so End(V) and GL(V) are also identified with Mn(F), the full matrix algebra of degree n over F, and GLn(F), the general linear group of degree n over F, respectively.

In the following, we fix an algebraically closed extension F of F once and for all. A subgroup G of GLn(F) is called a (linear) algebraic group defined over F, if G is the set of common zeros of a (finite) system of polynomial equations in n2 matrix entries with coefficients in F. If we embed G in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

G may be viewed as an affine variety in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The dimension of an algebraic group G is the dimension of the underlying affine variety. For instance, GLn(F) and SLn(F) (= {g[member of]GLn(F)|det(g)=1}) are algebraic groups of dimension n2 and n2 -1, respectively.

Let G and G' be algebraic groups defined over F in GLn(F) and GLn,(F), respectively. A group homomorphism φ: G -> G' is called a (rational) homomorphism defined over F, or for short, an F-homomorphism, if φ is a restriction to G of a polynomial map with coefficients in F of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. φ is called a (rational) isomorphism defined over F, or an F-isomorphism, if φ is a group isomorphism and both φ and φ-1 are F-homomorphisms. A bijective F-homomorphism is necessarily an F-isomorphism. (This is not true in general when the characteristic is positive.) Linear algebraic groups which are F-isomorphic to each other may be viewed as matrix expressions of one and the same (affine) algebraic group defined over F.

A subgroup H of an algebraic group G defined over F is called F-closed if H itself is an algebraic group defined over F, i. e., defined by polynomial equations with coefficients in F (for one and hence all matrix expression of G). When H is an F-closed normal subgroup of G, it is known (e. g., Borel [13]) that the factor group G/H has a natural structure of (linear) algebraic group defined over F (determined up to an F-isomorphism) such that the canonical homomorphism G -> G/K is an F-homomorphism; one then has the relation

dim G/H - dim G - dim H.

It is also known (Borel, loc. cit.) that, if φ: G->G' is an F-homomorphism, the image φ(G) is an F-closed subgroup of G', the kernel N = φ-1(e') is an F-closed normal subgroup of G, and one has an F-isomorphism G/N [congruent to] φ(G). The notions of a direct product G1 × G2 and a semi-direct product G1 · G2 of two algebraic groups G1 and G2 are defined in the natural manner.

For an algebraic group G defined over F, we denote by Gz the Zariski connected component of the underlying affine variety of G containing the unit element e. Then G2 is an F-closed normal subgroup of G of finite index, and the coset decomposition of G with respect to Gz coincides with the decomposition of the underlying affine variety of G into the union of irreducible components. Thus Gz is the unique irreducible component of G containing e; an algebraic group G is Zariski connected if and only if it is irreducible as affine variety.

For an algebraic group G defined over F in GLn(F), the subgroup G{F) = G[intersection] GLn(F) is called the group of "F-rational points" in G. Clearly the group G(F) is well-determined, independently of the matrix expression of G. More generally, if φ: G->G' is an F-homomorphism, φ induces an (abstract) group homomorphism φF:G(F)->G'(F). It is known that, when G is Zariski connected, or when F is algebraically closed, G(F) is Zariski dense in G.

In general, let G be an (abstract) subgroup of GLn(F) and let G be the Zariski closure of G in GLn(F). Then G is an algebraic group defined over F and one has G]subset]G(F). When we have the equality G=G(F), we call G an F-group and G the associated algebraic group. Let G and G' be F-groups with the associated algebraic groups G and G'. By definition, an "F-homomorphism" of G into G' is the restriction φF to G of an F-homomorphism φ:G->G' (which is uniquely determined by φF). The notions of "F-isomorphisms", "F-subgroups", etc. of F-groups are defined in a similar manner. For an F-group G with the associated algebraic group G, Gz=Gz(F) will be called the "Zariski connected component" of G; Gz is an F-group with the associated algebraic group Gz. It should be noted that some properties of F-homomorphisms of algebraic groups mentioned above break down in general for F-homomorphisms of F-groups. To see this, let φF: G->G' be an F-homomorphism of F-groups coming from an F-homomorphism φ: G->G' of the associated algebraic groups, and let N=Ker φF, N = Ker φ. Then it is clear that N = N [intersection] G, so that N is a normal F-subgroup of G. However, if N1 denotes the algebraic group associated with N, one has only Nz [subset] N1 [subset] N. On the other hand, φF(G) is Zariski dense in φ(G), but may not be equal to φ(G)(F). Thus we have only an injective homomorphism G/N->φ(G)(F) and, going to the Zariski closures, the induced F-homomorphism G/N1->φ(G), which may not be injective. For instance, for G=G'=GL1(R)=Rx and the R-homomorphism φ: x[??]x4, one has |N|=4, |N|=|N1|=2, and φ(G)(R)=Rx, φR(Rx)=Rx+ (the multiplicative group of positive real numbers).

When F=R or C, an F-group G in GLn(F) is a closed subgroup of GLn(F) in the usual topology, so that G has a unique structure of real (or complex) Lie group of the same dimension. We denote the identity connected component of G in the usual topology by G°. Then clearly and by a theorem of Whitney [1] the index (Gz:Gº) is finite. In particular, in the case F=C, one has G° =Gz, and so G is Zariski connected (i. e., the associated algebraic group G is Zariski connected) if and only if G is connected in the usual topology. This is false for F=R; e. g., G=Rx and SO(p, q) (p > 0, q > 0) are Zariski connected, but one has (G: G°)=2.

For a Zariski connected F-group G we denote by F[G] the "affine ring" of the associated algebraic group G over F, i. e., the ring of polynomial functions on G (viewed as an affine variety in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) with coefficients in F. F[G] is then an integral domain, finitely generated over F, and G acts on F[G] by the left translation λg1(g1 [member of] G):

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By definition, a derivation of F[G] is an F-linear transformation X of F[G] satisfying the condition

(1.2) X(f1 · f2) = (Xf1) · f2 + f1 · (Xf2) (f1, f2 [member of] F[G]).