Linear algebraic group

2020 Mathematics Subject Classification: Primary: 14-XX Secondary: 14L [MSN][ZBL]

A linear algebraic group is an algebraic group that is isomorphic to an algebraic subgroup of a general linear group. An algebraic group $G$ is linear if and only if the algebraic variety $G$ is affine, that is, isomorphic to a Zariski-closed subvariety of an affine space (cf. also Zariski topology).

The theory of linear algebraic groups arose in the context of the Galois theory of solving linear differential equations by quadratures at the end of 19th century (S. Lie, E. Picard, L. Maurer), and the study of linear algebraic groups over the field of complex numbers was originally carried out by analogy with the theory of Lie groups by the method of Lie algebras. A linear algebraic group $G$ over the field of complex numbers $\mathbb C$ can be regarded as an analytic subgroup (cf. Analytic group) of a group ${\rm GL}(n,\mathbb C)$ (of invertible complex $n\times n$-matrices). The Lie algebra of $G$, defined in the usual way, is then a subalgebra of the Lie algebra of ${\rm GL}(n,\mathbb C)$. An exhaustive answer can be given to the question arising here — which Lie subalgebras correspond to algebraic (and not only analytic) subgroups of ${\rm GL}(n,\mathbb C)$ (see Lie algebra, algebraic 1)).

In the middle of the 20th century the methods of the theory of Lie groups and Lie algebras were completed and generalized by C. Chevalley. He subsequently applied these methods to the study of linear algebraic groups over arbitrary fields of characteristic zero (see [Ch]). For fields of non-zero characteristic the method of Lie algebras is less effective, so there naturally arose the need for a global investigation of linear algebraic groups by means of methods of algebraic geometry. The foundations of a global investigation of linear algebraic groups were laid by A. Borel (see [Bo2]), after which the theory of linear algebraic groups acquired the form of an orderly discipline (see [Ch3]). One of the main problems in the theory of linear algebraic groups is that of classifying linear algebraic groups up to isomorphism. To some extent this problem reduces to the classification of two types of linear algebraic groups, semi-simple and solvable, since an arbitrary linear algebraic group has a maximal connected solvable normal subgroup $H$ such that the quotient group $G/H$ is a semi-simple linear algebraic group. One of the main results in the theory of linear algebraic groups is Chevalley's classification of connected semi-simple linear algebraic groups over an algebraically closed field of arbitrary characteristic (see [Ch2], [Hu]). This classification is analogous to the classification of Cartan–Killing of complex semi-simple Lie algebras (cf. Lie algebra, semi-simple). The classification of Chevalley is based on the fact that in a semi-simple algebraic group one can construct analogues to the elements of the theory of Cartan–Killing, namely the Cartan subgroups (cf. Cartan subgroup), roots, etc. An important role is played here by Borel subgroups (cf. Borel subgroup) and maximal tori (cf. Algebraic torus).

Let $G$ be a semi-simple linear algebraic group (cf. Semi-simple algebraic group), $T$ a maximal torus, $N_G(T)$ the normalizer of $T$ in $G$ (cf. Normalizer of a subset), and let $N_G(T)/T$ be the Weyl group of $G$. The torus $T$ is contained in only finitely many Borel subgroups of $G$, which are permuted transitively under conjugation by elements of $N_G(T)$. One can construct certain monomorphisms of the additive group of the field into the Borel subgroups (containing $T$), which play the role of roots. The Bruhat decomposition of $G$ gives insight into the purely group-theoretical aspects of semi-simple groups (see [Bo], [Ch2], [Hu]). The final classification of semi-simple groups does not depend on the characteristic of the ground field and therefore coincides with the classification of complex semi-simple algebraic groups.

The classification up to $k $-isomorphism of semi-simple linear algebraic groups defined over a non-algebraically closed field $k$ is significantly more complicated; it depends essentially on the field $k$ and reduces to the classification of $k$-forms of algebraic groups (see Form of an algebraic group). The set of $k$-forms of an algebraic group $G$ defined over an algebraic closure $K$ of $k$ is in one-to-one correspondence with the one-dimensional Galois cohomology set $H^1(k,{\rm Aut}(G))$, where ${\rm Aut}(G)$ is the group of automorphisms of $G$. In those cases when the Galois group ${\rm Gal}(K/k)$ of the extension $K/k$ is known (for example, if $k$ is the field of real numbers or a finite field), this gives significant information about the forms of $G$. But the main difficulty of investigating this deep problem, which is still (1989) far from being solved, lies in the study of the $k$-structures of semi-simple linear algebraic groups.

In the case of an arbitrary field $k$, the maximal $k$-split tori play a role analogous to that of maximal algebraic tori in a group $G$ over an algebraically closed field. A $k$-split torus is a torus that is $k$-isomorphic to a direct product of one-dimensional groups ${\rm GL(1,k)}$ which are defined over $k$. Maximal $k$-split tori in $G$ are conjugate by elements of the group of $k$-rational points $G_k$ and their dimension is called the $k$-rank of $G$ (denoted by ${\rm rank}_k\;G$). If ${\rm rank}_k\;G > 0$, then $G$ is said to be $k$-isotropic, otherwise $k$-anisotropic. A semi-simple group $G$ is $k$-isotropic if and only if $G_k$ has unipotent elements other than the identity. For a simple simply-connected group $G$ there was the Kneser–Tits hypothesis that if ${\rm rank}_k\;G > 0$, then $G_k$ is generated by unipotent elements (see [Ti2]). In general it has been proved false (see [Pl3]). The role of a Borel subgroup in the case of an arbitrary field $k$ is played by a minimal parabolic $k$-subgroup, that is, a subgroup of $G$ containing a Borel subgroup which is defined over $k$ and is minimal for these properties. One can define the root system with respect to a maximal $k$-split torus in $G$ and a relative Weyl group (see [Bo3]). If $G$ has a $k$-split maximal torus, then these structural elements do not depend on the field $k$ and also determine the group up to $k$-isomorphism. Groups that have $k$-split maximal tori are said to be $k$-split or Chevalley groups (cf. Chevalley group). Chevalley's classification of semi-simple groups over an algebraically closed field carries over to $k$-split groups. Chevalley proved that every semi-simple group has a split $k$-form (even "a form over Z" , see [Ch2], [St]).

In the general case Borel and J. Tits [Bo3] proved the existence of an analogue of the Bruhat decomposition for a group $G_k$, in which the role of the Borel subgroups is played by the groups of $k$-rational points of minimal parabolic $k$-subgroups. This makes it possible to reduce to a large extent the classification of semi-simple linear algebraic groups of positive $k$-rank to the classification of semi-simple groups of $k$-rank zero, that is, to the classification of $k$-anisotropic groups. Namely, a semi-simple group $G$ is defined up to $k$-isomorphism by its isomorphism class over the algebraic closure of $k$, its $k$-index and the semi-simple $k$-anisotropic kernel (see Anisotropic kernel) (see [Ti]). In some cases it has been possible to obtain a classification of $k$-anisotropic semi-simple groups. For example, over a finite field there are no anisotropic semi-simple groups, and for a wide class of local fields $k$ it has been proved (see ) that every $k$-anisotropic simple linear algebraic group $G$ is an inner form of type ${\rm A}_n$, that is, in the simply-connected case $G_k$ is isomorphic to ${\rm SL}(1,D)$ where $D$ is a skew-field of finite degree over $k$. The basis of these results is the concept of a Tits system, which is a deep axiomatic generalization of the Bruhat decomposition in the classical case (see [Ti2]).

Among other general results one should mention Grothendieck's results that a linear algebraic group defined over $k$ has maximal tori that are defined over $k$, and that a connected reductive linear algebraic group over $k$ is unirational over $k$, as an algebraic variety (see [Bo], [DeGr], [Pl2]).

If $G$ is defined over an algebraic number field or over a field of algebraic functions in one variable, then there arise problems about the arithmetic properties of $G$, the study of which is the subject of the arithmetic theory of linear algebraic groups (see Linear algebraic groups, arithmetic theory of, [Pl2]).

The ideas and techniques of linear algebraic groups have been used to study arbitrary linear groups, which has led to one of the fundamental methods in the theory of linear groups (see [Pl]).

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