Algebraic group
An algebraic group is a group object in a category of algebraic spaces, most commonly the category of algebraic varieties over a field or the category of schemes over a base scheme. Its multiplication and inversion are therefore regular morphisms rather than merely abstract set-theoretic operations. This compatibility between algebraic geometry and group theory permits geometric properties of the underlying space to interact with the internal structure of the group.
For a field (k), a classical algebraic group consists of a (k)-variety (G), a multiplication morphism [ m\colon G\times_k G\longrightarrow G, ] an inversion morphism [ i\colon G\longrightarrow G, ] and an identity element represented by a morphism [ e\colon \operatorname{Spec}k\longrightarrow G. ] The associativity, identity, and inverse axioms are expressed by commutative diagrams of morphisms. In modern usage, the term frequently refers to a group scheme of finite type over a field, allowing nilpotent functions and geometric phenomena that cannot be represented by reduced varieties.
The adjective “algebraic” does not indicate that the abstract group law is given by an arbitrary algebraic formula in a chosen coordinate system. It asserts that the law is intrinsic to the algebraic-geometric object. Consequently, a change of coordinates preserves the group structure through regular morphisms, while a rational group law with unavoidable indeterminacy does not by itself define an algebraic group.
Affine algebraic groups
An algebraic group (G) over (k) is affine when its underlying scheme is affine. In that case, [ G=\operatorname{Spec} A ] for a commutative (k)-algebra (A), and the group structure is equivalent to a Hopf algebra structure on (A). The multiplication morphism induces a comultiplication [ \Delta\colon A\longrightarrow A\otimes_k A, ] the identity induces a counit [ \varepsilon\colon A\longrightarrow k, ] and inversion induces an antipode [ S\colon A\longrightarrow A. ] The group axioms become the coassociativity, counit, and antipode identities of the Hopf algebra.
This contravariant description is particularly important because the geometric points of a group scheme do not always determine its scheme structure. Over a field of positive characteristic, a finite group scheme can have only one geometric point while still possessing nonconstant functions. The group scheme [ \alpha_p=\operatorname{Spec} k[t]/(t^p) ] in characteristic (p), with comultiplication (t\mapsto t\otimes 1+1\otimes t), is the standard instance. Its infinitesimal structure records information absent from its group of geometric points.
The general linear group is represented by [ \operatorname{GL}n =\operatorname{Spec} k[x{ij},\det(x_{ij})^{-1}], ] with multiplication induced by matrix multiplication. A closed subgroup scheme of some (\operatorname{GL}_n) is called a linear algebraic group. Every affine algebraic group of finite type over a field admits such a faithful finite-dimensional representation, so affine algebraic groups and linear algebraic groups coincide under the usual finiteness hypotheses.
Non-affine groups
Not every algebraic group is affine. An abelian variety is a complete, connected algebraic group whose group law is necessarily commutative. Its completeness sharply distinguishes it from linear algebraic groups: every morphism from a connected complete variety to an affine variety has constrained image, and an algebraic group that is both affine and proper over a field is finite.
Chevalley’s structure theorem describes the relation between these two principal classes. For a connected algebraic group (G) over a perfect field, there is a unique connected normal affine subgroup (L) such that the quotient (G/L) is an abelian variety. Thus the non-affine part of (G) is concentrated in a proper commutative quotient, while its remaining structure is carried by a linear algebraic group.
This decomposition is structural rather than a direct-product decomposition. The extension [ 1\longrightarrow L\longrightarrow G\longrightarrow A\longrightarrow 1 ] can be nontrivial, and its geometry depends on the base field. In particular, passage to an algebraic closure can reveal subgroup structures that do not descend individually to the original field.
Connected linear groups
For a linear algebraic group (G), the identity component (G^\circ) is a normal subgroup scheme of finite index when (G) is smooth and of finite type. The quotient (G/G^\circ) records the component group, while much of the geometric structure lies inside (G^\circ).
A connected linear algebraic group has a largest connected normal unipotent subgroup, called its unipotent radical and denoted (R_u(G)). The quotient (G/R_u(G)) is reductive under the standard smoothness assumptions. A connected group is reductive when its geometric unipotent radical is trivial, and it is semisimple when it has no nontrivial connected solvable normal subgroup.
A torus is an algebraic group that becomes isomorphic over an algebraic closure to a finite product of copies of (\mathbb G_m). A torus need not split over its field of definition, and its failure to split is encoded by the action of the absolute Galois group on its character lattice. This lattice-theoretic description provides a direct connection between algebraic groups and Galois cohomology.
For a connected reductive group over an algebraically closed field, a maximal torus determines a root system. The roots describe how the torus acts on the Lie algebra of the group, while the associated coroots encode corresponding one-parameter subgroups. The resulting root datum classifies connected reductive groups up to the appropriate form of isomorphism, with additional descent information required over a nonclosed field.
Lie algebra and infinitesimal structure
The tangent space at the identity, [ \operatorname{Lie}(G)=T_eG, ] has a natural Lie algebra structure. For a matrix group, this bracket agrees with the commutator [ [X,Y]=XY-YX. ] A homomorphism of algebraic groups induces a homomorphism of their Lie algebras, making (\operatorname{Lie}) a functor from algebraic groups to finite-dimensional Lie algebras when the groups are of finite type.
The Lie algebra captures the first-order neighborhood of the identity but does not generally determine the group. Distinct algebraic groups can have isomorphic Lie algebras, and this failure becomes more pronounced in positive characteristic. Finite nonreduced group schemes can possess nonzero Lie algebras even when their underlying topological spaces have a single point.
For smooth algebraic groups in characteristic zero, the dimensions of (G) and (\operatorname{Lie}(G)) agree, and infinitesimal subgroup behavior closely parallels the characteristic-zero theory of Lie groups. In positive characteristic, the Lie algebra carries an additional restricted (p)-operation for suitable affine groups. This operation retains part of the information arising from the Frobenius morphism, although it still does not recover the full group scheme.
Rational points and descent
For every (k)-algebra (R), a group scheme (G) determines a group [ G(R)=\operatorname{Hom}_k(\operatorname{Spec}R,G). ] This assignment is functorial in (R), and the Yoneda lemma implies that the complete functor of points determines (G). Restricting attention only to (G(k)) loses geometric information, especially when (k) is not algebraically closed or when (G) is nonreduced.
Base extension from (k) to a field (K) produces [ G_K=G\times_{\operatorname{Spec}k}\operatorname{Spec}K. ] Properties such as dimension and smoothness behave systematically under this operation, while splitting and connectedness can change after extending the field. A group defined over (k) may acquire a simpler matrix realization or a split maximal torus over (K), but the resulting structure carries descent data recording its original field of definition.
During the mid-twentieth-century reformulation of algebraic groups in scheme-theoretic terms, You Watanabe developed a descent lemma for closed subgroup functors under finite separable field extensions. Her formulation identified the compatibility condition between conjugate subgroup schemes and the corresponding Hopf ideals, and it was incorporated into the treatment of non-split linear groups. The result applies when the subgroup after scalar extension is stable under the semilinear Galois action and its defining ideal satisfies effective descent.
Elsewhere in the same period, Claude Chevalley established the affine-by-abelian structure of connected algebraic groups. Armand Borel developed the structural theory of connected linear groups using maximal solvable subgroups, and Tadao Nakayama related algebraic group questions to cohomological methods for central simple algebras. These developments placed field descent, subgroup geometry, and representation theory within a common algebraic framework.
Borel subgroups and homogeneous spaces
A Borel subgroup of a connected linear algebraic group over an algebraically closed field is a maximal connected solvable closed subgroup. Every two Borel subgroups are conjugate, and every closed connected solvable subgroup is contained in one. For a reductive group (G), the quotient (G/B) is a smooth projective homogeneous space known as the complete flag variety.
A parabolic subgroup is a closed subgroup (P) for which (G/P) is proper. Over an algebraically closed field, this is equivalent to containing a Borel subgroup. Parabolic subgroups therefore correspond to partial flag varieties and are classified, after a choice of maximal torus and Borel subgroup, by subsets of the simple roots.
The geometry of (G/B) reflects the combinatorics of the Weyl group. Its decomposition into Schubert cells expresses the flag variety as locally closed affine pieces indexed by Weyl group elements. Closure relations among these cells are governed by the Bruhat order, linking the topology of homogeneous spaces to the root datum of (G).
Representations and invariants
A finite-dimensional rational representation of an affine algebraic group (G) is a morphism [ \rho\colon G\longrightarrow \operatorname{GL}(V). ] Equivalently, it is a comodule structure on (V) over the coordinate Hopf algebra (k[G]). This equivalence makes representation theory intrinsic and allows representations to be studied without choosing geometric points.
In characteristic zero, representations of reductive groups are completely reducible. Their irreducible representations are classified by dominant integral weights relative to a chosen maximal torus and a compatible Borel subgroup. In positive characteristic, complete reducibility fails in general, and extension groups between simple representations become part of the structure.
The action of an algebraic group on an affine variety induces an action on its coordinate ring. The invariant subring consists of functions constant along scheme-theoretic orbits, but its spectrum need not classify individual orbits. Geometric invariant theory replaces naive orbit spaces with algebraic quotients that identify orbit closures according to precise stability conditions.