The picture illustrates Grothendieck's vision of a pinned reductive group: the body is a maximal torus T, the wings
are the opposite Borel subgroups B, and the pins rigidify the situation.
"Demazure nous indique que, derrière cette terminologie [épinglage], il y a l'image du papillon (que lui a fournie Grothendieck): le corps est un tore maximal T, les ailes sont deux sous-groupes de Borel opposées par rapport à T, on déploie le papillon en étalant les ailes, puis on fixe des éléments dans les groupes additifs (des épingles) pour rigidifier la situation (c.-à-d., pour éliminer les automorphismes)." (SGA 3, XXIII, p.177, new edition.)
The first eight chapters of the book study general algebraic group schemes over a field. They culminate in a proof of the Barsotti-Chevalley theorem stating that every algebraic group is an extension of an abelian variety by an affine algebraic group. The remaining chapters treat only affine algebraic groups. After a review of the Tannakian philosophy, there are short accounts of Lie algebras and finite group schemes. Solvable algebraic groups are studied in detail in Chapters 12-16. The final eight chapters treat the Borel-Chevalley structure theory of reductive algebraic groups over arbitrary fields. Three appendices review the algebraic geometry needed, the construction of very general quotients of algebraic groups, and the theory of root data.
The exposition incorporates simplifications to the theory by Springer, Steinberg, and others. Although the theory of algebraic groups can be considered a branch of algebraic geometry, most of those using it are not algebraic geometers. In the present work, prequisites have been kept to a minimum. The only requirement is a first course in algebraic geometry including basic commutative algebra.
-------------------------
*The only previous attempt to write such a book that I know of was
that of Demazure and Gabriel. As they wrote in the foreword of "Groupes
Algébriques (1971)" (my translation):
A. Grothendieck introduced two particularly useful tools into algebraic geometry: the functorial calculus and varieties with nilpotent elements. These tools allow us in particular to better understand phenomena arising from inseparability, by restoring differential calculus in nonzero characteristic, and they considerably simplify the general theory of algebraic groups. Hence we originally intended to develop the now classical theory of semisimple algebraic groups over algebraically closed fields (Borel-Chevalley) within the framework of schemes; it was simply a question of updating the 1956-58 seminar of Chevalley. But we soon realized that there did not exist a suitable book to serve as a reference for the general theory of algebraic groups, and moreover that it was impossible to refer a nonspecialist reader to the Grothendieck's Eléments de Géométrie Algébrique (EGA). Thus we were led to considerably modify and develop our original project, and we now present to the "mathematical community" a first volume devoted to the general theory of algebraic groups, where the case of semisimple groups is not entered into.Alas, no second volume was published.
a. Definition.
b. Basic properties of algebraic groups.
c. Algebraic subgroups.
d. Examples.
e. Kernels and exact sequences.
f. Group actions.
g. The homomorphism theorem for smooth groups.
h. Closed subfunctors: definitions and statements.
i. Transporters.
j. Normalizers.
k. Centralizers.
l. Closed subfunctors: proofs.
Exercises.
a. Affine algebraic groups.
b. Étale group schemes.
c. Anti-affine algebraic groups.
d. Homomorphisms of algebraic groups.
e. The Frobenius endomorphism.
f. Products.
g. Semidirect products.
h. The group of connected components.
i. The algebraic subgroup generated by a map.
j. Forms of algebraic groups.
k. Torsors and extensions
i. Restriction of scalars.
Exercises.
a. The comultiplication map.
b. Hopf algebras.
c. Hopf algebras and algebraic groups.
d. Hopf subalgebras.
e. Hopf subalgebras of O(G) versus algebraic subgroups of G.
f. Subgroups of G(k) versus algebraic subgroups of G.
g. Affine algebraic groups in characteristic zero are smooth.
h. Smoothness in characteristic p.
i. Faithful flatness for Hopf algebras.
j. The homomorphism theorem for affine algebraic groups.
Exercises
a. Representations and comodules.
b. Stabilizers.
c. Representations are unions of finite-dimensional representations.
d. Affine algebraic groups are linear.
e. Constructing all finite-dimensional representations.
f. Semisimple representations.
g. Characters and eigenspaces.
h. Chevalley's theorem.
i. The subspace fixed by a group.
Exercises.
a. The isomorphism theorems for abstract groups.
b. Quotient maps.
c. Existence of quotients.
d. Monomorphisms of algebraic groups.
e. The homomorphism theorem.
f. The isomorphism theorem.
g. The correspondence theorem.
h. The connected-étale exact sequence.
i. The category of commutative algebraic groups.
j. Sheaves.
k. The isomorphism theorems for group functors.
l. The isomorphism theorems for sheaves of groups.
m. The isomorphism theorems for algebraic groups.
n. Some category theory.
Exercises
a. Subnormal series.
b. Isogenies.
c. Composition series for algebraic groups.
d. Solvable and nilpotent algebraic groups.
e. The derived group of an algebraic group.
f. Nilpotent algebraic groups.
g. Existence of a largest algebraic subgroup with a given property.
h. Semisimple and reductive groups.
i. A standard example.
a. Group actions.
b. Linear actions: affine case.
c. The fixed subscheme.
d. Orbits and isotropy groups.
e. The functor defined by projective space.
f. Quotients: definition and properties.
g. Quotients: construction in the affine case.
h. Linear actions: projective case.
i. Complements.
j. Flag varieties.
k. Homogeneous spaces are quasi-projective.
Exercises.
a. Summary.
b. Generalities.
c. Local actions.
d. Anti-affine algebraic groups and abelian varieties.
e. Rosenlicht's decomposition theorem.
f. Rosenlicht's dichotomy.
g. The Barsotti-Chevalley theorem.
h. Anti-affine groups.
i. Extensions of abelian varieties by affine algebraic groups: a survey.
Exercises
a. Recovering a group from its representations.
b. Application to Jordan decompositions.
c. Characterizations of categories of representations.
d. Proof of Theorem 10.24.
e. Tannakian categories
f. Rosenlicht's dichotomy.
g. Properties of G versus those of Rep(G): a summary.
a. Definition.
b. The Lie algebra of an algebraic group.
c. Basic properties of Lie algebras.
d. The adjoint representation; definition of the bracket.
e. Description of the Lie algebra in terms of derivations.
f. Stabilizers.
g. Centres.
h. Centralizers.
i. An example of Chevalley.
j. The universal covering algebra.
k. The universal enveloping p-algebra.
Exercises.
a. Generalities.
b. Locally free finite group schemes over a base ring.
c. Cartier duality.
d. Finite group schemes of order p.
e. Derivations of Hopf algebras.
f. Structure of the underlying scheme of a finite group scheme.
g. Finite group schemes of order n are killed by n.
h. Finite group schemes of height at most one.
i. The Verschiebung morphism.
j. The Witt schemes Wn.
k. Commutative group schemes over a perfect field.
a. The characters of an algebraic group.
b. The algebraic group D(M).
c. Diagonalizable groups.
d. Diagonalizable representations.
e. Tori.
f. Groups of multiplicative type.
g. Classification of groups of multiplicative type.
h. Representations of groups of multiplicative type.
i. Density and rigidity.
j. Central tori as almost factors.
k. Maps to tori.
l. Linearly reductive groups.
m. Unirationality.
Exercises
a. The smoothness of the fixed subscheme.
b. Limits in affine and projective space.
c. Limits in affine algebraic varieties.
d. Limits in algebraic groups.
e. Tori acting on smooth affine varieties.
f. Luna maps.
g. Decomposition of a variety under the action of a torus.
h. Proof of the Bialynicki-Birula decomposition.
Exercises
a. Preliminaries from linear algebra.
b. Unipotent algebraic groups.
c. Unipotent algebraic groups in characteristic zero.
d. Unipotent algebraic groups in nonzero characteristic.
e. Split and wound unipotent groups.
Exercises.
a. Crossed homomorphisms.
b. Hochschild cohomology.
c. Hochschild extensions.
d. The cohomology of linear representations.
e. Linearly reductive groups.
f. Applications to homomorphisms.
g. Applications to centralizers.
h. Calculation of some extensions.
Exercises
a. Trigonalizable algebraic groups
b. Commutative algebraic groups.
c. Structure of trigonalizable algebraic groups.
d. Solvable algebraic groups.
e. Connectedness.
f. Nilpotent algebraic groups.
g. Split solvable groups.
h. Complements on unipotent algebraic groups.
i. Tori acting on algebraic groups.
j. Tori acting on solvable algebraic groups.
k. Summary.
Exercises.
a. The Borel fixed point theorem.
b. Borel subgroups and maximal tori.
c. The density theorem.
d. Centralizers of tori.
e. The normalizer of a Borel subgroup.
f. The variety of Borel subgroups.
g. Chevalley's description of the unipotent radical.
h. Proof of Chevalley's theorem.
i. Borel and parabolic subgroups over an arbitrary field.
j. Maximal tori and Cartan subgroups over an arbitrary field.
k. Algebraic groups over finite fields.
i. Split algebraic groups.
Exercises.
a. Definitions.
b. The universal covering.
c. Line bundles and characters.
d. Existence of a universal covering.
e. Applications.
f. Proof of Theorem 19.14.
Exercises.
a. Semisimple groups
b. Reductive groups.
c. The rank of a group variety.
d. Deconstructing reductive algebraic groups.
Exercises.
a. Group varieties of semisimple rank 0.
b. Homogeneous curves.
c. The automorphism group of the projective line.
d. A fixed point theorem for actions of tori.
e. Group varieties of semisimple rank 1.
f. Split reductive groups of semisimple rank 1.
g. Properties of SL_{2}.
h. Classification of the split reductive groups of semisimple rank 1.
i. The forms of SL_{2}, GL_{2}, PGL_{2}.
j. Classification of reductive groups of semisimple rank one.
k. Review of SL_{2}.
Exercises.
a. Split reductive groups and their roots.
b. The centre of G in terms of its roots.
c. Review of SL2.
d. The root datum of a split reductive group.
e. Borel subgroups; Weyl groups; Tits systems.
f. Complements on semisimple groups.
g. Complements on reductive groups.
h. Subgroups normalized by T.
j. The big cell.
k. Parabolic subgroups.
l. The root data of the classical semisimple groups.
Exercises.
a. The semisimple representations of a split reductive group.
b. Characters and Grothendieck groups.
c. Semisimplicity in characteristic zero.
d. The Chern class of a simple representation.
e. Construction of the simple representations of G.
f. Weyl's character formula.
g. Relation to the representations of Lie(G).
h. Hyperalgebras.
Exercises.
a. Isogenies of groups and root data.
b. Proof of the isogeny theorem.
c. Complements.
d. Pinnings.
e. Automorphisms.
f. Quasi-split forms
g. Statement of the existence theorem; applications
f. Proof of the existence theorem.
Exercises.
a. Deconstructing semisimple algebraic groups.
b. Generalities on forms of reductive groups
c. The centres of semisimple groups
d. Semisimple algebras
e. Algebras with involution.
f. The geometrically almost-simple groups of type A.
h. Clifford algebras.
i. The spin groups.
j. The geometrically almost-simple group of type B and D.
k. The classical groups in terms of sesquilinear forms.
l. The exceptional groups.
m. The trialitarian groups (groups of subtypes ^{3}D_{4}
and ^{6}D_{4})
Exercises.
a. Parabolic subgroups of reductive groups.
b. The small root system.
c. The Satake-Tits classification.
d. Representations theory.
e. Pseudo-reductive groups.
f. Nonreductive groups: Levi subgroups.
g. Galois cohomology.
Exercises.
a. Equivalence relations.
b. Existence of quotients in the finite affine case.
c. Existence of quotients in the finite case.
d. Existence of quotients in the presence of quasi-sections.
e. Existence generically of a quotient.
f. Existence of quotients of algebraic groups.
a. Preliminaries.
b. Reflection groups.
c. Root systems.
d. Root data.
e. Duals of root data.
f. Deconstructing root data.
g. Classification of reduced root systems.