Algebraic group
Algebraic structure → Group theory Group theory 



Infinite dimensional Lie group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.
Contents
Classes[edit]
Several important classes of groups are algebraic groups, including:
 Finite groups
 GL(n, C), the general linear group of invertible matrices over C
 Jet group
 Elliptic curves.
Two important classes of algebraic groups arise, that for the most part are studied separately: abelian varieties (the 'projective' theory) and linear algebraic groups (the 'affine' theory). There are certainly examples that are neither one nor the other — these occur for example in the modern theory of integrals of the second and third kinds such as the Weierstrass zeta function, or the theory of generalized Jacobians. But according to Chevalley's structure theorem any algebraic group is an extension of an abelian variety by a linear algebraic group. This is a result of Claude Chevalley: if K is a perfect field, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, such that H is a linear group and G/H an abelian variety.
According to another basic theorem, any group in the category of affine varieties has a faithful finitedimensional linear representation: we can consider it to be a matrix group over K, defined by polynomials over K and with matrix multiplication as the group operation. For that reason a concept of affine algebraic group is redundant over a field — we may as well use a very concrete definition. Note that this means that algebraic group is narrower than Lie group, when working over the field of real numbers: there are examples such as the universal cover of the 2×2 special linear group that are Lie groups, but have no faithful linear representation. A more obvious difference between the two concepts arises because the identity component of an affine algebraic group G is necessarily of finite index in G.
When one wants to work over a base ring R (commutative), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra. There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.
Algebraic subgroup[edit]
An algebraic subgroup of an algebraic group is a Zariskiclosed subgroup. Generally these are taken to be connected (or irreducible as a variety) as well.
Another way of expressing the condition is as a subgroup which is also a subvariety.
This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the connected component is of finite index > 1, is to admit nonreduced schemes, in characteristic p.
Coxeter groups[edit]
There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is , and the number of elements of the general linear group over a finite field is the qfactorial ; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.
Glossary of algebraic groups[edit]
There are a number of mathematical notions to study and classify algebraic groups.
In the sequel, G denotes an algebraic group over a field k.
notion  explanation  example  remarks 

linear algebraic group  A Zariski closed subgroup of for some n  Every affine algebraic group is isomorphic to a linear algebraic group, and vice versa  
affine algebraic group  An algebraic group which is an affine variety  , nonexample: elliptic curve  The notion of affine algebraic group stresses the independence from any embedding in 
commutative  The underlying (abstract) group is abelian.  (the additive group), (the multiplicative group),^{[1]} any complete algebraic group (see abelian variety)  
diagonalizable group  A closed subgroup of , the group of diagonal matrices (of size nbyn)  
simple algebraic group  A connected group which has no nontrivial connected normal subgroups  
semisimple group  An affine algebraic group with trivial radical  ,  In characteristic zero, the Lie algebra of a semisimple group is a semisimple Lie algebra 
reductive group  An affine algebraic group with trivial unipotent radical  Any finite group,  Any semisimple group is reductive 
unipotent group  An affine algebraic group such that all elements are unipotent  The group of uppertriangular nbyn matrices with all diagonal entries equal to 1  Any unipotent group is nilpotent 
torus  A group that becomes isomorphic to when passing to the algebraic closure of k.  G is said to be split by some bigger field k' , if G becomes isomorphic to G_{m}^{n} as an algebraic group over k'.  
character group X_{∗}(G)  The group of characters, i.e., group homomorphisms  
Lie algebra Lie(G)  The tangent space of G at the unit element.  is the space of all nbyn matrices  Equivalently, the space of all leftinvariant derivations. 
See also[edit]
 Algebraic topology (object)
 Borel subgroup
 Tame group
 Morley rank
 Cherlin–Zilber conjecture
 Adelic algebraic group
 Pseudoreductive group
References[edit]
 Chevalley, Claude, ed. (1958), Séminaire C. Chevalley, 19561958. Classification des groupes de Lie algébriques, 2 vols, Paris: Secrétariat Mathématique, MR 0106966, Reprinted as volume 3 of Chevalley's collected works., archived from the original on 20130830, retrieved 20120625
 Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Berlin, New York: SpringerVerlag, ISBN 9780387901084, MR 0396773
 Lang, Serge (1983), Abelian varieties, Berlin, New York: SpringerVerlag, ISBN 9780387908755
 Milne, J. S., Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups
 Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 9780195605280, OCLC 138290
 Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 9780817640217, MR 1642713
 Waterhouse, William C. (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Berlin, New York: SpringerVerlag, ISBN 9780387904214
 Weil, André (1971), Courbes algébriques et variétés abéliennes, Paris: Hermann, OCLC 322901
Further reading[edit]
 Algebraic groups and representations by M. Brion, B. Conrad, P. Gille, E. Letellier & B. Rémy