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Algebraic group

## Summary

In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.

Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.

An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called linear algebraic groups.[1] Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states that every algebraic group can be constructed from groups in those two families.

## Definition

Formally, an algebraic group over a field ${\displaystyle k}$  is an algebraic variety ${\displaystyle \mathrm {G} }$  over ${\displaystyle k}$ , together with a distinguished element ${\displaystyle e\in \mathrm {G} (k)}$  (the neutral element), and regular maps ${\displaystyle \mathrm {G} \times \mathrm {G} \to \mathrm {G} }$  (the multiplication operation) and ${\displaystyle \mathrm {G} \to \mathrm {G} }$  (the inversion operation) which satisfy the group axioms.[2]

A more sophisticated definition is that of a group scheme over ${\displaystyle k}$ . Yet another definition of the concept is to say that an algebraic group over ${\displaystyle k}$  is a group object in the category of algebraic varieties over ${\displaystyle k}$ .

## Classes

Several important classes of groups are algebraic groups, including:

There are other algebraic groups, but Chevalley's structure theorem asserts that every algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, 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 algebraic group and G/H an abelian variety.

According to another basic theorem[which?], any group that is also an affine variety has a faithful finite-dimensional linear representation: it is isomorphic to a matrix group, defined by polynomial equations.

Over the fields of real and complex numbers, every algebraic group is also a Lie group, but the converse is false.

A group scheme is a generalization of an algebraic group that allows, in particular, working over a commutative ring instead of a field.

## Algebraic subgroup

An algebraic subgroup of an algebraic group is a Zariski-closed 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 that 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 non-reduced schemes, in characteristic p.

## Coxeter groups

There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is ${\displaystyle n!}$ , and the number of elements of the general linear group over a finite field is the q-factorial ${\displaystyle [n]_{q}!}$ ; 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

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 ${\displaystyle {\rm {GL}}_{n}}$  for some n ${\displaystyle {\rm {SL}}_{n}}$  Every affine algebraic group is isomorphic to a linear algebraic group, and vice versa
affine algebraic group An algebraic group that is an affine variety ${\displaystyle {\rm {GL}}_{n}}$ , non-example: elliptic curve The notion of affine algebraic group stresses the independence from any embedding in ${\displaystyle {\rm {GL}}_{n}}$
commutative The underlying (abstract) group is abelian. ${\displaystyle {\mathbb {G} }_{a}}$  (the additive group), ${\displaystyle {\mathbb {G} }_{m}}$  (the multiplicative group),[3] any complete algebraic group (see abelian variety)
diagonalizable group A closed subgroup of ${\displaystyle (\mathbb {G} _{m})^{n}}$ , the group of diagonal matrices (of size n-by-n)
simple algebraic group A connected group that has no non-trivial connected normal subgroups ${\displaystyle {\rm {SL}}_{n}}$
semisimple group An affine algebraic group with trivial radical ${\displaystyle {\rm {SL}}_{n}}$ , ${\displaystyle {\rm {SO}}_{n}}$  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, ${\displaystyle {\rm {GL}}_{n}}$  Any semisimple group is reductive
unipotent group An affine algebraic group such that all elements are unipotent The group of upper-triangular n-by-n matrices with all diagonal entries equal to 1 Any unipotent group is nilpotent
torus A group that becomes isomorphic to ${\displaystyle (\mathbb {G} _{m})^{n}}$  when passing to the algebraic closure of k. ${\displaystyle {\rm {SO}}_{2}}$  G is said to be split by some bigger field k' , if G becomes isomorphic to Gmn as an algebraic group over k'.
character group X(G) The group of characters, i.e., group homomorphisms ${\displaystyle G\rightarrow {\mathbb {G} }_{m}}$  ${\displaystyle X^{*}(\mathbb {G} _{m})\cong \mathbb {Z} }$
Lie algebra Lie(G) The tangent space of G at the unit element. ${\displaystyle {\rm {Lie}}({\rm {GL}}_{n})}$  is the space of all n-by-n matrices Equivalently, the space of all left-invariant derivations.

## References

1. ^ Borel 1991, p.54.
2. ^ Borel 1991, p. 46.
3. ^ These two are the only connected one-dimensional linear groups, Springer 1998, Theorem 3.4.9
• Chevalley, Claude, ed. (1958), Séminaire C. Chevalley, 1956--1958. 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 2013-08-30, retrieved 2012-06-25
• Borel, Armand (1991). Linear algebraic groups. 2nd enlarged ed. Graduate Texts in Mathematics. Springer-Verlag. pp. x+288. Zbl 0726.20030.
• Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 21, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90108-4, MR 0396773
• Lang, Serge (1983), Abelian varieties, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90875-5
• Milne, J. S., Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups
• Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290
• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713
• Waterhouse, William C. (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90421-4
• Weil, André (1971), Courbes algébriques et variétés abéliennes, Paris: Hermann, OCLC 322901