Algebraic 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.

Classes

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 finite-dimensional 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

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 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 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 $n!$ , and the number of elements of the general linear group over a finite field is the q-factorial $[n]_{q}!$ ; 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 ${{\rm {GL}}}_{n}$ for some n	${{\rm {SL}}}_{n}$	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	${{\rm {GL}}}_{n}$ , non-example: elliptic curve	The notion of affine algebraic group stresses the independence from any embedding in ${{\rm {GL}}}_{n}$
commutative	The underlying (abstract) group is abelian.	${{\mathbb G}}_{a}$ (the additive group), ${{\mathbb G}}_{m}$ (the multiplicative group),^[1] any complete algebraic group (see abelian variety)
diagonalizable group	A closed subgroup of $({\mathbb {G}}_{m})^{n}$ , the group of diagonal matrices (of size n-by-n)
simple algebraic group	A connected group which has no non-trivial connected normal subgroups	${{\rm {SL}}}_{n}$
semisimple group	An affine algebraic group with trivial radical	${{\rm {SL}}}_{n}$ , ${{\rm {SO}}}_{n}$	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, ${{\rm {GL}}}_{n}$	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 $({\mathbb {G}}_{m})^{n}$ when passing to the algebraic closure of k.	${{\rm {SO}}}_{2}$	G is said to be split by some bigger field k' , if G becomes isomorphic to G_mⁿ as an algebraic group over k'.
character group X_∗(G)	The group of characters, i.e., group homomorphisms $G\rightarrow {{\mathbb G}}_{m}$	$X^{*}({\mathbb {G}}_{m})\cong {\mathbb {Z}}$
Lie algebra Lie(G)	The tangent space of G at the unit element.	${{\rm {Lie}}}({{\rm {GL}}}_{n})$ is the space of all n-by-n matrices	Equivalently, the space of all left-invariant derivations.

References

↑ 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
Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics, 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, 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, 66, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90421-4
Weil, André (1971), Courbes algébriques et variétés abéliennes, Paris: Hermann, OCLC 322901