Algebraic group

Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. In category theoretic terms, 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
* 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 a basic 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. [Chevalley's result is from 1960 and difficult. Contemporary treatment by Brian Conrad: [http://www.math.lsa.umich.edu/~bdconrad/papers/chev.pdf PDF] .]

According to another basic theorem, any group in the category of affine varieties has a faithful 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".

ee also

*Tame group
*Morley rank
*Cherlin-Zilber conjecture
*Adelic algebraic group

References

* | year=1972 | volume=21
*
* Milne, J. S., " [http://www.jmilne.org/math/ Algebraic and Arithmetic Groups.] "
*
* | year=1998 | volume=9
*
*

Notes


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Algebraic-group factorisation algorithm — Algebraic group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the reduced groups obtained by performing the equations defining the… …   Wikipedia

  • Linear algebraic group — In mathematics, a linear algebraic group is a subgroup of the group of invertible n times; n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation MTM = I where …   Wikipedia

  • Adelic algebraic group — In mathematics, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K , and the adele ring A = A ( K ) of K . It consists of the points of G having values in A ; the definition of the appropriate… …   Wikipedia

  • Semisimple algebraic group — In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.… …   Wikipedia

  • Differential algebraic group — In mathematics, a differential algebraic group is a differential algebraic variety with a compatible group structure. Differential algebraic groups were introduced by Cassidy (1972). References Cassidy, Phyllis Joan (1972), Differential algebraic …   Wikipedia

  • Radical of an algebraic group — In mathematics, an algebraic group G contains a unique maximal normal solvable subgroup; and this subgroup is closed. Its identity component is called the radical of G . External links * [http://eom.springer.de/R/r077110.htm Radical of a group ] …   Wikipedia

  • Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… …   Wikipedia

  • Algebraic — may refer to:* Algebraic chess notation mdash; a method used to record and describe the play of chess games. * Algebraic data types. * Algebraic group mdash; a term from algebraic geometry. * Algebraic numbers (those which are roots of certain… …   Wikipedia

  • Algebraic torus — In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory (see maximal torus). The theory of tori is in some sense opposite to that of… …   Wikipedia

  • Group of Lie type — In mathematics, a group of Lie type G(k) is a (not necessarily finite) group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”