Linear algebraic group

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In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation MTM = I where MT is the transpose of M.

The main examples of linear algebraic groups are certain Lie groups, where the underlying field is the real or complex field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by the Peter–Weyl theorem.) These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. Compact Lie groups were considered by Élie Cartan, Ludwig Maurer, Wilhelm Killing, and Sophus Lie in the 1880s and 1890s in the context of differential equations and Galois theory. However, a purely algebraic theory was first developed by Kolchin (1948), with Armand Borel as one of its pioneers. The Picard–Vessiot theory did lead to algebraic groups.

The first basic theorem of the subject is that any affine algebraic group is a linear algebraic group: that is, any affine variety V that has an algebraic group law has a faithful linear representation, over the same field, which is also a morphism of varieties. For example the additive group of an n-dimensional vector space has a faithful representation as (n+1)×(n+1) matrices.

One can define the Lie algebra of an algebraic group purely algebraically (it consists of the dual number points based at the identity element); and this theorem shows that we get a matrix Lie algebra. A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one Go containing the identity will be a normal subgroup of G.

One of the first uses for the theory was to define the Chevalley groups.

Examples[edit]

Since \mathbb{G}_m = GL_1, \mathbb{G}_m is a linear algebraic group. The embedding x \mapsto \begin{bmatrix}
 1 & x \\
 0 & 1
\end{bmatrix} shows that \mathbb{G}_a is a unipotent group.

The deeper structure theory applies to connected linear algebraic groups G, and begins with the definition of Borel subgroups B. These turn out to be maximal as connected solvable subgroups (i.e., subgroups with composition series having as factors one-dimensional subgroups, all of which are groups of additive or multiplicative type); and also minimal such that G/B is a projective variety.

The most important subgroups of a linear algebraic group, besides its Borel subgroups, are its tori, especially the maximal ones (similar to the study of maximal tori in Lie groups). If there is a maximal torus which splits (i.e. is isomorphic to a product of multiplicative groups), one calls the linear group split as well. If there is no splitting maximal torus, one studies the splitting tori and the maximal ones of them. If there is a rank at least 1 split torus in the group, the group is called isotropic and anisotropic if this is not the case. Any anisotropic or isotropic linear algebraic group over a field becomes split over the algebraic closure, so this distinction is interesting from the point of view of Algebraic number theory.

Group actions[edit]

Let G be a unipotent group acting on an affine variety. Then every G-orbit in the variety is closed.

The Borel fixed-point theorem states that a connected solvable group acting on a non-empty complete variety admits a fixed point. The classical Lie–Kolchin theorem follows from the theorem applied to the flag variety.

Non-algebraic Lie groups[edit]

There are several classes of examples of Lie groups that aren't the real or complex points of an algebraic group.

  • Any Lie group with an infinite group of components G/Go cannot be realized as an algebraic group (see identity component).
  • The center of a linear algebraic group is again a linear algebraic group. Thus, any group whose center has infinitely many components is not a linear algebraic group. An interesting example is the universal cover of SL2(R). This is a Lie group that maps infinite-to-one to SL2(R), since the fundamental group is here infinite cyclic - and in fact the cover has no faithful matrix representation.
  • The general solvable Lie group need not have a group law expressible by polynomials.

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