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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 , 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 == Since , is a linear algebraic group. The embedding shows that 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「linear algebraic group」の詳細全文を読む スポンサード リンク
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