Linear algebraic group

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a linear algebraic group is a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of the <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> of <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> (under <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a>) that is defined by <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> equations. An example is the <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a>, defined by the relation 
 
 
 
 
 M
 
 T
 
 
 M
 =
 
 I
 
 n
 
 
 
 
 {\displaystyle M^{T}M=I_{n}}
 
 where 
 
 
 
 
 M
 
 T
 
 
 
 
 {\displaystyle M^{T}}
 
 is the <a href="/facts/Transpose/8wmsagGS">transpose</a> of 
 
 
 
 M
 
 
 {\displaystyle M}
 
.
Many <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a> can be viewed as linear algebraic groups over the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers. (For example, every <a href="/facts/Compact_Lie_group/Itl2HJaR">compact Lie group</a> can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the <a href="/facts/Simple_Lie_group/LkHTNYU9">simple Lie group</a> <a href="/facts/Special_linear_group/rsahah9w">SL(n,R)</a>.) The simple Lie groups were classified by <a href="/facts/Wilhelm_Killing/W27PB2MJ">Wilhelm Killing</a> and <a href="/facts/%25C3%2589lie_Cartan/Uvhkp66S">Élie Cartan</a> in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include <a href="/facts/Ludwig_Maurer/YR4yy9U6">Maurer</a>, <a href="/facts/Claude_Chevalley/MJvMYE34">Chevalley</a>, and <a href="/facts/Ellis_Kolchin/13Uqv1CI">Kolchin</a> (1948). In the 1950s, <a href="/facts/Armand_Borel/QkRUS4ej">Armand Borel</a> constructed much of the theory of algebraic groups as it exists today.
One of the first uses for the theory was to define the <a href="/facts/Chevalley_group/gmboug78">Chevalley groups</a>.

Linear algebraic group open-in-new

Linear algebraic group