Editing Linear algebraic group (section)

==Examples==
For a [[positive integer]] <math>n</math>, the [[general linear group]] <math>GL(n)</math> over a field <math>k</math>, consisting of all invertible <math>n\times n</math> matrices, is a linear algebraic group over <math>k</math>. It contains the subgroups 
:<math>U \subset B \subset GL(n)</math>
consisting of matrices of the form, resp.,
:<math>\left ( \begin{array}{cccc} 1 & * & \dots & * \\ 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & * \\ 0 & \dots & 0 & 1\end{array} \right )</math> and <math>\left ( \begin{array}{cccc} * & * & \dots & * \\ 0 & * & \ddots & \vdots \\ \vdots & \ddots & \ddots & * \\ 0 & \dots & 0 & *\end{array} \right )</math>.
The group <math>U</math> is an example of a [[unipotent]] linear algebraic group, the group <math>B</math> is an example of a [[Solvable group|solvable]] algebraic group called the [[Borel subgroup]] of <math>GL(n)</math>. It is a consequence of the [[Lie-Kolchin theorem]] that any connected solvable subgroup of <math>\mathrm{GL}(n)</math> is conjugated into <math>B</math>. Any unipotent subgroup can be conjugated into <math>U</math>.

Another algebraic subgroup of <math>\mathrm{GL}(n)</math> is the [[special linear group]] <math>\mathrm{SL}(n)</math> of matrices with determinant 1.

The group <math>\mathrm{GL}(1)</math> is called the '''[[multiplicative group]]''', usually denoted by <math> \mathbf G_{\mathrm m}</math>. The group of <math>k</math>-points <math>\mathbf G_{\mathrm m}(k)</math> is the multiplicative group <math>k^*</math> of nonzero elements of the field <math>k</math>. The '''additive group''' <math>\mathbf G_{\mathrm a}</math>, whose <math>k</math>-points are isomorphic to the additive group of <math>k</math>, can also be expressed as a matrix group, for example as the subgroup <math>U</math> in <math>\mathrm{GL}(2)</math> : 
:<math>\begin{pmatrix}
 1 & * \\
 0 & 1
\end{pmatrix}.</math>

These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their [[linear representation]]s (as algebraic groups). Every representation of the multiplicative group <math>\mathbf G_{\mathrm m}</math> is a [[direct sum]] of [[irreducible representation]]s. (Its irreducible representations all have dimension 1, of the form <math>x \mapsto x^n</math> for an integer <math>n</math>.) By contrast, the only irreducible representation of the additive group <math>\mathbf G_{\mathrm a}</math> is the trivial representation. So every representation of <math>\mathbf G_{\mathrm a}</math> (such as the 2-dimensional representation above) is an iterated [[composition series|extension]] of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.