In algebraic geometry, a Cartan subgroup of a connected
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 M^TM = I ...
over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate.
Examples
* For a finite field ''F'', the group of diagonal matrices
where ''a'' and ''b'' are elements of ''F
*''. This is called the split Cartan subgroup of GL
2(''F'').
* For a finite field ''F'', every maximal commutative semisimple subgroup of GL
2(''F'') is a Cartan subgroup (and conversely).
See also
*
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
References
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{{algebra-stub
Algebraic geometry
Linear algebraic groups