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Diagonalizable group
In mathematics, an affine group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over k is said to split over k or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an algebraic group. Every diagonalizable group splits over ks.[1] Any closed subgroup and image of diagonalizable groups are diagonalizable. The torsion subgroup of a diagonalizable group is dense.
The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian group with \( \Gamma \)[2]-equivariant morphisms without p-torsion. This is an analog of Poincaré duality and motivated the terminology.
A diagonalizable k-group is said to be anisotropic if it has no nontrivial k-valued character.
The so-called "rigidity" states that the identity component of the centralizer of a diagonalizable group coincides with the identity component of the normalizer of the group. The fact plays a crucial role in the structure theory of solvable groups.
A connected diagonalizable group is called an algebraic torus (which is not necessarily compact, in contrast to a complex torus). A k-torus is a torus defined over k. The centralizer of a maximal torus is called a Cartan subgroup.
Notes
^ The separable closure of k.
^ \( \Gamma=\operatorname{Gal}(k_s / k) \)
References
Borel, A. Linear algebraic groups, 2nd ed.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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