Cartan subgroup

 ( Algebraic Geometry ) 

In the theory of , a Cartan subgroup of a connected linear algebraic group $G$ over a (not necessarily algebraically closed) field $k$ is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If $k$ is algebraically closed, they are all conjugate to each other.

Notice that in the context of algebraic groups a torus is an algebraic group $T$ such that the base extension $T\_\{(\backslash bar\{k\})\}$ (where $\backslash bar\{k\}$ is the algebraic closure of $k$) is isomorphic to the product of a finite number of copies of the $\backslash mathbf\{G\}\_m=\backslash mathbf\{GL\}\_1$. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal torus in the theory of .

If $G$ is reductive group (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser and thus Cartan subgroups of $G$ are precisely the maximal tori.

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Example The general linear groups $\backslash mathbf\{GL\}\_n$ are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of $\backslash mathbf\{G\}\_m$ already before any base extension), and it can be shown to be maximal. Since $\backslash mathbf\{GL\}\_n$ is reductive, the diagonal subgroup is a Cartan subgroup.

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See also

• Borel subgroup
• Algebraic group
• Algebraic torus

• Armand Borel (1991). 9783540973706 ISBN 9783540973706
• Serge Lang (2025). 9780387953854, Springer. ISBN 9780387953854

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