User:Maximilian Janisch/latexlist/Algebraic Groups/Regular torus

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An algebraic torus in a connected algebraic group $k$ (over an algebraically closed field) that is contained in only a finite number of Borel subgroups (cf. Borel subgroup). The maximal tori in $k$ are always regular (cf. Maximal torus). In general, a torus $S \subset G$ is regular if and only if its centralizer $C _ { G } ( S )$ is a solvable group. One-dimensional regular tori $5$ and their corresponding one-parameter subgroups $\lambda : G _ { m } \rightarrow S$ (also called regular) play an important role in algebraic group theory. A torus that is not regular is called singular. For reductive groups $k$ (cf. Reductive group), a criterion for the singularity of a torus $S \subset G$ can be given in terms of root systems. Thus, if $T$ is a maximal torus in $k$ containing $5$ and $\phi ( T , G )$ is the corresponding root system, then $5$ is singular if and only if $S \subset \operatorname { Ker } \alpha$ for some $\alpha \in \phi ( T , G )$.

A regular torus in $k$ is sometimes defined as a torus $5$ that contains a regular element (an element $s \in S$ is regular if the dimension of the centralizer $C _ { G } ( s )$ in $k$ is minimal), and it is then called a semi-regular torus if it is regular in the sense of the original definition (see, for example, [1]). Both these definitions are equivalent for reductive groups.

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039