Regular torus

An algebraic torus in a connected algebraic group $ G $ ( over an algebraically closed field) that is contained in only a finite number of Borel subgroups (cf. Borel subgroup). The maximal tori in $ G $ 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 $ S $ 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 $ G $ ( 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 $ G $ containing $ S $ and $ \phi (T,\ G) $ is the corresponding root system, then $ S $ is singular if and only if $ S \subset \mathop{\rm Ker}\nolimits \ \alpha $ for some $ \alpha \in \phi (T,\ G) $ .

A regular torus in $ G $ is sometimes defined as a torus $ S $ that contains a regular element (an element $ s \in S $ is regular if the dimension of the centralizer $ C _{G} (s) $ in $ G $ 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