# Regular torus

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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
How to Cite This Entry:
Regular torus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_torus&oldid=44283
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article