Regular torus
An algebraic torus in a connected algebraic group (over an algebraically closed field) that is contained in only a finite number of Borel subgroups (cf. Borel subgroup). The maximal tori in
are always regular (cf. Maximal torus). In general, a torus
is regular if and only if its centralizer
is a solvable group. One-dimensional regular tori
and their corresponding one-parameter subgroups
(also called regular) play an important role in algebraic group theory. A torus that is not regular is called singular. For reductive groups
(cf. Reductive group), a criterion for the singularity of a torus
can be given in terms of root systems. Thus, if
is a maximal torus in
containing
and
is the corresponding root system, then
is singular if and only if
for some
.
A regular torus in is sometimes defined as a torus
that contains a regular element (an element
is regular if the dimension of the centralizer
in
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 |
Regular torus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_torus&oldid=44283