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An [[Algebraic torus|algebraic torus]] in a connected [[Algebraic group|algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809001.png" /> (over an algebraically closed field) that is contained in only a finite number of Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]). The maximal tori in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809002.png" /> are always regular (cf. [[Maximal torus|Maximal torus]]). In general, a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809003.png" /> is regular if and only if its centralizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809004.png" /> is a [[Solvable group|solvable group]]. One-dimensional regular tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809005.png" /> and their corresponding one-parameter subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809006.png" /> (also called regular) play an important role in algebraic group theory. A torus that is not regular is called singular. For reductive groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809007.png" /> (cf. [[Reductive group|Reductive group]]), a criterion for the singularity of a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809008.png" /> can be given in terms of root systems. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r0809009.png" /> is a maximal torus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090010.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090012.png" /> is the corresponding root system, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090013.png" /> is singular if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090014.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090015.png" />.
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An [[Algebraic torus|algebraic torus]] in a connected [[Algebraic group|algebraic group]] $  G $ (
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over an algebraically closed field) that is contained in only a finite number of Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]). The maximal tori in $  G $
 +
are always regular (cf. [[Maximal torus|Maximal torus]]). In general, a torus $  S \subset G $
 +
is regular if and only if its centralizer $  C _{G} (S) $
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is a [[Solvable group|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|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 $
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090016.png" /> is sometimes defined as a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090017.png" /> that contains a regular element (an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090018.png" /> is regular if the dimension of the centralizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080900/r08090020.png" /> 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, [[#References|[1]]]). Both these definitions are equivalent for reductive groups.
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A regular torus in $  G $
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is sometimes defined as a torus $  S $
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that contains a regular element (an element $  s \in S $
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is regular if the dimension of the centralizer $  C _{G} (s) $
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in $  G $
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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, [[#References|[1]]]). Both these definitions are equivalent for reductive groups.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel,   "Linear algebraic groups" , Benjamin (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys,   "Linear algebraic groups" , Springer (1975)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>

Latest revision as of 15:07, 17 December 2019

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=18431
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article