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Difference between revisions of "Quasi-split group"

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''quasi-splittable group, over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767101.png" />''
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''quasi-splittable group, over a field $k$''
  
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An affine algebraic group (cf. [[Affine group|Affine group]]; [[Algebraic group|Algebraic group]]) defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767102.png" /> containing a [[Borel subgroup|Borel subgroup]] defined over the same field. Every affine algebraic group becomes a quasi-split group for some extension of the ground field, for example, over the algebraic closure of this field. Every affine algebraic group defined over a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767103.png" /> is quasi-split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767104.png" />. See also [[Split group|Split group]].
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An affine algebraic group (cf. [[Affine group|Affine group]]; [[Algebraic group|Algebraic group]]) defined over $k$ containing a [[Borel subgroup|Borel subgroup]] defined over the same field. Every affine algebraic group becomes a quasi-split group for some extension of the ground field, for example, over the algebraic closure of this field. Every affine algebraic group defined over a finite field $k$ is quasi-split over $k$. See also [[Split group|Split group]].
  
  
  
 
====Comments====
 
====Comments====
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A group can be quasi-split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767105.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767107.png" />-quasi-split, without being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076710/q0767108.png" />-split.
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A group can be quasi-split over $k$, or $k$-quasi-split, without being $k$-split.
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Humphreys,   "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table>

Latest revision as of 16:48, 7 July 2014

quasi-splittable group, over a field $k$

An affine algebraic group (cf. Affine group; Algebraic group) defined over $k$ containing a Borel subgroup defined over the same field. Every affine algebraic group becomes a quasi-split group for some extension of the ground field, for example, over the algebraic closure of this field. Every affine algebraic group defined over a finite field $k$ is quasi-split over $k$. See also Split group.


Comments

A group can be quasi-split over $k$, or $k$-quasi-split, without being $k$-split.

References

[a1] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039
How to Cite This Entry:
Quasi-split group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-split_group&oldid=13662
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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