Difference between revisions of "Split group"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.I. Merzlyakov, "Rational groups" , Moscow (1980) (In Russian) {{MR|0602700}} {{ZBL|0518.20032}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR></table> |
Revision as of 14:52, 24 March 2012
over a field ,
-split group
A linear algebraic group defined over and containing a Borel subgroup that is split over
. Here a connected solvable linear algebraic group
is called split over
if it is defined over
and has a composition series (cf. Composition sequence)
such that the
are connected algebraic subgroups defined over
and each quotient group
is isomorphic over
to either a one-dimensional torus
or to the additive one-dimensional group
. In particular, an algebraic torus is split over
if and only if it is defined over
and is isomorphic over
to the direct product of copies of the group
. For connected solvable
-split groups the Borel fixed-point theorem holds. A reductive linear algebraic group defined over
is split over
if and only if it has a maximal torus split over
, that is, if its
-rank coincides with its rank (see Rank of an algebraic group; Reductive group). The image of a
-split group under any rational homomorphism defined over
is a
-split group. Every linear algebraic group
defined over a field
is split over an algebraic closure of
; if
is also reductive or solvable and connected, then it is split over some finite extension of
. If
is a perfect field, then a connected solvable linear algebraic group defined over
is split over
if and only if it can be reduced to triangular form over
. If
, then a linear algebraic group defined over
is split over
if and only if its Lie algebra
is a split (or decomposable) Lie algebra over
; by definition, the latter means that the Lie algebra
has a split Cartan subalgebra, that is, a Cartan subalgebra
for which all eigenvalues of every operator
,
, belong to
.
If is the real Lie group of real points of a semi-simple
-split algebraic group
and if
is the complexification of the Lie group
, then
is called a normal real form of the complex Lie group
.
There exist quasi-split groups (cf. Quasi-split group) over a field that are not split groups over
; the group
is an example for
.
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
[2] | A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402 |
[3] | Yu.I. Merzlyakov, "Rational groups" , Moscow (1980) (In Russian) MR0602700 Zbl 0518.20032 |
[4] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039 |
Split group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Split_group&oldid=21942