Split algebraic group: Difference between revisions
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An [[algebraic group]] over a field <math>K</math> (not necessarily algebraically closed) is termed a '''split algebraic group''' if its [[Borel subgroup]] has a [[composition series of an algebraic group|composition series]] such that all the composition factors (i.e., all the successive group quotients) are isomorphic to either the additive group of <math>K</math> or the multiplicative group of <math>K</math>. | An [[algebraic group]] over a field <math>K</math> (not necessarily algebraically closed) is termed a '''split algebraic group''' if its [[Borel subgroup]] has a [[composition series of an algebraic group|composition series]] such that all the composition factors (i.e., all the successive group quotients) are isomorphic to either the additive group of <math>K</math> or the multiplicative group of <math>K</math>. | ||
Note that if <math>K</math> is an [[algebraically closed field]], then every [[affine algebraic group]] over <math>K</math> is split. | Note that if <math>K</math> is an [[algebraically closed field]], then every [[linear algebraic group]] (and hence, every [[affine algebraic group]]) over <math>K</math> is split. | ||
Revision as of 16:36, 1 January 2012
This article defines a property that can be evaluated for an algebraic group. it is probably not a property that can directly be evaluated, or make sense, for an abstract group|View other properties of algebraic groups
Definition
An algebraic group over a field (not necessarily algebraically closed) is termed a split algebraic group if its Borel subgroup has a composition series such that all the composition factors (i.e., all the successive group quotients) are isomorphic to either the additive group of or the multiplicative group of .
Note that if is an algebraically closed field, then every linear algebraic group (and hence, every affine algebraic group) over is split.