Split group

over a field $ k $ , $ k $ - split group

A linear algebraic group defined over $ k $ and containing a Borel subgroup that is split over $ k $ . Here a connected solvable linear algebraic group $ B $ is called split over $ k $ if it is defined over $ k $ and has a composition series (cf. Composition sequence) $ B = B _{0} \supset B _{1} \supset \dots \supset B _{t} = \{ 1 \} $ such that the $ B _{i} $ are connected algebraic subgroups defined over $ k $ and each quotient group $ B _{i} /B _ {i + 1} $ is isomorphic over $ k $ to either a one-dimensional torus $ G _{m} \cong \mathop{\rm GL}\nolimits _{1} $ or to the additive one-dimensional group $ G _{a} $ . In particular, an algebraic torus is split over $ k $ if and only if it is defined over $ k $ and is isomorphic over $ k $ to the direct product of copies of the group $ G _{m} $ . For connected solvable $ k $ - split groups the Borel fixed-point theorem holds. A reductive linear algebraic group defined over $ k $ is split over $ k $ if and only if it has a maximal torus split over $ k $ , that is, if its $ k $ - rank coincides with its rank (see Rank of an algebraic group; Reductive group). The image of a $ k $ - split group under any rational homomorphism defined over $ k $ is a $ k $ - split group. Every linear algebraic group $ G $ defined over a field $ k $ is split over an algebraic closure of $ k $ ; if $ G $ is also reductive or solvable and connected, then it is split over some finite extension of $ k $ . If $ k $ is a perfect field, then a connected solvable linear algebraic group defined over $ k $ is split over $ k $ if and only if it can be reduced to triangular form over $ k $ . If $ \mathop{\rm char}\nolimits \ k = 0 $ , then a linear algebraic group defined over $ k $ is split over $ k $ if and only if its Lie algebra $ L $ is a split (or decomposable) Lie algebra over $ k $ ; by definition, the latter means that the Lie algebra $ L $ has a split Cartan subalgebra, that is, a Cartan subalgebra $ H \subset L $ for which all eigenvalues of every operator $ \mathop{\rm ad}\nolimits _{L} \ h $ , $ h \in H $ , belong to $ k $ .

If $ G _ {\mathbf R} $ is the real Lie group of real points of a semi-simple $ \mathbf R $ - split algebraic group $ G $ and if $ G _ {\mathbf C} $ is the complexification of the Lie group $ G _ {\mathbf R} $ , then $ G _ {\mathbf R} $ is called a normal real form of the complex Lie group $ G _ {\mathbf C} $ .

There exist quasi-split groups (cf. Quasi-split group) over a field $ k $ that are not split groups over $ k $ ; the group $ \mathop{\rm SO}\nolimits (3,\ 1) $ is an example for $ k = \mathbf R $ .

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