# Split group

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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
How to Cite This Entry:
Split group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Split_group&oldid=44286
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article