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The decomposition G = HR,(G) described in the theorem is called the Levi decomposition. One can use it to reduce many problems to reductive groups. 3 is the analog of the theorem for Lie groups obtained by Levi and Maltsev (cf. Maltsev [1,2]). 10. Reductive groups. The basic properties of reductive groups are listed in the following theorem. 4. Let G be a reductive K-group. e. H n S is finite); (4) if char K = 0, then any algebraic representation f : G -+ GL, ( 0 ) is completely reducible. A deeper analysis of reductive, and especially semisimple, groups is based on the concept of a root system.

Note that not every K-morphism f : RLIK(G) 4 RLIK(H) has the form j = RLIK(f) for a suitable Lmorphism f : G 4 H. 4) it is easy to obtain the equality X ( R L I K ( G ) ) ~ = X(G)L for the groups of rational characters (cf. 61). Restriction of scalars has two noteworthy arithmetic properties. Let LIK be an extension of an algebraic number field and let v E VK. 3. The Lie algebra of an algebraic group. e. for any two points gl, g2 E G the translation map x t+ g2g;1x is a morphism of G as an algebraic variety, sending gl to g2.

N - 1. Note that most of the above statements do not carry over for positive characteristic. We shall require a technical assertion about unipotent groups. such that the factors Gi/Gi+1 are isomorphic to G, or (6,. 5) of K-subgroups such that the Gi/Gi+l are K-isomorphic to 6 , or G,, then G is said to be K-split. Indeed, this is equivalent to the existence of a maximal K-split torus T C G, and then any K-torus in G is K-split. 5) are K-isomorphic to 6,. 9. Connected groups. , maximal connected solvable subgroups).