Reductive group

In mathematics, a reductive group is an algebraic group "G" such that the unipotent radical of the identity component of "G" is trivial. Any semisimple algebraic group and any algebraic torus is reductive, as is any general linear group.

The name comes from the complete reducibility of linear representations of such a group, which is a property in fact holding over fields of characteristic zero. Haboush's theorem shows that a certain rather weaker property holds for reductive groups in the general case.

Lie group case

More generally, in the case of Lie groups, a reductive Lie group "G" is sometimes defined as one such that its Lie algebra "g" is the Lie algebra of a real algebraic group that is reductive, in other words a Lie algebra that is the sum of an abelian and a semisimple Lie algebra. Sometimes the condition that identity component "G"⁰ of "G" is of finite index is added.

A Lie algebra is reductive if and only if its adjoint representation is completely reducible, but this does not imply that all of its finite dimensional representations are completely reducible. The concept of reductive is not quite the same for Lie groups as it is for algebraic groups because a reductive Lie group can be the group of real points of a unipotent algebraic group.

For example, the one-dimensional, abelian Lie algebra R is obviously reductive, and is the Lie algebra of both a reductive algebraic group "G"_"m" (the multiplicative group of nonzero real numbers) and also a unipotent (non-reductive) algebraic group "G"_"a" (the additive group of real numbers). These are not isomorphic as "algebraic groups"; at the Lie algebra level we see the same structure, but this is not enough to make any stronger assertion (essentially because the exponential map is not an algebraic function).

ee also

*Root datum

References

*Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
*A. Borel, J. Tits, [http://www.numdam.org/item?id=PMIHES_1965__27__55_0 "Groupes réductifs"] Publ. Math. IHES , 27 (1965) pp. 55–150; [http://www.numdam.org/item?id=PMIHES_1972__41__253_0 "Compléments à l'article «Groupes réductifs»."] Publications Mathématiques de l'IHÉS, 41 (1972), p. 253-276
*Bruhat, François; Tits, Jacques "Groupes réductifs sur un corps local" : [http://www.numdam.org/item?id=PMIHES_1972__41__5_0 I. Données radicielles valuées.] Publications Mathématiques de l'IHÉS, 41 (1972), p. 5-251 [http://www.numdam.org/item?id=PMIHES_1984__60__5_0 II. Schémas en groupes. Existence d'une donnée radicielle valuée.] Publications Mathématiques de l'IHÉS, 60 (1984), p. 5-184
*springer|id=R/r080440|title=Reductive group|author=V.L. Popov
*springer|id=l/l058500|title=Lie algebra, reductive|author=A.L. Onishchik
*T. A. Springer, [http://www.ams.org/online_bks/pspum331/pspum331-ptI-1.pdf "Reductive groups"] , in [http://www.ams.org/online_bks/pspum331/ "Automorphic forms, representations, and L-functions" vol 1] ISBN 0-8218-3347-2