Thursday, July 19th, 2012 | Author: Konrad Voelkel
If you don't have a solid education in linear algebraic groups, you might nevertheless encounter the term "reductive groups" now and then. People keep telling you to think about , as an algebraic group, and that's a good first approximation. If you want to go a step further, some confusion can happen, since the definition of reductive groups can be given for groups over the complex numbers in two quite different ways and only one of them generalizes to reductive groups over other fields, and if one wants to do non-perfect fields, it gets even more complicated.
I want to give some very short explanations on how to think about reductive groups and semisimple algebraic groups.
Short warning: I'm not going to discuss reductive Lie groups or reductive p-adic, adelic, whatsoever Lie groups. These occur, for example, by taking the points in a topological ring and obtaining a new topology, and have a rich structure theory by themselves. To complete the warning, I mention that the concept of reductive groups is not the same for Lie groups as it is for algebraic groups, since the real points of a unipotent algebraic group (which is like the opposite of reductive) can be a reductive Lie group. The correct bridge to understand this is the Lie algebra.
Short explanation: connected linear algebraic groups are those where you have a useful theory of Borel subgroups and parabolic subgroups; reductive groups are a special case where you have a useful theory of root systems and Bruhat decomposition. Much of the theory of reductive groups becomes easier in the special cases of split groups (this can be done by base change to an algebraically closed field), of (étale) simply connected groups (this can be done by covering with a unique simply connected cover), or of semi-simple groups (this can be done by quotienting out the so-called radical).
- Elliptic curves or more generally Abelian varieties are by definition projective algebraic group varieties, so they are not affine algebraic groups.
- , the additive group, with , is an affine algebraic group, but it is not reductive.
- is called a vector group (not reductive).
- For any commutative affine algebraic group decomposes as , a product of the closed subgroup of semisimple elements with the closed subgroup of unipotent elements.
- is reductive (and split). Special case: , the multiplicative group, with .
- is called a torus (it's reductive).
- is semi-simple (hence reductive) and simply connected.
- is semi-simple but not simply connected; is its simply connected cover.
- for a quadratic form over a field is a reductive group, defined over , but not necessarily split over (depending on ). If is a splitting field of , the base change of to is a split reductive group.
- Linear algebraic groups which are connected, semi-simple, split and simply-connected are called "Chevalley groups of universal type" and they can be constructed explicitly from their root system (and are classified by their root data). They are all defined over the integers. The groups and are particular examples.
I'm sure that there are more important examples that one should mention here, so I welcome any suggestions in the comment box below.
Reductive Groups Over The Complex Numbers
First of all, a reductive group is a linear algebraic group (with extra properties), which by definition means it is an affine group scheme, and one can show that it admits a map for some n. The most important objects to study a linear algebraic group are the spaces it acts on, especially the linear ones, namely representations, and the subobjects, i.e. subgroups and quotients, which correspond to normal subgroups.
Linear algebraic group over the complex numbers means that the affine group scheme is defined over the complex numbers, i.e. by polynomial equations with complex coefficients.
In this case, one can define: a linear algebraic group over the complex numbers is reductive if its representation category (the category of all finite dimensional complex representations) is completely reducible, i.e. every representation decomposes as a direct sum of simple objects. Warning: For representations, the term "completely reducible" is a synonym of "semisimple", whereas for linear algebraic groups, "reductive" and "semisimple" have a different meaning. To get this straight, we will later on discuss semisimple linear algebraic groups, which are all reductive, but not the other way around.
On one hand, this definition is short and neat, but on the other hand it is not attached to subobjects or quotients of the group, but to its representation category, which might seem a little bit away from the group. While it is true that one can recover a linear algebraic group from its representation category by the Tannakian formalism, one could hope for something more direct.
In fact, for fields of characteristics 0, one can express the reductivity property by the Lie algebra (no wonder, since the representation theory is also governed by the representation theory of the Lie algebra): A finite dimensional Lie algebra over a field of char. 0 is reductive if its adjoint representation is completely reducible (i.e. is a direct sum of simple representations).
One defines a reductive Lie group by requiring its Lie algebra to be reductive.
One can prove that a Lie algebra over a field of char. 0 is reductive iff it is the Lie algebra of a reductive algebraic group over this field.
Warning: it does not follow that the representation category of the reductive Lie algebra itself is completely reducible.
Reductive Algebraic Groups, in Terms of Radicals
Now suppose we have an algebraically closed field k and all the groups we consider in this paragraph are defined over this field k.
We define the radical of a linear algebraic group as a maximal subgroup among all subgroups that are closed, connected, normal and solvable, denoted .
I want to remind you of the solvability condition in the discrete case:
A discrete group G is called solvable if the descending derived normal series
stabilizes to the trivial group in finitely many steps, so there is some n such that . Here we use the notation for the iterated commutator subgroup. Abelian groups have , so solvable generalizes abelian.
More importantly, nilpotent groups are solvable. Nilpotent (discrete) groups are those which have a finite length lower central series
We define the unipotent radical of a linear algebraic group as a maximal subgroup among all subgroups that are closed, connected, normal and unipotent, denoted .
A group is unipotent if all its elements are unipotent. One characterization is that under any embedding a unipotent element of G is mapped to a unipotent element of . An element of is called unipotent if is a nilpotent endomorphism of . Now you might think that's a bad definition, and yes, there are other definitions, but they are not shorter.
Any unipotent group is solvable, so and for quotients we also get
an epimorphism .
The definition of a reductive group in terms of radicals is now:
An affine algebraic group G is called reductive if it is connected and . It is called semi-simple if furthermore .
For any reductive group G there is the semi-simple quotient . A nice characterization of the radical is that it is precisely the intersection of all Borel subgroups.
As an example, look at , which has no closed connected normal unipotent subgroup, hence vanishing unipotent radical. Any subgroup of the diagonal matrices with roots of unity as entries is a closed solvable subgroup, and all of them (together with all permutations which are diagonal in another basis) form the radical. The quotient is precisely , since you can represent the determinant of every (nxn)-matrix by a diagonal matrix with entries (over an algebraically closed field, of course).
Reductive Groups Over Perfect Fields
Now we take a field k (any field) and let our groups be defined over that field k.
You can take the definition in terms of radicals and unipotent radicals, to define what a reductive or a semisimple linear algebraic group over a field k is, and just use it with the maybe non-closed field k. One calls these notions k-reductive and k-semisimple. One obvious question is, if the base change to an algebraic closure will be -reductive or -semisimple again. In general, this is wrong. One calls a group defined over some field k reductive (resp. semisimple) if the base change to a separable closure is -reductive (resp. -semisimple).
A perfect field is a field with the property that all finite field extensions are separable. Examples: finite fields an algebraically closed fields. Counterexample: .
One advantage of perfect fields is that you can replace any separable closure with an algebraic closure (it's just the same). Much of Galois theory, and therefore étale topology, becomes complicated at the first encounter because of this issue.
For affine algebraic groups defined over perfect fields k, the notion k-reductive coincides with the notion from the previous paragraph. There are affine algebraic groups defined over non-perfect fields k which are k-reductive but not reductive, so you have to be super-careful. Look at the WP article on pseudo-reductive groups for more info.
Reductive Groups Over Schemes
If you look at affine algebraic groups defined over a local ring, you can take the base change to the residue field and thus talk about reductive and semisimple groups over local rings.
In SGA3, Exposé XIX, 2.7 a reductive group over a scheme S is defined as a smooth affine group scheme over S such that all the geometric ﬁbers are connected reductive algebraic groups. That specializes to S being the spectrum of a ring and becomes less scheme-theoretic for a local ring (if you want to work out some examples ...).
- T. Springer: Linear Algebraic Groups (Vol 9 in Progress in Mathematics, or republished by Birkhäuser; make sure to read 2nd edition instead of 1st)
- J. Milne: Course Notes on Reductive Groups
- Encyclopedia of Mathematics: Reductive Group (really worth reading! short!)
- Encyclopedia of Mathematics: Reductive Lie Algebra
- Wikipedia: Reductive Group