# Algebraic Groups III (structure theorems)

This post will likely contain no proofs (might change later). I just want to collect some more substantial results about algebraic groups.

The last post left off with the definition of semisimple groups and a basic result about them. Continuing in this direction:

Thm: If is a semisimple group over a field then is the connected component of . Further .

It is classical that any element can be decomposed uniquely as . invertible implies is invertible. Then where is unipotent.

Thm: If is any affine algebraic group then any can be decomposed into . An analogous decomposition applies to the Lie algebra. Further is closed.

The key to proving this is verifying it for and noting that any affine algebraic group can be embedded as a closed subgroup of . For the statement about note that as is closed so is the intersection of two closed subset.

Thm (structure thm): Let is any commutative affine algebraic group. If is connected then so are . Further the multiplication map is an isomorphism and and .

A torus is an algebraic group isomorphic to the diagonal matrices in some . A d-group is an affine algebraic group such that has a basis consisting of characters. E.g. consider . Then . The character group of is isomorphic to ; i.e. is . This also gives all monomials in , i.e. a basis.

Thm: If is a d-group then where is a torus and is a finite group whose order is not divisible by the characteristic of the field.

### Solvable and Nilpotent

Analogously to how these notions are defined with Lie algebras there is the upper series and the lower series where is the group generated by all commutators .

is solvable if $D^nG = e$ for some . It is nilpotent if for some .

Thm: If is a positive dimensional nilpotent group then is also positive dimensional. Further if is a proper closed subgroup then .

proof : This is not so hard to prove once its established that

Lemma: If are closed subgroups of with connected then .

this in turn is not hard to establish modulo

Prop. Let be the inverse morphism. If is any family of morphisms such that

1) is also part of the family

2) the are irreducible varieties.

3)

then the group generated by the is closed and connected. Further there is a finite sequence such that .

This is proved in Humphrey’s book, section 7.5. The proof is a little technical but also another good example of what Chevalley’s thm is good for. No proof for now.

proof (of lemma): Simply consider the family of morphisms for given by . Apply the prop. QED.

let be the largest number such that . By the lemma its connected and hence hence positive dimensional.

For the second statement use induction on . Set . Either in which case replace with and use induction hypothesis or showing the dimensions are not equal. QED.

The following is the group theoretic analogue of Lie and Engel’s theorem.

Thm: If is finite dimensional and it is unipotent or solvable then has a common eigenvector.

It implies any connected solvable is a subset of , the upper triangular matrices over a vector space of dimension over a field . There’s a split short exact sequence

where are the unipotent matrices in vector space of dimension over a field . Intersecting with gives

is a closed connected subgroup of hence also a torus. This is some intuition for the following result

Thm: Let be a connected solvable group. Then is closed connected normal and contains . Set . The maximal tori of are all conjugate under and fixing a maximal torus: .

Thm(fixed point): If is connected and solvable and acts on a complete variety then it has a fixed point.

### Borel and Root subgroups

Let be an affine algebraic group. The identity component of the maximal normal solvable subgroup is called the radical of . Then is the unipotent radical. If is connected, then a borel subgroup is a maximal solvable closed subgroup.

Suppose is connected. It is semisimple if . It is reductive if .

Thm: Let be a borel subgroup. Then is projective and all other borel subgroups of are conjugate. Further iff is projective.

I think the proof of this statement is quite enlightening but this post is already too long…

The groups in the thm are called parabolic subgroups. Reductive groups have the following nice property

Thm: Let be reductive; a maximal torus and the set of roots; let . There exists a unique -stable subgroup of having . There is an isomorphism such that . Note .

In the case of matrix lie groups the first claim is easy to see: The group in question is roughly . Verify that’s is -stable.

If then

which is in since .

proof (of the assertion): the group is one dimensional and the only such groups are . So general theory says there is some isomorphism . For and for consider

this is an automorphism . So gives a character of . More explicitly for a character . Or

So there is a commutative diagram formed by and by . Looking at the map on differentials and writing it follows that

.

### Bruhat Decomposition

Let be a reductive group. Fix a maximal torus and a borel subgroup containing it. = the unipotent elements of , and the Weyl group; for I use the convention to say that maps to .

More notation

; a root subgroup.

is independent of choice of lift.

Technical proposition:

One version of Bruhat decomposition says for fixed and and for any there are unique such that . Using and I see also that

a more common version is that .

this actually has a short axiomatic proof, but that’s for another time.

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- July 10, 2010 / 11:30 pm

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