In Highlights 12, we used some of the equivalent conditions for a connected algebraic group $ G$ over a field $ k=\overline{k}$ to have semisimple rank 1 in the study of reductive groups (these are the groups whose unipotent radical $ R(G)_u$ is trivial).

Precisely, we showed that such a $ G$ must have a semisimple commutator $ [G,G]$ subgroup whose dimension is three, and that we can write

$ G = Z(G)^\circ\cdot [G,G]$

where the $ -\cdot-$ denotes that this is an almost direct product: in other words, the multiplication map $ Z(G)^\circ\times[G,G]\to G$ is surjective with finite kernel.

Let $ T\subseteq G$ be a maximal torus. We will show in this post that $ C_G(T) = T$ for a connected reductive group $ G$ of semisimple rank 1.
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To analyse the structure of a group G
you will need the radical and a torus T.
The group of Weyl may also may also suit
to prevent the scattering of many a root.
Functors are nice including the one of Lie
Parabolics bring in the ge-o-metry!
The theory of weights may seem oh so eerie
Until you start representation theory!

The structure of reductive and semisimple groups over an algebraically closed field will be pinnacle of this post series. After we have finished with this, this series will end and we will start to learn about algebraic groups from the perspective of group schemes, and we shall use some of the results we have seen so far by using that we really have just been studying the $ \overline{k}$-points of group schemes (classical algebraic geometry).

The topic for today is the radical and unipotent radical, that will allow us to define the concept of semisimple and reductive group. We will then use the roots, which are certain characters of a maximal torus. These will give us a root system, so we will take a break to study these, and classification of root systems will enable us to classify algebraic groups.
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Last time in Wild Spectral Squences 2, we saw how to prove the five lemma using a spectral sequence. Today, we’ll see a very simple application of spectral sequences to the concept of cohomological dimension in group cohomology.

We will define the well-known concept of cohomological dimension of a group $ G$, and then show how the dimension of $ G$ relates to the dimension of $ G/N$ and $ N$ for a normal subgroup $ N$. We do this with a spectral sequence. Although this application will appear to be very simple, it might be a good exercise for those just learning about spectral sequences.

The Category

For the sake of conreteness, let us work in the category of $ G$-modules where $ G$ is a profinite group. The $ G$-modules are the $ \mathbb{Z}G$-modules $ A$ with a continuous action of $ G$ where $ A$ is given the discrete topology, but we could be working with any group $ G$ with suitable, minor modifications, or Lie algebras, etc.

Like in all homology theories, there is the notion of cohomological dimension: a profinite group $ G$ has cohomological dimension $ n\in \mathbb{N}$ if for every $ r > n$ and every torsion $ G$-module $ A$, the group $ H^r(G,A)$ is trivial. If no such $ n$ exists, we say that $ G$ has cohomological dimension $ \infty$. The usual arithmetic rules in working with $ \infty$ apply.
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Posted by Jason Polak on 15. April 2013 · 2 comments · Categories: algebraic-geometry, group-theory · Tags: ,

highlights11In order to understand the structure of reductive groups, we will first look at some “base cases” of groups that are quite small. These are the groups of so-called semisimple rank 1, which by definition are the algebraic groups $ G$ such that $ G/R(G)$ has rank 1, where $ R(G)$ is the connected component of the unique largest normal solvable subgroup. In this post, we shall see a detailed proof of a theorem that gives several different characterisations of these groups.

As usual, we consider algebraic groups over an algebraically closed field $ k$. The proof we follow will be Theorem 25.3 in Humphrey’s book “Linear Algebraic Groups”. The reason I will go through it here is because in the book, I found the proof a bit terse and in a few points the proof relies on exercises, so it should be instructive to write down a more self-contained proof in my own words.
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In Highlights 9 of this series, we showed that for an algebraic group $ G$ and a closed subgroup $ H\subseteq G$, we can always choose a representation $ G\to\rm{GL}(V)$ with a line $ L\subseteq V$ whose stabiliser is $ H$. In turn, this allows us to identify the quotient $ G/H$ with the orbit of the class $ [L]$ in the projective space $ \mathbf{P}(V)$, which satisfies the universal property for quotients, thereby giving us a sensible variety structure on $ G/H$.

In this post, we specialise to the case of a Borel subgroup $ B\leq G$; that is $ B$ is maximal amongst the connected solvable groups. Such a subgroup is necessarily closed!

The fact that will allow us to study Borel subgroups is the fixed point theorem: a connected solvable group that acts on a nonempty complete variety has a fixed point. By choosing a representation $ G\to \rm{GL}(V)$ with a line $ L\subseteq V$ whose stabiliser is $ B$, we get identify $ G/B$ with a quasiprojective variety. However, in this case $ G/B$ is actually projective. Here is a short sketch:
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Conventions: an algebraic group here is a linear algebraic group over a fixed algebraically closed field $ k$.

In Highlights 6 and Highlights 7 in this series on algebraic groups, we saw that given any finite dimensional $ k$-subspace $ F\subseteq k[G]$, we can find a finite dimensional subspace $ E\supseteq F$ such that $ E$ is stable under all left-translations. (The same method shows that we can choose $ E$ invariant under all right translations in the case of $ G\times G\to G$. Explicitly, if $ f\in k[G]$ and $ x\in G$ then the right translation $ \rho_x$ is defined by $ (\rho_xf)(y) = f(yx)$.)

Using either left or right translations, we can use the techniques we have seen to construct for any closed subgroup $ H\subseteq G$ a representation $ \phi:G\to \rm{GL}(V)$ that contains a line $ L$ whose stabiliser in $ G$ is exactly $ H$. This is an extremely important technique that will allow us to realise the homogeneous space $ G/H$ as a quasiprojective variety! This will come in handy later for studying Borel subgroups $ B\subseteq G$. In this case we will see that $ G/B$ is actually projective (and in our setting, equivalently, complete), which is a key observation for deducing the structure of reductive groups.

In this post, we shall go through the construction of this representation.
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Borel subgroups are an important type of subgroup that will allow us to gain insight into the mysterious structure of algebraic groups. We shall look at the definition and some basic examples in this post. As usual, algebraic group means some linear algebraic group defined over an algebraically closed field $ k$.

A Borel subgroup $ B\subseteq G$ of an algebraic group $ G$ is a maximal connected solvable subgroup amongst the solvable subgroups of $ G$. Notice the absense of the adjective “closed” here: although $ G$ may contain solvable groups that are not closed, one that is maximal amongst the connected solvable ones must be closed. Indeed, if $ B$ is a subgroup maximal amongst the connected solvable subgroups of $ G$ then its closure $ \overline{B}$ is also connected and solvable.

Before we go any further, it’s helpful to have an example. Since all linear algebraic groups are closed subgroups of some $ \mathrm{GL}_n(k)$, let’s do $ \mathrm{GL}_n(k)$. We claim that a Borel subgroup of $ \mathrm{GL}_n(k)$ is $ B = \mathrm{T}_n(k)$, the closed subgroup of upper triangular matrices. Let’s now sketch the proof that $ B$ is actually a Borel.
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Let $ H\subseteq G$ be a subgroup of a topological group $ G$ (henceforth abbreviated “group”). If the induced topology on $ H$ is discrete, then we say that $ H$ is a discrete subgroup of $ G$. A commonplace example is the subgroup $ \mathbb{Z}\subseteq \mathbb{R}$: the integers are normal subgroup of the real numbers (with the standard topology).

Observe that in $ \mathbb{R}$, the subset $ \mathbb{Z}$ is also a closed set. What about an arbitrary discrete subgroup $ H\subseteq\mathbb{R}$? In other words, if $ H$ is discrete, is $ H$ necessarily closed?
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In the previous post, we saw that if $ G\times X\to X$ is an algebraic group acting on a variety $ X$ and $ F\subseteq k[X]$ is a finite-dimensional subspace then there exists a finite dimensional subspace $ E\subseteq k[X]$ with $ E\supseteq F$ such that $ E$ is invariant under translations.

Recall that if $ g\in G$ and $ f\in k[X]$ then the translation $ \tau_g(f)(y) = f(g^{-1}y)$ so that $ E$ being invariant under translations means that $ \tau_g(E) = E$ for all $ g\in G$. Now, let’s use the method outlined in the previous post to actually construct a three-dimensional representation of $ G = \mathrm{SL}_2(k)$.

The Example

In this setting we specialise to the case where $ G$ is an algebraic group acting on itself via multiplication: $ m:G\times G\to G$ is given by a Hopf algebra homomorphism $ \Delta:k[G]\to k[G]\otimes_kk[G]$. Of course, in this case we will need to actually choose some finitely generated Hopf $ k$-algebra as our ring of functions of $ \mathrm{SL}_2(k)$. Let’s use $ k[\mathrm{SL}_2] = k[T_1,T_2,T_3,T_4]/(T_1T_4 – T_2T_3 – 1)$. Thus we think of elements of $ \rm{SL}_2(k)$ as homomorphisms $ k[\mathrm{SL}_2]\to k$ corresponding to the matrix:

$ \begin{pmatrix}T_1 & T_2 \\ T_3 & T_4\end{pmatrix}$

The comultiplication map is then easily checked to be given by
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Posted by Jason Polak on 18. March 2013 · Write a comment · Categories: algebraic-geometry, group-theory · Tags: ,

Soon it will be time to explore some aspects of root systems and structure theory for reductive groups. Our goal is to understand everything in the classical setting over an algebraically closed field, and then explore reductive groups over arbitrary base schemes.

Before we do this, I will give a few examples for some of the technical machinery we shall rely on. In this post, we shall see how an algebraic group acting on a variety $ X$ and a function $ f\in k[X]$ gives rise to a representation of $ G$, and in the next post we shall see an example. I learnt the material in this section mainly from Jim Humphrey’s book “Linear Algebraic Groups”.

Finite Dimensional, Infinite Dimensional

Our setting is an arbitrary algebraic group $ G$ over an algebraically closed field $ k$.

Let $ X$ be a variety over $ k$ on which $ G$ acts, so that $ G\times X\to X$ is a group action and a morphism of varieties. If $ g\in G$ then there is a translation algebra homomorphism $ \tau_g:k[X]\to k[X]$ defined by $ \tau_g(f)(y) = f(g^{-1}y)$. The inverse is there so that

$ [\tau_g(\tau_h(f))](y) = \tau_h(f)(g^{-1}y) = f(h^{-1}g^{-1}y) = f((gh)^{-1}y) = \tau_{gh}(y)$

In other words, $ G\to\mathrm{Aut}_{k}(k[X])$ is actually a group homomorphism. Now, $ k[X]$ is a $ k$-vector space and thus this gives a representation of $ G$, but it is infinite dimensional. How can we get finite dimensional representations?
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