Posted by Jason Polak on 23. August 2016 · Write a comment · Categories: group-theory · Tags: ,


Let $k$ be a commutative ring. Let $\G_a$ be group functor $\G_a(R) = R$ and $\G_m$ be the group functor $\G_m(R) = R^\times$, both over the base ring $k$. What are the homomorphisms $\G_a\to \G_m$? In other words, what are the characters of $\G_a$? This depends on the ring, of course!

The representing Hopf algebra for $\G_a$ is $k[x]$. And, the representing Hopf algebra for $\G_m$ is $k[x,x^{-1}]$. Homomorphisms $\G_a\to \G_m$ correspond to Hopf algebra maps $k[x,x^{-1}]\to k[x]$. Such a map is a $k$-algebra homomorphism that satisfies the additional conditions for being a Hopf algebra homomorphism.
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Suppose we have a $2\times 2$ matrix
M = \begin{pmatrix}
x_{11} & x_{12}\\
x_{21} & x_{22}
with entries in a field $F$. The characteristic polynomial of this matrix is $p(t) := {\rm det}(tI_2 – M) = t^2 – (x_{11} + x_{22})t + x_{11}x_{22} – x_{21}x_{12}$. One might ask: how can we produce a matrix with a given characteristic polynomial? This can be accomplished using the rational canonical form:
t^2 + at + b\mapsto
0 & -b\\
1 & -a
We can calculate that the characteristic polynomial of this matrix to be $t^2 + at + b$. This map gives a bijection between quadratic monic polynomials in $F[t]$ and matrices of the above form. One way to understand this phenomenon is through algebraic groups. To explain, let's stick with $F$ having characteristic zero, though much of what we do can be done in characteristic $p$ for $p$ sufficiently large as well using very different techniques.
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The previous series on algebraic groups is over. Actually, I barely got to the root system and root datum of a reductive group, but I found that the whole slew of material was getting too complex to organise on this blog, which I feel is better for more self-contained posts. Instead, I have begun to write a set of notes in real LaTeX describing the root datum of a reductive group and root systems in general, most of which I have completed, and I will release a draft soon.

In the mean time, I will continue posts on algebraic groups by reviewing some theorem and then illustrating it with an application such an an exercises from some textbook. Let's start by reviewing a simple one: let $ k$ be an algebraically closed field. Then an irreducible finite set in the Zariski topology on $ k^n$ (as a classical variety) necessarily has one element. (A topological space is irreducible if it cannot be written as the union of two proper closed subsets.)

We added the disclaimer "as a classical variety" since over an algebraically closed field $ k$ it is sufficient to consider the closed points of $ \mathrm{Spec}(k[x_i])$—otherwise the statement would be false.
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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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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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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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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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Posted by Jason Polak on 14. June 2012 · 1 comment · Categories: algebraic-geometry · Tags: , , ,

I've decided to start this series with a few posts on the Lie algebra of an algebraic group. This seems to me the first real technical aspect of the classical theory that arises in Humphreys' book. We shall loosely follow this book as a guide, but we shall also deviate and look at more scheme-theoretic treatments.

We will define a functor from the category of algebraic groups over a commutative ring $ k$ to Lie algebras over $ k$. The idea is that Lie algebras are often easier to work with than algebraic groups directly, so the Lie algebra will help us with things (such as classification problems). For now, what we do will be general enough so that we do not need to assume that $ k$ is a field.

We shall look at several definitions of a Lie algebra of an algebraic group, and prove that they are all equivalent. After this, we shall examine what this functor does to morphisms (the "differential of a morphism") and then give a few examples of why this process is useful.
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