Highlights in Linear Algebraic Groups 7: Representations II

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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Highlights in Linear Algebraic Groups 6: Representations I

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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