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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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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As in many mathematics departments, graduate students in McGill’s Department of Mathematics and Statistics have to take a comprehensive examination comprising of two parts: a written part (Part A) and an oral part (Part B). The Part B exam is based on two topics related to the student’s field of research. One of my Part B topics is algebraic groups.

The algebraic groups section will consist of some classical material found in usual sources, for which I am mainly using Humphreys’ book, and Brian Conrad’s notes on reductive group schemes, which can be found on his website.
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