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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