Isomorphism of affine group schemes of rank 2

Question

I was reading Waterhouse's book "Introduction to Affine Group Schemes", when I found, in Chapter 2, an exercise about classification of Affine Group Schemes of Rank 2. I proved essentially that every Affine Group Scheme of Rank 2 over a ring $R$ is of the kind $G_{a, b}=$Spec$A$, where $A$ is the $R$-Hopf Algebra given by $A=R[X]/(X^{2}+aX)$, with coproduct given by $\Delta(x)=x\otimes 1+1\otimes x+bx\otimes x$, where $x$ is the image of $X$ under the quotient map, and $a,b\in R$ such that $ab=2$. Now it remains to prove that $G_{a,b}$ is isomorphic to $G_{a',b'}$ if and only if there exists $u\in R$ invertible, such that $a=ua'$ and $b=u^{-1}b'$. I'd like to prove this using essentially the induced iso on the Hopf Algebras representing $G_{a,b}$ and $G_{a',b'}$, but I really don't find a way to realize the conditions on $a$ and $b$. Do you have any idea?

Answer 1

It seems that the naïve approach works, as follows. Let$$A = R[X]/(X^2+aX) = R[x], \quad B = R[Y]/(Y^2+a'Y) = R[y],$$with $$ \Delta(x) = x \otimes 1 + 1 \otimes x + bx \otimes x , \quad \Delta(y) = y \otimes 1 + 1 \otimes y + b'y \otimes y. $$ Let $\varphi: A \to B$ be a Hopf algebra isomorphism, and write $\varphi(x) = \lambda + \mu y$ for some $\lambda \in R$ and some $\mu \in R^\times$. Now, simply express that $\Delta(\varphi(x)) = (\varphi \otimes \varphi)(\Delta(x))$, and notice that$$\{ 1 \otimes 1, y \otimes 1, 1 \otimes y, y \otimes y \}$$forms a basis for the free $R$-module $A \otimes A$. Equating coefficients yields $$b\mu^2 = b'\mu, \quad b\lambda\mu = 0, \quad \lambda = 2\lambda + b\lambda^2.$$ Since $\mu$ is invertible in $R$, this yields $b' = b\mu$ and $\lambda=0$. Finally, expressing that $\varphi(x^2) = \varphi(x)^2$ yields $a = a'\mu$.