$\DeclareMathOperator\SL{SL}$As explained in [this question][1] , there are two $\lambda$-ring structures on ${\mathbb Z}[x]$. In layman's terms, both come from a realization of ${\mathbb Z}[x]$ as the representation ring of an algebraic (semi)-group: it is either $R({\mathbb C},\times)$, or $R(\SL_2({\mathbb C}))$.

**Question 1:** What are $\lambda$-ring homomorphisms between these? As we can choose any of the two structures for the range and the domain, it is four separate questions.

In particular, I am interested in the semi-group of endomorphisms of $R(\SL_2({\mathbb C}))$. I can see 3 of its elements: zero, identity and 
$$P: R(\SL_2({\mathbb C}))\cong R(\operatorname{PSL}_2({\mathbb C}))\hookrightarrow R(\SL_2({\mathbb C})).$$ 
If one thinks of $x$ as the standard 2-dimensional representation of $\SL_2$, they are given by
$$0: x\mapsto 2, \ \ 1:x\mapsto x, \ \ P:x\mapsto x^2-2.$$
**Question 2:** Is this semigroup cyclic? Note that $P$ will be the generator.

  [1]: https://mathoverflow.net/questions/109642/lambda-ring-structures-on-mathbb-a2