be *observable*.

In the introduction of ERA the authors write:

[Notations and prerequisites.]{} In this paper we assume that the reader is familiar with the basic results and notations of the theory of affine algebraic groups its actions and representations which appear –eventually with slight differences– in the intial chapters of the standard textbooks on the subject such as: A. Borel’s [@kn:borelbook], C. Chevalley’s [@kn:Che], G. Hochschild’s [@kn:hobook], J. E. Humphrey’s [@kn:humph], T.A. Springer’s [@kn:sprbook] or the more recent monograph [@kn:nosotros2]. We work with groups and varieties defined over an algebraically closed field that will be denoted as $\Bbbk$.

If $G$ is an affine algebraic group then the algebra $\Bbbk[G]$ of polynomial functions on $G$ (with pointwise operations of sum and product) is in fact a Hopf algebra and its operations are defined as follows. The comultiplication $\Delta:\Bbbk[G] \to \Bbbk[G] \otimes
\Bbbk[G] \text{ written as } \Delta(f)=\sum f_1 \otimes f_2$ –Sweedler’s notation– is characterized by the fact that for all $x,y
\in G$: $\sum f_1(x)f_2(y)=f(xy)$. The antipode $S: \Bbbk[G] \to
\Bbbk[G]$ is defined for all $x \in G$ as $S(f)(x)=f(x^{-1})$ and the counit $\varepsilon: \Bbbk[G] \to \Bbbk$ is $\varepsilon(f)=f(e)$. In particular de left and right translations of $f$ by an 