Invariants of polynomial algebras on infinite G-modules Is there a reasonable characterization of the ring of $G$-invariants
($G$ - a finite dimensional simple Lie group) in the polynomial algebra
on direct sum of infinite number of irreducible $G$-modules?
To be more specific: is it possible to determine in
an explicit way the ring of $SL(2)$-invariants of the polynomial ring
$\mathrm{Pol}[sl(2)\oplus sl(2)\oplus sl(2)\oplus ...]$,
where $sl(2)$ is regarded as a copy of 3-dimensional adjoint representation
of $SL(2)$ (say, over complex numbers)?
(I'm asking this for a friend who hasn't a mathoverflow account yet and hope this is not a violation of the rules.)
 A: I assume your ground field is ${\mathbb C}$. Any invariant polynomial on ${\rm Pol}[\sum_1^\infty\mathfrak{sl}(2)]$ is an invariant polynomial on some finite sum, say the first $r$ copies $\sum_1^r\mathfrak{sl}(2)$. Also, the image of ${\rm SL}(2)$ as a linear group acting on $V={\mathbb C}^3\cong\mathfrak{sl}(2)$ is the special orthogonal group ${\rm SO}(3)$.
Now by Kraft-Procesi, Classical Invariant Theory, A Primer, Thm. 10.2$^*$, the invariants ${\rm Pol}[V^{\oplus r}]^{{\rm SO}(3)}$ are generated by the quadratic and cubic invariants. Specifically, since $V$ is isomorphic to its dual as ${\rm SO}(3)$-representation, we have a non-degenerate ${\rm SO}(3)$-equivariant map $V\otimes V\rightarrow{\mathbb C}$, which gives us a quadratic invariant for each pair $i,j$ with $1\leq i\leq j\leq r$. For the cubics, by identifying $V\oplus V\oplus V$ with the space of $3\times 3$ matrices, the determinant gives us an invariant for every $i,j,k$ with $1\leq i\leq j\leq k\leq r$.
$^*$I am referring to the preliminary version which can be downloaded at Kraft's homepage.
