TITLE: Structure of $[S(\mathfrak{g})\otimes S(\mathfrak{g})]^G$ for semisimple $\mathfrak{g}$
QUESTION [5 upvotes]: Let  $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. $S(\mathfrak{g})^G$ is a polynomial algebra with rank $\mathfrak{g}$ generators. Call them $c_i(x)$, where $x\in \mathfrak{g}$ and $i=1,\ldots,\mathrm{rank}\ g$.
Now, consider $G$ acting diagonally on $\mathfrak{g}\oplus\mathfrak{g}$. What is the ring of invariants $R=[S(\mathfrak{g})\otimes S(\mathfrak{g})]^G$ ? (I guess this is a textbook material; sorry for not doing the homework of looking up the books, but it's a Saturday today and it's easier to ask here than to go to the library at my institute in a cold weather...)
For $x\oplus y \in \mathfrak{g}\oplus\mathfrak{g}$, I can easily see $R\ni c_i(x + t y)$ for a scalar $t$. Do they generate $R$ (polynomially)?

REPLY [7 votes]: Most of what is known in the case of $GL(n,K)$ acting diagonally by conjugation on $r$-tuples of $n \times n$ matrices can be found in Chapters 8-10 of my CBMS notes "The Polynomial Identities and Invariants of $n \times n$ matrices".  There $K$ is a field of characteristic zero, the fixed ring is called the ring of invariants of 
$n \times n$ matrices, and it is denoted $C(n,r)$.
First Fundamental theorem (H. Weyl):  $C(n,r)$ is generated by traces of monomials.
Second Fundamental Theorem (B. Konstant, C. Procesi, Y. P. Razmyslov):  Let $S_r$ be the symmetric group on $r$ letters, and let $J(n,r)$ be the two-sided ideal of the group ring
$KS_r$ corresponding to Young diagrams with $\geq n+1$ rows. Then there is an easily defined
map from $KS_r$ onto the multilinear elements of degree $r$ in $C(n,r)$ whose kernel is $J(n,r)$.
Nagata-Higman Theorem:  If an algebra $A$ (noncommutative, without a unit) satisfies 
$x^n = 0$ for every $x \in A$, then there is an integer $N = N(n)$ such that 
$x_1 \cdots x_N = 0$ for all $x_1, \dots ,x_N \in A$.  Procesi made the brilliant 
observation that the least such $N$ valid for all $A$ is also the least integer such that $C(n,r)$ is generated by traces of monomials of length $\leq N$.  Using the Second 
Fundamental Theorem Procesi and Razmyslov showed that $N(n) \leq n^2$.  Studying the
Nagata-Higman Theorem, E. N. Kuzmin showed that $N(n) \geq n(n+1)/2$ and conjectured that 
$N(n) = n(n+1)/2$.  His conjecture has only been verified for $n \leq 4$.
General theorems of M. Hochster - J. L. Roberts and M. P. Murthy imply that $C(n,r)$ is a
unique factorization domain and Gorenstein, and M. van den Bergh proved that it is
Cohen-Macaulay.
Results of several authors show that $C(n,r)$ is a polynomial ring if and only if $n = 1$,
or $r = 1$, or $(n,r) = (2,2)$, and is a complete intersection if and only it is a polynomial ring or $(n,r) = (2,3)$, or $(n,r) = (3,2)$.