Let $M\_n$ be the set of $n$-by-$n$ matrices with complex entries, viewed as a variety over $k=\mathbb{C}$. Equip $M\_n$ with the conjugation action of $\mathrm{GL}(n)=\mathrm{GL}(n,\mathbb{C})$. Consider $A:=\mathrm{Mor}^{\mathrm{GL}(n)}(M\_n \oplus M\_n, M\_n)$, the set of $\mathrm{GL}(n)$-equivariant maps (of algebraic varieties) $M\_n \oplus M\_n \to M\_n$, where $GL(n)$ acts diagonally on $M\_n \oplus M\_n$. Then the (standard) algebra structure of $M\_n$ (with multiplication given by matrix multiplication) induces an algebra structure on $A$. 


The following maps belong to $A$:


1. $M\_n \oplus M\_n \to M\_n \colon (A,B) \mapsto A$;
2. $M\_n \oplus M\_n \to M\_n \colon (A,B) \mapsto B$;
3. $M\_n \oplus M\_n \to M\_n \colon (A,B) \mapsto f(A,B)I\_n$, where $I\_n$ is the identity matrix and $f$ is an element of the ring of invariants $k[M\_n \oplus M\_n]^{\mathrm{GL}(n)}$.


Do they generate $A$ as an algebra?



Generators for the algebra of GL(n)-equivariant maps from M_n + M_n to M_n
The answer is affirmative not only in the case of 2 matrices, but also in the case of any number of matrices; in fact, an analogous statement is true for quiver representations (in characteristic 0).


The original question can be restated as follows.



> 
> Let $P$ be the space of polynomial functions of 2 $n\times n$ matrices, with the adjoint action of $GL\_n$ and the ring of invariants $I.$ Consider the space $\text{Hom}\_{GL\_n}(M\_n,P)$ as an $I$-module. Is it true that it is generated by the products of matrices?
> 
> 
> 


For the case of any number of generic matrices $A\_1,\ldots,A\_k,$ Procesi proved that over a field $k$ of characteristic 0, $I$ is spanned by the traces of the products of matrices. Formally, consider words in the free monoid with $k$ generators, substitute the generic matrices, and take a trace. 


Procesi, C. *The invariant theory of n×n matrices*. Advances in Math. 19 (1976), no. 3, 306–381


The statement follows by adjoining an extra generic matrix $A\_0$ and converting an $M\_n$-space into a $GL\_n$-invariant forming a product with $A\_0$ and taking the trace, then undoing the trace of the term in the trace polynomial from Procesi's theorem containing $A\_0.$ 




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Here is a vast generalization due to Le Bruyn and Procesi. Given a finite quiver $Q$ and a dimension vector $\alpha,$ consider the corresponding representation space $R(Q,\alpha)$ with the action of the algebraic group $GL(\alpha)$ and the space $P$ of polynomial functions on $R.$ (If the quiver consists of a single vertex with $k$ loops and $\alpha=n$ then the representation space is given by $k$ generic $n\times n$ matrices with the simultaneous conjugation action by $GL\_n.$) Then, over a field of characteristic 0, the algebra $I$ of polynomial invariants is spanned by the traces of matrix products over oriented cycles in $Q$ and for any pair of vertices $(i,j)$ of $Q,$ the space $\text{Hom}\_{GL(\alpha)}(\text{Hom}\_k(V\_i,V\_j),P)$ is generated as an $I$-module by the products over oriented paths connecting $i$ with $j.$


Lieven Le Bruyn, Claudio Procesi, *Semisimple representations of quivers*.
Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598


