Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$.

Question: *What can we say about the algebra structure of $Ext^{\ast}(E,E)$?*

Since this is a fairly general question, let me be more precise.

Let us for simplicity assume that $E$ is a smooth point in the moduli spaces of stable sheaves $M_H(v)$ with $v=v(E)$ its Mukai vector. Are there some general results on the algebra structure $Ext^{\ast}(E,E)$? In particular, do we know how the (Yoneda) product looks like or what properties it has?

For curves and Fano surfaces for example the product is trivial, whereas for K3 surfaces the pairing $Ext^1(E,E) \times Ext^1(E,E) \to Ext^2(E,E)\cong Hom(E,E) \cong \mathbb{C}$ is perfect and skew-symmetric. I was wondering if there are any known structural results in higher dimensions?

For example, is the product (e.g. on Calabi-Yau varieties, as suggested by the K3 case) graded commutative? Since $E$ is assumed a smooth point in moduli, the obstructions to deform $E$ vanish and so the Maurer-Cartan equation gives that the product $Ext^1(E,E) \times Ext^1(E,E) \to Ext^2(E,E)$ is skew-symmetric

More generally, one can consider $RHom(E,E)$ as a differential-graded Lie algebra and as such one can associate to it a deformation functor which in this case controls the local deformation theory of $E$. This can be quite complicated, so for simplicity let us assume that this differential-graded Lie algebra is formal, i.e. quasi-isomorphic to its cohomology which is precisely $Ext^{\ast}(E,E)$. Are there on a smooth projective $X$ any a priori constraints on the algebra structure of $Ext^{\ast}(E,E)$?