The representation theory of quivers and finite-dimensional algebras deals with representations of products of general linear groups, and it is hence a «type $\mathsf{A}$ situation». There are many attempts to drag into the picture groups of other nature. In this talk I will talk about a new attempt to get actions of groups of types $\mathsf{B}$, $\mathsf{C}$, and $\mathsf{D}$ on the representation varieties associated to algebras with self-duality based on joint works with Magdalena Boos and partially with Francesco Esposito. For hereditary algebras this reduces to the approach due to Derksen and Weyman in 2002 when they introduced the so-called «symmetric quivers».

In the first part I will mostly talk about quivers of type $\mathsf{A}$ and their symmetric representation theory, and state one of our main result with Lena which states that the symmetric orbit closures are induced by non-symmetric ones for symmetric quivers of finite type. Then I will talk about the connection with 2-nilpotent Borel orbits in classical Lie algebras worked out with Lena and Francesco, and give an example that shows that in this context is not true the orbit closures of type $\mathsf{D}$ are induced by type $\mathsf{A}$. I will close the talk by stating various conjectures and open problems concerning the problem of when symmetric orbit closures are induced by type $\mathsf{A}$.