rt.representation theory – Algebras with a simple preserving duality and finite global dimension


Algebras with a simple preserving duality (an anti-automorphism preserving pointwise a primitive full set of ortohogonal idempotents) and finite global dimension include important classes of algebras such as Schur algebras and blocks of category $mathcal{O}$, which are also quasi-hereditary.

Question: Is there a (natural) example of an algebra with a simple preserving duality and finite global dimension that is not quasi-hereditary?