  In this paper we study symmetric Leibniz and related algebras, namely
symmetric dialgebras and symmetric Perm-algebras. We also calculate their
Koszul duals, if not known. This will give us Lie-admissible algebras and new
types of algebras, which we call commutative and associative admissible
algebras. Finally we prove that all operads describing the considered algebras
are Koszul.
