On dual fermionic Novikov algebras

Fermionic Novikov algebras are a special class of pre-Lie algebras whose right multiplication operators are anti-commutative and are closely linked to certain Hamiltonian superoperators. In this paper, we introduce the notion of a dual fermionic Novikov algebra and show that its associated operad, the dual fermionic Novikov operad, is the Koszul dual of the operad of fermionic Novikov algebras. Furthermore, applying the Ginzburg-Kapranov criterion, we prove that both the fermionic Novikov operad and its Koszul dual fail to be Koszul. Moreover, a Lie algebra can be naturally constructed via the tensor product of a fermionic Novikov algebra and a dual fermionic Novikov algebra. Then we study two distinct bilinear forms on dual fermionic Novikov algebras. A dual fermionic Novikov algebra equipped with a nondegenerate antisymmetric invariant bilinear form is called quadratic, and it naturally gives an equivalence between its regular and coregular representations. On the other hand, a dual fermionic Novikov algebra endowed with a nondegenerate symmetric closed bilinear form is called quasi-Frobenius. Such algebras can be characterized by relative Rota-Baxter operators associated to the coregular representation. Finally, we further introduce pre-dual fermionic Novikov algebras, which serve as the underlying algebraic structure of quasi-Frobenius dual fermionic Novikov algebras.