Koszul-Vinberg structures and compatible structures on left-symmetric algebroids

In this paper, we introduce the notion of Koszul-Vinberg-Nijenhuis structures on a left-symmetric algebroid as analogues of Poisson-Nijenhuis structures on a Lie algebroid, and show that a Koszul-Vinberg-Nijenhuis structure gives rise to a hierarchy of Koszul-Vinberg structures. We introduce the notions of ${\rm KV\Omega}$-structures, pseudo-Hessian-Nijenhuis structures and complementary symmetric $2$-tensors for Koszul-Vinberg structures on left-symmetric algebroids, which are analogues of ${\rm P\Omega}$-structures, symplectic-Nijenhuis structures and complementary $2$-forms for Poisson structures. We also study the relationships between these various structures.