Pseudo‐Hessian Structures on Associative Algebras

ABSTRACT Let be an associative algebra. The commutator then induces a Lie algebra structure on , denoted by . Let be any connected Lie group whose Lie algebra is . In this paper, we study pseudo‐Hessian metrics on , which are in bijective correspondence with left‐invariant pseudo‐Hessian metrics on the affine Lie group endowed with its canonical bi‐invariant, flat, and torsion‐free connection associated with the product “”. We establish structure theorems for associative algebras admitting such metrics, including nilpotent, semisimple, and matrix subalgebras. Our results complement previous work by Yagi on Hessian structures and extend their scope to arbitrary signatures. In particular, we classify pseudo‐Hessian structures of Lorentzian signature in the unital case and derive obstruction results for the existence of such structures on non‐unital associative algebras.