Bulletin of the
Korean Mathematical Society BKMS

In this paper, we focus on pseudo-Riemannian associative fermionic Novikov algebras. We prove that the underlying Lie algebras of pseudo-Riemannian associative fermionic Novikov algebras are $2$-step nilpotent and that pseudo-Riemannian associative fermionic Novikov algebras are $3$-step nilpotent. Moreover, we construct a pseudo-Riemannian associative fermionic Novikov algebra in dimension 14, which is not a Novikov algebra. It implies that the inverse proposition of Corollary 2 in the paper ``Pseudo-Riemannian Novikov algebras" [J. Phys. A: Math. Theor. $\bf 41$ (2008), 315207] does not hold.

Keywords: Novikov algebra, fermionic Novikov algebra, pseudo-Riemannian Lie algebra

MSC numbers: 17B60, 17A30, 17D25