Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and the Hamiltonian operators in formal variational calculus. In this note we prove that the underlying Lie algebras of quadratic Novikov algebras are 2-step nilpotent. Moreover, we give the classification up to dimension $10$.

In this article, we provide an algorithm with Wolfram Mathematica code that gives a unified computational power in classification of finite dimensional nilpotent algebras using Skjelbred-Sund method. To illustrate the code, we obtain new finite dimensional Moufang algebras.

This paper is primarily concerned with pseudo-Riemannian superalgebras, which are superalgebras endowed with pseudo-Riemannian non-degenerate supersymmetric consistent bilinear forms. Decompositions of pseudo-Riemannian superalgebras whose left centers are isotropic and whose left centers are not isotropic are investigated.

We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $({ℒ}_i,{\alpha }_{{ℒ}_i})$ is (isomorphic to) the direct limit of universal central extensions of $({ℒ}_i,{\alpha }_{{ℒ}_i})$. As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras ${\left\{({\mathrm{sl}}_k\left(å\right),{\alpha }_k)\right\}}_{k\in I}$ and describe the universal central extension of its direct limit.