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AbstractThis paper is devoted to the study of local derivations and automorphisms of nilpotent Lie algebras. Namely, we proved that nilpotent Lie algebras with indices of nilpotency 3 and 4 admit a local derivation (local automorphism) which is not a derivation (automorphism). Further, it is presented a sufficient condition under which a nilpotent Lie algebra admits a local derivation which is not a derivation. With the same condition, it is proved the existence of pure local automorphism on a nilpotent Lie algebra. Finally, we present an n-dimensional nilpotent non-associative algebra for which the space of local derivations coincides with the space of derivations. [ABSTRACT FROM AUTHOR]

In this paper we introduce skew-symmetric n-ary brackets on an associative commutative algebra, obtained from Jacobian and Wronskian by adding one more column. For the n-algebras obtained we find the necessary and sufficient conditions to be n-Lie algebras. Thereby we give several new examples of n-Lie algebras. Some of the examples happen to be simple n-Lie algebras. [ABSTRACT FROM AUTHOR]

We give algebraic and geometric classifications of complex four-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only 1 8 non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of three rigid algebras and two one-parametric families of algebras. [ABSTRACT FROM AUTHOR]