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In this paper, we develop a method to obtain the algebraic classification of noncommutative Jordan algebras from the classification of Jordan algebras of the same dimension. We use this method to obtain the algebraic classification of complex $3$-dimensional noncommutative Jordan algebras. As a byproduct, we obtain the classification of complex $3$-dimensional Kokoris, standard, generic Poisson, and generic Poisson--Jordan algebras; and also complex $4$-dimensional nilpotent Kokoris and standard algebras. In addition, we consider the geometric classification of varieties of cited algebras, that is the description of its irreducible components., Comment: arXiv admin note: text overlap with arXiv:2406.10947

This article will discussing on $\frac{1}{2}$-derivations of quasi-filiform Lie algebras of maximum length. The non-trivial transposed Poisson algebras with the quasi-filiform Lie algebras of maximum length are constructed by using $\frac{1}{2}$-derivations of Lie algebras. We have established commutative associative multiplication to construct a transposed Poisson algebra with an associated given Lie algebra., Comment: arXiv admin note: text overlap with arXiv:2401.04443

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. They are characterized as algebras with two operations whose sum is associative. In particular, the paper is devoted to classifying anti-dendriform algebras associated with null-filiform associative algebras and three-dimensional algebras. It is known that any finite-dimensional associative algebra without a non-zero idempotent element is nilpotent. If an associative algebra has a non-zero idempotent element, then there does not exist a compatible anti-dendriform algebra structure associated with associative algebras., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2209.08962 by other authors

This paper is devoted to the complete algebraic classification of complex 5-dimensional nilpotent bicommutative algebras., Comment: arXiv admin note: substantial text overlap with arXiv:2402.00419, arXiv:1903.08997, arXiv:2307.00289, arXiv:2107.05440, arXiv:2101.07375, arXiv:2308.00374

This paper is devoted to the complete algebraic classification of complex $5$-dimensional nilpotent Novikov algebras., Comment: arXiv admin note: substantial text overlap with arXiv:2307.00289, arXiv:2004.03598, arXiv:2205.00444, arXiv:2206.00315, arXiv:1812.01442, arXiv:2106.00336, arXiv:2111.00785, arXiv:2308.00374

In this article, we described 1/2-derivations of solvable Lie algebras with a thread-like nilradical. Nontrivial transposed Poisson algebras with solvable Lie algebras are constructed. That is, by using 1/2-derivations of Lie algebras, we have established commutative associative multiplication to construct a transposed Poisson algebra with an associated given Lie algebra., Comment: 43 pages

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional nilpotent binary Leibniz and $4$-dimensional nilpotent mono Leibniz algebras. As a corollary, we have the complete algebraic and geometric classification of complex $4$-dimensional nilpotent algebras of nil-index $3$., Comment: arXiv admin note: substantial text overlap with arXiv:2205.00444, arXiv:2206.00315, arXiv:2106.00336, arXiv:1912.02691, arXiv:1812.01442, arXiv:2307.00289

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional nilpotent Leibniz algebras. In particular, the variety of complex $5$-dimensional nilpotent Leibniz algebras has dimension $24$ it has $10$ irreducible components (there is only one rigid algebra in this variety)., Comment: arXiv admin note: substantial text overlap with arXiv:2205.00444, arXiv:2206.00315, arXiv:2106.00336, arXiv:1912.02691, arXiv:1812.01442, arXiv:2111.00785, arXiv:2109.00483, arXiv:2004.03598

This paper is devoted to the study of local and 2-local derivations of nullfiliform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such that the codimension of nilradical is equal to the number of generators of the nilradical.

The present article is a part of the study of solvable Leibniz algebras with a given nilradical. In this paper solvable Leibniz algebras, whose nilradicals is naturally graded quasi-filiform algebra and the complemented space to the nilradical has one dimension, are described up to isomorphism., Comment: 16 pages

The paper is devoted studying solvable Leibniz algebras with a nilradical possessing the codimension equals the number of its generators. We describe this class in non-split nilradical case. Then the case of split nilradical is worked out. We show that the results obtained earlier on this class of Leibniz algebras come as particular cases of the results of this paper. It is shown that such a solvable extension is unique. Finally, we prove that the solvable Leibniz algebras considered are complete.

In this paper we introduce the notion of pure non-characteristically nilpotent Lie algebra and under a condition we prove that a complex maximal extension of a finite-dimensional pure non-characteristically nilpotent Lie algebra is isomorphic to a semidirect sum of the nilradical and its maximal torus. We also prove that such solvable Lie algebras are complete and we specify a subclass of the maximal solvable extensions of pure non-characteristically nilpotent Lie algebras that have trivial cohomology group. Some comparisons with the results obtained earlier are given.

In the paper the class of all solvable extensions of a filiform Leibniz algebra in the infinite-dimensional case is classified. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of residually solvable Leibniz algebra. It is proven that the second cohomology group of the extension is trivial., Comment: 16 pages

In the paper we describe the derivations of two $\mathbb{N}$-graded infinity-dimensional Lie algebras $\mathbf{n}_1$ and $\mathbf{n}_1$ what are positive parts of affine Kats-Moody algebras $A^{(1)}_1$ and $A^{(2)}_2$, respectively. Then we construct all pro-solvable Lie algebras whose potential nilpotent ideals are $\mathbf{n}_1$ and $\mathbf{n}_2$. For two specific representatives of these classes low-dimensional (co)homology groups are computed.

In the present paper we prove the existence outer derivations finite-dimensional solvable Lie algebras with nilradical of maximal rank and complementary subspace to nilardical of dimension less than rank of the nilradical.

The paper is devoted to the study of pro-solvable Lie algebras whose maximal pro-nilpotent ideal is either $\mathfrak{m}_0$ or $\mathfrak{m}_2$. Namely, we describe such Lie algebras and establish their completeness. Triviality of the second cohomology group for one of the obtained algebra is established.

In this paper we investigate pre-derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three non-intersected families. We describe the pre-derivation of filiform Leibniz algebras for the first and second families. We found sufficient conditions under which filiform Leibniz algebras are strongly nilpotent. Moreover, for the first and second families, we give the description of characteristically nilpotent algebras which are non-strongly nilpotent.

We describe solvable Leibniz algebras whose nilradical is a quasi-filiform Leibniz algebra of maximum length., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1801.08935