The concept of (α,β)-fuzzy Lie algebras over an (α,β)-fuzzy field is introduced. We provide characterizations of an $(\in,\in\vee q)$-fuzzy Lie algebra over an $(\in,\in\vee q)$-fuzzy field.

We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras.

In this work, the nul-filiform and filiform Zinbiel algebras are described up to isomorphism. Moreover, the classification of complex Zinbiel algebras dimensions $\leq 3$ is extended up to dimension 4.

We use the general method of quantization by Drinfel'd twist element to quantize explicitly the Lie bialgebra structures on the $q$-analog Virasoro-like algebras studied in Comm. Algebra, 37 (2009), 1264-1274.

We are attempting to give a new proof to the problem of characterization of the support of the product of conjugacy classes in the compact Lie group SU$(n)$ without any reference to the Mehta-Seshadri theorem in algebraic geometry as it was the case in [1]

Left Cheban loops are loops that satisfy the identity $x(xy \cdot z) = yx \cdot xz$. Right Cheban loops satisfy the mirror identity $(z \cdot yx)x = zx \cdot xy$. Loops that are both left and right Cheban are called Cheban loops. Cheban loops can also be characterized as those loops that satisfy the identity $x(xy \cdot z) = (y \cdot zx)x$. These loops were introduced by A. M. Cheban. Here we initiate a study of their structural properties. Left Cheban loops are left conjugacy closed. Cheban loops are weak inverse property, power associative, conjugacy closed loops; they are centrally nilpotent of class at most two.

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials $C[X_1,... ,X_n]$ using exterior calculus. After presenting some non-homogenous Poisson brackets on this algebra, we compute Poisson cohomological spaces when the Poisson structure corresponds to a bracket of a rigid Lie algebra.

In this paper, we give general definitions of non-commutative jets in the local and global situation using square zero extensions and derivations. We study the functors $\operatorname{Exan}_k(A,I)$, where $A$ is any $k$-algebra, and $I$ is any left and right $A$-module and use this to construct affine non-commutative jets. We also study the Kodaira-Spencer class $\operatorname{KS}(\mathcal{L})$ and relate it to the Atiyah class.

We show that every Lie algebra is equipped with a natural (1,1)-variant tensor field, the "canonical endomorphism field", determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We show its relevance for classical mechanics, in particular for Lax equations. We show that the space of Lax vector fields is closed under Lie bracket and we introduce a new bracket for vector fields on a Lie algebra. This bracket defines a new Lie structure on the space of vector fields.

In this article, we give a formula for the number of Gelfand-Zetlin patterns, using dimensions of the symmetry classes of tensors.

We establish connection between product of two matrices of order $k\times k$ over a field and the product of the k-mappings corresponding to the $k$-operations, defined by these matrices. It is proved that, in contrast to the binary case, for arity $k\geq 3$ the components of the $k$-permutation inverse to a $k$-permutation, all components of which are polynomial $k$-quasigroups, are not necessarily $k$-quasigroups although are invertible at least in two places. Some transformations with the help of permutations of orthogonal systems of polynomial $k$-operations over a field are considered.

Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative types and study the relations between the obtained algebras. The associative case is richer since it admits the notion of unit element. We use this fact to find sufficient conditions for Hom-associative algebras to be associative and classify the implications between the Hom-associative types of unital algebras.