The Cayley-Dickson $Q_n$ loop is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley-Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, and sedenions).We discuss properties of the Cayley-Dickson loops, show that these loops are Hamiltonian, and describe the structure of their automorphism groups.

A theory of grading preserving contractions of representations of Lie algebras has been developed. In this theory, grading of the given Lie algebra is characterized by two sets of parameters satisfying a derived set of equations. Here we introduce a list of resulting 3-dimensional representations for the $\mathbb{Z}_3$-grading of the $\mathfrak{sl}(2)$ Lie algebra.

In this paper, we apply the concepts of fuzzy sets to Lie algebras in order to introduce and to study the notions of solvable and nilpotent fuzzy radicals. We present conditions to prove the existence and uniqueness of such radicals.

We describe quantizations on monoidal categories of modules over finite groups. Those are given by quantizers which are elements of a group algebra. Over the complex numbers we find these explicitly. For modules over $S_3$ and $A_4$ we give explicit forms for all quantizations.

In this paper, we describe the derivations of complex $n$-dimensional naturally graded filiform Leibniz algebras $\mathrm{NGF_1}$, $\mathrm{NGF_2}$, and $\mathrm{NGF_3}$.We show that the dimension of the derivation algebras of $\mathrm{NGF_1}$ and $\mathrm{NGF_2}$ equals $n+1$ and $n+2$, respectively, while the dimension of the derivation algebra of $\mathrm{NGF_3}$ is equal to $2n−1$. The second part of the paper deals with the description of the derivations of complex $n$-dimensional filiform non Lie Leibniz algebras, obtained from naturally graded non Lie filiform Leibniz algebras. It is well known that this class is split into two classes denoted by $\mathrm{FLb}_n$ and $\mathrm{SLb}_n$. Here we found that for $L ∈ \mathrm{FLb}_n$, we have $n−1≤\mathrm{dimDer}(L)≤n+1$ and for algebras $L$ from $\mathrm{SLb}_n$, the inequality $n−1 ≤ \mathrm{dimDer}(L) ≤ n+2$ holds true.