జర్నల్ ఆఫ్ జెనరలైజ్డ్ లై థియరీ అండ్ అప్లికేషన్స్

In this paper, we describe the derivations of complex n-dimensional naturally graded filiform Leibniz algebras NGF1, NGF2, and NGF3.We show that the dimension of the derivation algebras of NGF1 and NGF2 equals n+1 and n+2, respectively, while the dimension of the derivation algebra of NGF3 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 FLbn and SLbn. Here we found that for L ∈ FLbn, we have n−1≤dimDer(L)≤n+1 and for algebras L from SLbn, the inequality n−1 ≤ dimDer(L) ≤ n+2 holds true.

 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 S3 and A4 we give explicit forms for all quantizations.

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.

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 Z3-grading of the sl(2) Lie algebra.

The Cayley-Dickson loop Qn 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.