This paper is a sequel to the paper [7] in which generalized Cartan type S Lie algebras tzS(A, T, φ) over a field F of characteristic 0 were studied. We have tried to make this paper independent of other papers. So in Section 2, we give a description of relevant Lie algebras and some basic facts which will be used in this paper. In Section 3 we introduce a class of Lie algebras which are subalg...

Quantum Lie algebras (an important class of quadratic algebras arising in the Woronowicz calculus on quantum groups) are generalizations of Lie (super) algebras. Many notions from the theory of Lie (super)algebras admit “quantum” generalizations. In particular, there is a BRST operator Q (Q2 = 0) which generates the differential in the Woronowicz theory and gives information about (co)homologie...

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this paper, we use Hodge Laplacian to study the coho...

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie alg...

The Leibniz algebras appear as a generalization of the Lie algebras [8]. The classification of naturally graded p-filiform Lie algebras is known [3], [4], [5], [9]. In this work we deal with the classification of 2-filiform Leibniz algebras. The study of p-filiform Leibniz non Lie algebras is solved for p = 0 (trivial) and p = 1 [1]. In this work we get the classification of naturally graded no...

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules Lh(g) of the quantized enveloping algebras Uh(g). On them the quantum Lie bracket is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie alge...

The Connes-Kreimer renormalization Hopf algebras are examples of a canonical quantization procedure for pre-Lie algebras. We give a simple construction and explanation of this quantization using the universal enveloping algebra for so-called twisted Lie algebras (Lie algebras in the category of symmetric sequences of k-modules). As an application, we obtain a simple proof of the (strengthened) ...

A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory pairs characterize their linear deformations by second group. Then introduce notion relative Rota-Baxter-Nijenhuis structures on pairs, investigate properties, prove that structure gives rise to compatible Rota-Baxter operators under certain condition. Finally, show equiva...

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules Lh(g) of the quantized enveloping algebras Uh(g). On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie alge...