We study free centre-by-metabelian Lie algebras over a field of characteristic 2. By using homological methods, we determine the dimensions of the fine homogeneous components of the second-derived algebra. In conjunction with earlier results by Mansuroǧlu and the second author, this leads to a complete description of the additive structure of the second-derived ideal in the free centre-by-metab...

‎Let $L$ be a Lie algebra‎, ‎$mathrm{Der}(L)$ be the set of all derivations of $L$ and $mathrm{Der}_c(L)$ denote the set of all derivations $alphainmathrm{Der}(L)$ for which $alpha(x)in [x,L]:={[x,y]vert yin L}$ for all $xin L$‎. ‎We obtain an upper bound for dimension of $mathrm{Der}_c(L)$ of the finite dimensional nilpotent Lie algebra $L$ over algebraically closed fields‎. ‎Also‎, ‎we classi...

In [8] de Concini, Kac and Procesi introduced the simply connected quantised universal enveloping algebra U = Uε,P (g) over C at a primitive lth root of unity ε associated to a simple finite-dimensional complex Lie algebra g. The importance of the study of the centre Z of U and its spectrum Maxspec(Z) is pointed out in [7,8]. In this article we consider the following two conjectures concerning ...

in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

‎in this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎we show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a lie groupoid‎. ‎using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the lie algebra of all vector fields on the smooth...

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

‎in this paper lie symmetry analysis is applied in order to find new solutions for fokker plank equation of ornstein-uhlenbeck process‎. ‎this analysis classifies the solutions format of the fokker plank equation by using the lie algebra of the symmetries of our considered stochastic process‎.

‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

Let $\mathbb C_Q$ denote a rational quantum torus with $d$ variables, and $\mathcal Z$ be the centre of C_Q$. In this paper we give explicit description structure cuspidal modules for derivation Lie algebra D$ over C_Q$, an extra associative Z$-action.