Analogue to commutants in the theory of associative algebras, one can construct a new subalgebra of vertex algebra known as a vertex algebra commutant. In this paper, for the adjoint representation V of Lie algebra sl(2,C), we describe a commutant of βγSystem S(V ) by giving its generators, moreover, we get a new Howe pair of vertex algebras.

‎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...

A unital $C^*$ -- algebra $mathcal A,$ endowed withthe Lie product $[x,y]=xy- yx$ on $mathcal A,$ is called a Lie$C^*$ -- algebra. Let $mathcal A$ be a Lie $C^*$ -- algebra and$g,h:mathcal A to mathcal A$ be $Bbb C$ -- linear mappings. A$Bbb C$ -- linear mapping $f:mathcal A to mathcal A$ is calleda Lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

In this paper, we discuss the pair problem of generators in affine Kac-Moody Lie algebras. For any affine Kac-Moody algebra g(A) of X l type and arbitrary nonzero imaginary root vector x, we prove that there exists some y ∈ g(A), such that g′(A) is contained in the Lie algebra generated by x and y.

We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L∞ structures, we give an explicit description of the differential graded Lie algebra that controls this problem.

in this paper, by using of frobenius theorem a relation between lie subalgebras of the lie algebra of a top space t and lie subgroups of t(as a lie group) is determined. as a result we can consider these spaces by their lie algebras. we show that a top space with the finite number of identity elements is a c^{∞} principal fiber bundle, by this method we can characterize top spaces.

A nonzero element x of a Lie algebra L over a field F is called extremal if [x, [x,L]] ⊆ Fx. Extremal elements are a well-studied class of elements in simple finite-dimensional Lie algebras of Chevalley type: they are the long root elements. In [CSUW01], Cohen, Steinbach, Ushirobira and Wales have studied Lie algebras generated by extremal elements, in particular those of Chevalley type. The au...

A nonzero element x of a Lie algebra L over a field F is called extremal if [x, [x,L]] ⊆ Fx. Extremal elements are a well-studied class of elements in simple finite-dimensional Lie algebras of Chevalley type: they are the long root elements. In [CSUW01], Cohen, Steinbach, Ushirobira and Wales have studied Lie algebras generated by extremal elements, in particular those of Chevalley type. The au...