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Mainly motivated by Pirashvili's spectral sequences on a Leibniz algebra, a cohomological characterization of Leibniz central extensions of Lie algebras is given. In particular, as applications, we obtain the cohomological version of Gao's main theorem for Kac-Moody algebras and answer a question in an earlier paper by Liu and Hu (2004).

In this paper, we study the derivations, the central extensions and the automorphism group of the extended Schrödinger-Virasoro Lie algebra sv, introduced by J. Unterberger [25] in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that sv is an infinite-dimensional complete Lie algebra and the universal central extension of sv in the category of Leibniz algebras is the same as that in the category of Lie algebras. 2 , Y 1 2 , M 0 . The structure and representation theory of sv have been extensively studied by C. Roger and J. Unterberger. We refer the reader to [22] for more details. Recently, in order to investigate vertex representations of sv, J. Unterberger [25] introduced a class of new infinite-dimensional Lie algebras sv called the extended Schrödinger-Virasoro algebra (see section 2), which can be viewed as an extension of sv by a conformal current with conformal weight 1.where sv n = span{L n , M n , N n } and sv n+ 1 2 = span{Y n+ 1 2 } for all n ∈ Z.Definition 2.3. ([5]) Let g be a Lie algebra and V a g-module. A linear map D : g −→ V is called a derivation, if for any x, y ∈ g, we have D[x, y] = x.D(y) − y.D(x).If there exists some v ∈ V such that D : x → x.v, then D is called an inner derivation.Let g be a Lie algebra, V a module of g. Denote by Der(g, V ) the vector space of all derivations, Inn(g, V ) the vector space of all inner derivations. SetDenote by Der(g) the derivation algebra of g, Inn(g) the vector space of all inner derivations of g. We will prove that all the derivations of sv are inner derivations.By Proposition 1.1 in [5], we have the following lemma.

In this paper, we define and study Whittaker modules for the super-Viraoro algebras, including the Neveu-Schwarz algebra and the Ramond algebra. We classify the simple Whittaker modules and obtain necessary and sufficient conditions for irreducibility of these modules.

In this paper, we describe the structure of Verma modules over the W-algebra W(2,2). We show that either a Verma module over W(2,2) is irreducible or its maximal submodule is cyclic.