Leibniz 2－cocycles of the Central Extension of Lie algebra W（0，1）＊

GAO Shoulan，YAO Chenhui

（School of Science，Huzhou University，Huzhou 313000）

0 Introduction

In［1］，Jean－Louis Loday firstly introduced the concept of Leibniz algebra in his study of the socalled Leibniz homology as a noncommutative analog of Lie algebra homology.A vector space L equipped with a C－bilinear map［－，－］：L×L→L is called a Leibniz algebra if the following Leibniz identity satisfies

Obviously，Lie algebras are Leibniz algebras.A Leibniz algebra L is a Lie algebra if and only if［x，x］＝0 for all x∈L.

Jean－Louis Loday and Teimuraz Pirashvili established the concept of universal enveloping algebras of Leibniz algebras and interpreted the Leibniz（co）homology HL＊（resp.HL＊）as a Tor－functor（resp.Ext－functor）in［1］.A bilinear C－valued formφon L is called a Leibniz 2－cocycle if

Similar to the 2－cocycle on Lie algebras，a linear function f on L can induce a Leibniz 2－cocycleφf(shuō)，that is，

Such a Leibniz 2－cocycle is called trival.The one－dimensional Leibniz central extension corresponding to a trivial Leibniz 2－cocycle is also trivial.

View a Lie algebra as a Leibniz algebra，it is a natural question to compare its Leibniz and Lie central extensions.For many well－known Lie algebras such as the Witt algebra，Kac－Moody algebras，and the Lie algebras of differential operators，this question has already been answered（see for［1，2，3，4］）.In this spaper，we determine the second Leibniz cohomology group HL2（L，C）of L（see defini－tion 1 for detail）in the category of Leibniz algebras.

Throughout the paper，we denote by Z the set of integers and all the vector spaces are assumed over the complex field C.

1 The Leibniz Central Extensions of L

Definition 1［4］The Lie algebra L is a vector space spanned by a basis｛Lm，Im，C｜m ∈Z｝with the following brackets：for all m，n∈Z.

L is one－dimensional central extension of W（0，1）Lie algebra（see［4］for detail）.And it is not perfect because I0can not be generated by others elements in L.

Letφbe a Leibniz 2－cocycle on L.Define a linear function f on L by

［1］Loday J L，Pirashvili T.Universal enveloping algebras of Leibniz algebras and （co）－h(huán)omology［J］.Math Ann，1993（296）：138－158.

［2］Gao S，Jiang C，Pei Y.Structure of the extended Schrodinger－Virasoro Lie alg ~ebra e sv［J］.Algebra Colloq，2009，16（4）：549－566.

［3］Hu N，Pei Y，Liu D.A cohomological characterization of Leibniz central extensions of Lie algebras［J］.Proc Amer Math Soc，2008（136）：437－447.

［4］Gao S，Jiang C，Pei Y.Low－dimensional cohomology groups of the Lie algebras W（a，b）［J］.Communication in Algebra，2011，39（2）：397－423.

MSC 2000：17B40