Low-Dimensional Cohomology Groups of the Lie Algebras W(a,b)

In this article, we determine one-dimensional (Leibniz) central extensions, the derivation algebra, and the automorphism group of the infinite-dimensional Lie algebra W(a, b) (for a, b ∈ ℂ) which is the semidirect product Lie algebra of the Witt algebra and its tensor density module I(a, b). Key Words: Central extensionDerivation algebraWitt algebra2000 Mathematics Subject Classification: 17B6517B6717B68 ACKNOWLEDGMENTS We are very grateful to the referee for his (or her) valuable suggestions and comments. We also thank Dr. Qifen Jiang for her careful reading so that some typos in the manuscript have been eliminated. This work was supported in part by the NSFC grants 10871125, 10811120445, 10931006, the leading academic discipline project of Shanghai normal university (DZL803), the grant of Science and Technology Commission of Shanghai Municipality (No. 09XD1402500) and the Innovation Program of Shanghai Municipal Education Commission (11ZZ18, 11YC85). Notes Communicated by K. Misra.