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Low dimensional cohomology of Hom-Lie algebras and q-deformed W(2, 2) algebra

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Abstract

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the q-deformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also, we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras. As application, we compute all α k-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.

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Authors and Affiliations

  1. Science Research Center, Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, Harbin, 150080, P. R. China

    La Mei Yuan & Hong You

  2. School of Mathematical Sciences, Suzhou University, Suzhou, 215006, P. R. China

    Hong You

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  1. La Mei Yuan
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  2. Hong You
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Correspondence to La Mei Yuan.

Additional information

Supported by China Scholarship Council (Grant No. 201206125047), China Postdoctoral Science Foundation Funded Project (Grant No. 2012M520715) and the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 201462)

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Yuan, L.M., You, H. Low dimensional cohomology of Hom-Lie algebras and q-deformed W(2, 2) algebra. Acta. Math. Sin.-English Ser. 30, 1073–1082 (2014). https://doi.org/10.1007/s10114-013-2614-1

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  • DOI: https://doi.org/10.1007/s10114-013-2614-1

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