Structure of Degenerate Block Algebras

Abstract

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map \(\phi\) : A \(\times\) A \(\rightarrow\) F, we define a Lie algebra \({\cal L}\) = \({\cal L}\) (A, \(\phi\)) over F with basis {ex | x \(\in\) A/{0}} and Lie product [ex,ey] = \(\phi\)(x,y)ex+y. We show that \({\cal L}\) is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der \({\cal L}\) of \({\cal L}\) is a complete Lie algebra. We describe the double extension D(\({\cal L}\), T) of \({\cal L}\) by T, where T is spanned by the locally finite derivations of \({\cal L}\), and determine the second cohomology group H2(D(\({\cal L}\), T),F) using anti-derivations related to the form on D(\({\cal L}\), T). Finally, we compute the second Leibniz cohomology groups HL2(\({\cal L}\), F) and HL2(D(\({\cal L}\), T), F).

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Correspondence to Linsheng Zhu.

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2000 Mathematics Subject Classification: 17B05, 17B30

This work was supported by the NNSF of China (19971044), the Doctoral Programme Foundation of Institution of Higher Education (97005511), and the Foundation of Jiangsu Educational Committee.

Communicated by Nanqing Ding

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Zhu, L., Meng, D. Structure of Degenerate Block Algebras. Algebra Colloq. 10, 53–62 (2003). https://doi.org/10.1007/s100110300007

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Keywords

  • quadratic Lie algebra
  • double extension
  • derivation
  • Leibniz cohomology