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Communications in Mathematical Physics

, Volume 298, Issue 3, pp 585–611 | Cite as

Deformation Quasi-Hopf Algebras of Non-semisimple Type from Cochain Twists

  • C. A. S. Young
  • R. Zegers
Article

Abstract

Given a symmetric decomposition \({\mathfrak g=\mathfrak h\oplus \mathfrak p}\) of a semisimple Lie algebra \({\mathfrak g}\), we define the notion of a \({\mathfrak p}\)-contractible quantized universal enveloping algebra (QUEA): for these QUEAs the contraction \({\mathfrak g\rightarrow\mathfrak g_0}\) making \({\mathfrak p}\) abelian is nonsingular and yields a QUEA of \({\mathfrak g_0}\). For a certain class of symmetric decompositions, we prove, by refining cohomological arguments due to Drinfel’d, that every QUEA of \({\mathfrak g_0}\) so obtained is isomorphic to a cochain twist of the undeformed envelope \({\mathcal U(\mathfrak g_0)}\). To do so we introduce the \({\mathfrak p}\)-contractible Chevalley-Eilenberg complex and prove, for this class of symmetric decompositions, a version of Whitehead’s lemma for this complex. By virtue of the existence of the cochain twist, there exist triangular quasi-Hopf algebras based on these contracted QUEAs and, in the approach due to Beggs and Majid, the dual quantized coordinate algebras admit quasi-associative differential calculi of classical dimensions. As examples, we consider κ-Poincaré in 3 and 4 spacetime dimensions.

Keywords

Hopf Algebra Hochschild Cohomology Restrictive Type Hopf Algebra Structure Quasitriangular Hopf Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DurhamDurhamUK

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