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

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JournalCommunications in Mathematical Physics
DatePublished - Sep 2010
Issue number3
Number of pages27
Pages (from-to)585-611
Original languageEnglish


Given a symmetric decomposition of a semisimple Lie algebra , we define the notion of a -contractible quantized universal enveloping algebra (QUEA): for these QUEAs the contraction making abelian is nonsingular and yields a QUEA of . For a certain class of symmetric decompositions, we prove, by refining cohomological arguments due to Drinfel’d, that every QUEA of so obtained is isomorphic to a cochain twist of the undeformed envelope . To do so we introduce the -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 ¿-Poincaré in 3 and 4 spacetime dimensions.

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