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

Charles Young; R Zegers

doi:10.1007/s00220-010-1086-8

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

Charles Young, R Zegers

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	585-611
Number of pages	27
Journal	Communications in Mathematical Physics
Volume	298
Issue number	3
DOIs	https://doi.org/10.1007/s00220-010-1086-8
Publication status	Published - Sept 2010

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abstract = "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{\textquoteright}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{\textquoteright}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 ¿-Poincar{\'e} in 3 and 4 spacetime dimensions. ",

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

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