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

TY - JOUR

T1 - Deformation Quasi-Hopf Algebras of Non-semisimple Type from Cochain Twists

AU - Young, Charles

AU - Zegers, R

PY - 2010/9

Y1 - 2010/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77954952972&partnerID=8YFLogxK

U2 - 10.1007/s00220-010-1086-8

DO - 10.1007/s00220-010-1086-8

M3 - Article

SN - 0010-3616

VL - 298

SP - 585

EP - 611

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -