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

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.