Construction of quantized enveloping algebras by cocycle deformation

Abstract

By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known construction by (generalized) quantum doubles, but unlike in the known situation, it saves us from difficulties in checking complicated defining relations.

Simple modules over finite quantum groups and their Drinfel'd doubles

“…Essential for their result are the Hopf-algebra objects, called Nichols algebras, in the braided tensor category G G YD of Yetter-Drinfel'd modules over a finite abelian group G. In fact, their principle is first to classify the finite-dimensional Nichols algebras L, and then to classify those Hopf algebras which turn, after gr applied, into the bosonizations L >⋖ kG of L by kG. The finite-dimensional pointed Hopf algebras of our concern are those H which are deformed from (L ⊗ R) >⋖ kG by two-cocycle, as in [14] (see also [15]), so that H admits (obviously by construction) the triangular decomposition…”

“…which generalizes (1.1). Here, L and R are finite-dimensional Nichols algebras (in positive characteristic, they may be pre-Nichols algebras [14], more generally) which are mutually symmetric in that sense that the braidings between L ⊗ R and R ⊗ L are mutually inverse. The explicit construction of H is given by Proposition 3.3.…”

“…We actually modify the construction in [14], taking the R above to be a (pre-) Nichols algebra in the category YD G G of the opposite-sided Yetter-Drinfel'd modules; it gives rise to the opposite-sided bosonization kG ⋗< R. This modification indeed realizes the triangular decomposition (1.2), and makes a more conceptual treatment of D(H) possible. Our parametrization pattern for simple modules (see Theorem 2.3) renews that by Krop and Radford so as to connect directly to the triangular decomposition.…”