Cohomological construction of quantized universal enveloping algebras

Abstract

Given an associative algebra $A$, and the category, $\cC$, of its finite dimensional modules, additional structures on the algebra $A$ induce corresponding ones on the category $\cC$. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on $Rep_A$ is induced by an algebra homomorphism $A\to A\otimes A$ (comultiplication), coassociative up to conjugation by $Φ\in A^{\otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $\cR\in A^{\otimes 2}$ (commutativity constraint), together with an antiautomorphism (antipode), $S$, of $A$ satisfying the certain compatibility conditions. A morphism of quasi-tensor structures is given by an element $F\in A^{\otimes 2}$ with suitable induced actions on $Φ$, $\cR$ and $S$. Drinfeld defined such a structure on $A=U(\cG)[[h]]$ for any semisimple Lie algebra $\cG$ with the usual comultiplication and antipode but nontrivial $\cR$ and $Φ$ and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), $U_h(\cG)$. In the paper we give a direct cohomological construction of the $F$ which reduces $Φ$ to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that $F$ can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of $U(\cG)[[h]]$, in particular, $F$ gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements $F$, $R$ and $Φ$ can be chosen to be