Cohomological construction of quantized universal enveloping algebras

Abstract:

Given an associative algebra $A$ and the category $\mathcal C$ of its finite dimensional modules, additional structures on the algebra $A$ induce corresponding ones on the category $\mathcal C$. 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 $\Phi \in A^{\otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $\mathcal R \in A^{\otimes 2}$ (commutativity constraint), together with an antiautomorphism (antipode) $S$ of $A$ satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element $F\in A^{\otimes 2}$ with suitable induced actions on $\Phi$, $\mathcal R$ and $S$. Drinfeld defined such a structure on $A=U(\mathcal G )[[h]]$ for any semisimple Lie algebra $\mathcal {G}$ with the usual comultiplication and antipode but nontrivial $\mathcal R$ and $\Phi$, 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), $U_{h}(\mathcal G )$. In the paper we give a direct cohomological construction of the $F$ which reduces $\Phi$ 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 $U(\mathcal G )[[h]]$, 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 $\Phi$ can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to $\mathcal G$ with $\Phi$ depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories.