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- W3165331096 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 $Ato Aotimes A$ (comultiplication), coassociative up to conjugation by $Phiin A^{otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $cRin A^{otimes 2}$ (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 $Fin A^{otimes 2}$ with suitable induced actions on $Phi$, $cR$ 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 $Phi$ 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), $U_h(cG)$. 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 $U(cG)[[h]]$, 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 $Phi$ can be chosen to be" @default.
- W3165331096 created "2021-06-07" @default.
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- W3165331096 date "1995-06-01" @default.
- W3165331096 modified "2023-09-27" @default.
- W3165331096 title "Cohomological construction of quantized universal enveloping algebras" @default.
- W3165331096 hasPublicationYear "1995" @default.
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