dorsal/arxiv
View SchemaCohomological construction of quantized universal enveloping algebras
| Authors | J. Donin, S. Shnider |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9506013 |
| URL | https://arxiv.org/abs/q-alg/9506013 |
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 $\Phi\in A^{\otimes 3}$ (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 $\Phi$, $\cR$ 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
{
"annotation_id": "a9430cee-3c81-4b8a-b5d1-26e8d5350b9b",
"date_created": "2026-03-02T18:01:25.350000Z",
"date_modified": "2026-03-02T18:01:25.350000Z",
"file_hash": "72c9e2ce0ae799c9a0d9c023098b9389ebc09c2e00dbb90a5190039fac4910d5",
"private": false,
"record": {
"abstract": "Given an associative algebra $A$, and the category, $\\cC$, of its finite\ndimensional modules, additional structures on the algebra $A$ induce\ncorresponding ones on the category $\\cC$. Thus, the structure of a rigid\nquasi-tensor (braided monoidal) category on $Rep_A$ is induced by an algebra\nhomomorphism $A\\to A\\otimes A$ (comultiplication), coassociative up to\nconjugation by $\\Phi\\in A^{\\otimes 3}$ (associativity constraint) and\ncocommutative up to conjugation by $\\cR\\in A^{\\otimes 2}$ (commutativity\nconstraint), together with an antiautomorphism (antipode), $S$, of $A$\nsatisfying the certain compatibility conditions. A morphism of quasi-tensor\nstructures is given by an element $F\\in A^{\\otimes 2}$ with suitable induced\nactions on $\\Phi$, $\\cR$ and $S$. Drinfeld defined such a structure on\n$A=U(\\cG)[[h]]$ for any semisimple Lie algebra $\\cG$ with the usual\ncomultiplication and antipode but nontrivial $\\cR$ and $\\Phi$ and proved that\nthe corresponding quasi-tensor category is isomomorphic to the category of\nrepresentations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra\n(QUE), $U_h(\\cG)$.\n In the paper we give a direct cohomological construction of the $F$ which\nreduces $\\Phi$ to the trivial associativity constraint, without any assumption\non the prior existence of a strictly coassociative QUE. Thus we get a new\napproach to the DJ quantization. We prove that $F$ can be chosen to satisfy\nsome additional invariance conditions under (anti)automorphisms of\n$U(\\cG)[[h]]$, in particular, $F$ gives an isomorphism of rigid quasi-tensor\ncategories. Moreover, we prove that for pure imaginary values of the\ndeformation parameter, the elements $F$, $R$ and $\\Phi$ can be chosen to be",
"arxiv_id": "q-alg/9506013",
"authors": [
"J. Donin",
"S. Shnider"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Cohomological construction of quantized universal enveloping algebras",
"url": "https://arxiv.org/abs/q-alg/9506013"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "1f24c852-b231-4696-af43-f8a279de8500",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}