dorsal/arxiv
View SchemaSemiinfinite cohomology of associative algebras and bar duality
| Authors | Sergey Arkhipov |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9602013 |
| URL | https://arxiv.org/abs/q-alg/9602013 |
Abstract
We describe semiinfinite cohomology of associative algebras in terms of Koszul (or bar) duality. Consider an associative algebra $A$ and two its subalgebras $B$ and $N$ such that $A=B\otimes N$ as a vector space. We prove that the endomorphism algebra of the semiregular $A$-module appears naturally in semiinfinite cohomology theory as a ``two times Koszul dual'' to the algebra $A$. We compare semiinfinite cohomology of universal enveloping algebras with the well-known Lie algebra semiinfinite cohomology. A new description of the critical 2-cocycle is provided. As a consequence we obtain another proof of the fact that additive categories generated by Verma modules over affine Lie algebras on dual levels are antiequivalent.
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"abstract": "We describe semiinfinite cohomology of associative algebras in terms of\nKoszul (or bar) duality. Consider an associative algebra $A$ and two its\nsubalgebras $B$ and $N$ such that $A=B\\otimes N$ as a vector space. We prove\nthat the endomorphism algebra of the semiregular $A$-module appears naturally\nin semiinfinite cohomology theory as a ``two times Koszul dual\u0027\u0027 to the algebra\n$A$. We compare semiinfinite cohomology of universal enveloping algebras with\nthe well-known Lie algebra semiinfinite cohomology. A new description of the\ncritical 2-cocycle is provided. As a consequence we obtain another proof of the\nfact that additive categories generated by Verma modules over affine Lie\nalgebras on dual levels are antiequivalent.",
"arxiv_id": "q-alg/9602013",
"authors": [
"Sergey Arkhipov"
],
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"q-alg",
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"title": "Semiinfinite cohomology of associative algebras and bar duality",
"url": "https://arxiv.org/abs/q-alg/9602013"
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