Semiinfinite cohomology of associative algebras and bar duality

Details Semiinfinite cohomology of associative algebras and bar duality Semiinfinite cohomology of associative algebras and bar duality

1996-02-05

arxiv.org/abs/q-alg/9602013v1

Mathematics

Quantum Algebra

21 pages

Scientific paper

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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