Integration of the lifting formulas and the cyclic homology of the algebras of differential operators |
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Authors: | B Shoikhet |
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Institution: | IUM, 11 Bol'shoj Vlas'evskij per., Moscow 121002, Russia, e-mail: borya@mccme.ru, RU
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Abstract: | We integrate the Lifting cocycles Y2n+1, Y2n+3, Y2n+5,? (Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,(\rm Sh1,2]) on the Lie algebra Difn of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem FT1]:¶¶ H·Lie(\frak g\frak lfin¥(Difn);\Bbb C) = ù·(Y2n+1, Y2n+3, Y2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) . |
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