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Symmetric (co)homologies of Lie algebras
Institution:1. State Grid Ganzhou Power Supply Company, Zanxianlu No.1, Ganzhou 341001, China;2. State Grid Jiangxi Electric Power Company, Hubindonglu No.666, Nanchang 330077, China;3. Southeast University, Sipailou No.2, Nanjing 210096, China
Abstract:Cohomologies of Lie algebras are usually calculated using the Chevalley-Eilenberg cochain complex of skew-symmetric forms. We consider two cochain complexes consisting of forms with some symmetric properties. First, cochains C*(L) are symmetric in the last 2 arguments, skew-symmetric in the others and satify moreover some kind of Jacobi condition in the last 3 arguments. In characteristic 0, its cohomologies are isomorphic to the cohomologies of the factor-complex C*(L,L’)/C*+1(L,K). Second, a symmetric version Cλ*(A) is defined for an associative algebra A. It is a subcomplex of the cyclic cochain complex. These symmetric cochain complexes are used for the calculation of 3-cohomologies of Cartan Type Lie algebras with trivial coefficients.
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