Homology And Cohomology Groups Of Associative Color Algebras

Lie color algebras is a natural extension of Lie algebras and Lie super algebras. And it is a topic of research and application in mathematics and physics. As is well known, algebraic homology and co homology theory may be considered as an extension of ordinary representation theory. At present, some parts of the homology and co homology theory of Lie algebras, Lie super algebras, associative algebras, and modular Lie super algebras have been studied. In particular, the relations between the co homology groups of Lie color algebras L and the co homology of universal enveloping algebras of Lie color algebras (U ( L ),ξ)have been solved by the theory of extension functor. As is known that L is a sub algebra of U ( L )which is a associative color algebra, so the homology and co homology groups of associative algebras inspire us to study the homology and co homology groups of the associative color algebras.First of all, the standard complex of associative color algebras is reduced by some basic properties, and two important identities 1)δγα+γαδ=μα2)δ′γα+γαδ′=μαholds.The structure of A? -module C + ( A, M) and C ? ( A, M), the trivial representation between homology and co homology are given, too.Secondly, the relations between vanishing theorems of the associative color algebras and central elements of the augmentation ideal of the associative color algebras is as following, Theorem 3.7 Let M be aΓ? graded A-module and suppose thatThen the following statements hold:(1) lu F? n = 0, ?n≥0(2) If u acts invertibly on M, then F? n = 0, ?n≥0Theorem 3.10 Let M be aΓ? graded A- module and suppose that u is an element of( ) ( ( ))Then the following statements hold:(1) Fn ? ru = 0, ?n≥0(2) If u acts invertibly on M, then Fn ? = 0, ?n≥0Corollary 3.11 Let M be aΓ? graded A-module,Then the following statements hold: (1) Put P := Ker (τ)∩C ( Ae), if there is u∈P which acts invertibly on M. Then H n ( A, M ) = 0, ?n≥0( H n( A, M ) = 0, ?n≥0)(2) If M is irreducible and P ? M≠0, then H n ( A, M ) = 0, ?n≥0( H n( A, M ) = 0, ?n≥0). Finally, the relations between vanishing theorems of the associative color algebras and local nilpotence of the associative color algebras is as following:Theorem 4.9 Let M be aΓ? graded A-module If there is u∈Asuch that: (1) adu :A ?→A?is local nilpotent, adu ( a ) = [u , a ] = ua ?ξ(u , a ) au , a∈hg ( A?) (2) The mapping M→M , m→u ? m ?ξ(u , m )m ? uis invertible. Then H n( A, M ) = 0, ?n≥0. K...