Lie環의 Cohomology = The Cohomology of Lie Algebras

다국어 초록 (Multilingual Abstract)

For a commutative ring K and a K-algebra we make the enveloping ring Λ( )Λ( )=Λ( ), where Λ( ) is the oposite ring of Λ. In this case, for a projective restlution of Λ: …→X( )→X( )→…→X( )→Λ→0 (exact) and both side additive groups A and B of Λ( ).
We can define the homology and cohomology groups of Λ such that {H(A ( ), X( )}={H( )(A, Λ)}, {H(Hom Λ( )(X( ), C)}={H( )(A, C)}, where H( )(A,Λ)=Tor( )(A,Λ) and H( )(H,C)=Ext( )(Λ,C).
Because of there is the augmentation epimorphism ρ:Λ( )→Λ and ε:Λ→K, by the standard complex S(Λ)and the normalized standard complex N(Λ), we define the homology and cohomology groups of A such that ρ:Λ( )→Λ H( )(Λ,A)=H(A ( )S( )(Λ)) H( )(Λ,C)=H(Hom( )(S( )(Λ),C) ε:Λ→K Tor( )(A,K)=H(A ( )( )(Λ)) Exi( )(K,C)=H(Hom( )( )(Λ)C)) (See Chap. I) × × ×
In all cases, Lie algebra g is a subring of Die algebra g(A) that consists of all representation from a g-additiv group A to g itself. And ore assure it satisfies the following relation. g∋x, x → [x. x] = xy-yx see(2.1)
In order to contruct the cohomology group on a Lie algebra g we should should define the tensor algebra T(g) of such that T( )(g)=K, T( )(g)=g,…… T( )(g)=g ( )……g ( ) T(g)=∑( )T(g) (direct sum)
When T(g) is a K-algebra.
Let us consider a ideal U(g) of T(g) which consists of all elements x ( ) y x x〔x,y〕, If we put K(g)=T(g)/U(g), then k(g) is K-bree(see theorem 9) By the homomorphism ε:K(g)→K and the proceding statement the homalogy and cohomology group of a Lie algebrag is defined as follows. H(A ( )(K(g)=Tor( )(A, K) H(Hom( )(( )(K(g),c)=Exr( )(K, C)
Also, by the outer ring E(g) of a K-free Lie algeora g, the cohomology theory of g is definded snch that Ext( )(K, C)=H(Hom( )(En(g) C) (See§§5.6 chap. Ⅱ)
For a element f of Hom( )(En(g), C) and a element, <x( ), ……, x( )> of E(g), if we defined such that δf<x( )……, x( )>= ∑( )(-1)( )x( )f<x( ), ……, x( )>+ ∑( )f <〔x( ),x( )〕, x( ), ……, x( ), ……x( )> (n>0) δf<x>=xf (f∈Hom( )(K, C)=C, then δis a boundary operator of the Cohomology the ory of g.
In this paper, g-homomorphism θ(x) and ρ(x) are defined as follows θ(x): (θ(x)f) <x( ),……, x( )>=x(f<x( ),……, x( )>-∑f<x( ),……〔x( ), x( )〕, x( )> (f∈Hom( )(E( )(g), C) θ(x)f=xf (<x( ),……, x( )>E( )(g)) ρ(x) : (ρ(x)f)<x( ),……, x( )>=f<x( )x( ),……, x( )> (f∈Hom( )(E( )(g), C)=C), (ρ(x)f)=0 (f∈Hom( )(E( )(g), C) =C). where < x( ),……, x( )> ∈E( )(g) and E( )(g)=0
Theorem: If g is a sems-simple ring and a left g-module A is g-isreducible, then the above cohomology groups areallzero, wehere gA≠0 (See theorem 14). (i.e E.t( )(K. A)=H(Hom( )(E( )(g), A)=0 for q=0.1.2.……)

목차 (Table of Contents)

• 1. 序 論
• 2. Lie 環과 Cohomology
• 3. 半單純 Lie 環