Lie Algebra Cohomology
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Lie Algebra Cohomology Juan Camilo Fiallo R December 4, 2013 Abstract We start with a Lie Algebra g over a Field K, then we construct the universal enveloping algebra Ug and define the cohomology groups Hn(g;A) for every (left) g-module A, by regarding A as a Ug- module. Then we give proofs of the Weyl Theorem and deduce the Whitehead Lemmas as corollaries. 1 Introduction The origin of Cohomology theory of Lie Algebras lies in algebraic topology. Chevalley-Eilenberg (see [1]) have shown that the real cohomology of the underlying topological space of a compact connected Lie group is isomorphic to the real cohomology of its algebra. We are going to give cohomological proofs of the two main theorems in the theory of Lie algebras over a field of characteristic 0. The first of these theorem is that the finite-dimensional representations of a semi-simple Lie algebra are completely reducible. The main step in that proof will be to show that the first cohomology group of a semi-simple Lie algebra with arbitrary finite-dimensional coefficient module is trivial. This is known as the first Whitehead Lemma. Then we are going to prove that every finite dimensional Lie algebra g is the split extension of a semi-simple Lie algebra by the radical of g: The main step in the proof of this result will be to show that the second cohomology group of a semi-simple Lie algebra with arbitrary finite-dimensional coefficient module is trivial. This is known as the second Whitehead Lemma. 2 The Universal enveloping algebra Let K be a field. A Lie algebra g over K is a vector space over K together with a bilinear map [ ; ]: g g g; × ! satisfying the following two identities [x;x] = 0; x g 2 [x;[y;z]] + [y;[z;x]] + [z;[x;y]] = 0; x;y;z g: 2 The second equation is known as the Jacobi identity. A Lie algebra homomorphism f : g h is a K-linear ! map compatible with the Lie Bracket. A Lie subalgebra h of g is a linear subspace closed under the bracket. A Lie subalgebra h is called a Lie ideal of g, if [x;y] h for every x g and every y h. If h is a 2 2 2 Lie ideal of g, then the quotient space g=h inherits the Lie algebra structure from g. A Lie algebra is called abelian if [x;y] = 0 for all x;y g. This is a special case, because any K- 2 vector space can be regarded as an abelian Lie algebra. In fact, given any K-algebra Λ we can associate 1 (functorially) a Lie Algebra LΛ with the same underlying vector space as Λ, the Lie bracket being defined by [x;y] = xy yx: − In order to define a the cohomology group of a Lie algebra we need to find a left adjoint to the functor L (i.e. a functor U determined by the Hom( , ) functor, going from the category of Lie Algebras to the category of Algebras). Such functor indeed exists and the image of the Lie algebra g is called the universal enveloping algebra of g and is denoted by Ug: Let us start with the notion of a tensor algebra over the K-vector space M. Definition 2.1. Denote, for n 1, the n-fold tensor product of M by T M, ≥ n T M = M M ::: M; n times: n ⊗K ⊗K ⊗K L 1 Set T0M = K. Then, the tensor algebra TM is n=0 TnM; with the multiplication induced by (m ::: m ) (m0 :::m0 ) = m ::: m m0 :::m0 1 ⊗ ⊗ p · 1 ⊗ q 1 ⊗ ⊗ p ⊗ 1 ⊗ q where m ;m M for 1 i p; 1 j q: i j 2 ≤ ≤ ≤ ≤ Note that TM is the free K-algebra over M; More precisely: To any K-algebra Λ and any K-linear map f : M Λ there exists a unique algebra homomorphism f : TM Λ extending f. In other words, the ! 0 ! functor T is left-adjoint to the underlying functor to K-vector spaces which forgets the algebra structure. This is because f (m ::: m ) may and must be defined as 0 1 ⊗ ⊗ p f (m1)f (m2):::f (mp): Definition 2.2. Given a K-Lie algebra g, we define the universal enveloping algebra Ug of g to be the quotient of the tensor algebra T g by the ideal I generated by the elements of the form x y y x [x;y]; x;y g: ⊗ − ⊗ − 2 Then Ug = T g=I: Clearly, we have a canonical mapping of K-vector spaces i : g Ug defined by the projection g ! ⊂ T g Ug which can be easily turned into a Lie algebra homomorphism ! i : g LUg: ! It is now easy to see that any Lie algebra homomorphism f : g LΛ induces a unique homomorphism ! f : Ug Λ since the homomorphism f : T g Λ vanishes on the ideal I. We may summarize this in the 1 ! 0 ! following proposition: Proposition 2.3. The universal enveloping algebra functor is a left adjoint to the functor L. The next step of our construction is to study the structure of Ug: Let e ; i J be a K-basis of g f ig 2 indexed by a simply-ordered set J. Let I = i ;:::;i denote an increasing sequence of elements in J. Then f 1 ng define e = e e Ug to be the projection of e ::: e T g. I i1 ··· in 2 i1 ⊗ ⊗ in 2 Theorem 2.4. (Birkhoff-Witt) Let ei ; i J, be a K-basis of g: Then the elements eI corresponding to all finite f g 2 increasing sequence I (including the empty one) form a K-basis of Ug: 2 Proof. We are going to prove this in several steps. We begin defining the index of a monomial e ::: e j1 ⊗ ⊗ jn as follows. For any 1 i < k n, set ≤ ≤ ( 0 if ji jk ηik = ≤ 1 if ji > jk and define the index X ind(e ::: e ) = η : j1 ⊗ ⊗ jn ik i<k Note that ind=0 if and only if j ::: j . Monomials having this property will be called standard. We 1 ≤ ≤ n now suppose jk > jk+1 and we wish to compare ind(e ::: e ) and (1) j1 ⊗ ⊗ jn ind(e ::: e e ::: e ) (2) j1 ⊗ ⊗ jk+1 ⊗ jk ⊗ ⊗ jn where the second monomial is obtained by interchanging ek and ek+1: Let ηik0 be denotes the η0s for the second monomial. Then we have η0 = η if i;j , k;k + 1; η0 = η ; η0 = η if i < k; η = η ; ij ij ik i;k+1 i;k+1 ik 0k;j k+1;j η = η if j > k + 1 and η = 0, η = 1: Hence, 0k+1;j k;j 0k;k+1 0k;k+1 ind(e ::: e ) = 1 + ind(e ::: e e ::: e ): j1 ⊗ ⊗ jn j1 ⊗ ⊗ k+1 ⊗ k ⊗ ⊗ jn We use these remarks to the following claims. Claim 2.5. Every element of T g is congruent mod I to a K-linear combination of 1 and standard monomials. Proof. It suffices to prove the statement for monomials. We order these by degree and for a given degree by the index. To prove the assertion for a monomial e ::: e it suffices to assume it for monomials j1 ⊗ ⊗ jn of lower degree and for those of the same degree n which are of lower index than the given monomial. Assume the monomial is not standard and suppose jk > jk+1: We have, h i e ::: e = e ::: e e ::: e + e ::: e ;e ::: e ; (mod I): j1 ⊗ ⊗ jn j1 ⊗ ⊗ k+1 ⊗ k ⊗ ⊗ jn j1 ⊗ ⊗ jk jk+1 ⊗ ⊗ jn The first term of the right-hand side is of lower index than the given monomial while the second is a linear combination of monomials of lower degree. The result follows from the induction hypothesis. We wish to show that the the cosets of 1 and the standard monomials are linearly independent and so form a basis for Ug. For this purpose we introduce the vector space V with the basis e e ; i ::: i n j1 ··· jn 1 ≤ ≤ n and the vector space V = K1 V V ::: The next claim will help us prove the independence we want. ⊕ 1 ⊕ 2 ⊕ Claim 2.6. There exists a linear mapping σ of T g onto V such that σ(1) = 1 (3) σ(ei1 ei2 ::: ein ) = ei1 ei2 ein ; if i1 ::: in (4) ⊗ ⊗ ⊗ ··· h ≤ i ≤ σ(e ::: e e e e ::: e ) = σ(e ::: e ;e ::: e ) (5) j1 ⊗ ⊗ jn − j1 ⊗ jk+1 ⊗ jk ⊗ ⊗ jn j1 ⊗ ⊗ jk jk+1 ⊗ ⊗ jn Proof. Set σ(1) = 1 and let Tn;j g be the subspace of Tng spanned by the monomials of degree n and index less or equal to j. Suppose a linear mapping σ has already been defined for K1 T1gg T1g ::: Tn 1g ⊕ ⊕ ⊕ ⊕ − satisfying equations 4 and 5 for the monomials in this space. We extend σ linearly to K1 T g T g ⊕ 1 ⊕ 1 ⊕ ::: Tn 1g Tn;j g by requiring that σ(ei ei ::: ej ) = ei ei for the standard monomials of degree n. ⊕ − ⊕ 1 ⊗ 2 ⊗ ⊗ n 1 ··· n 3 Next assume thatσ has been defined for K1 T1g T1g ::: Tn 1g Tn;i 1g, satisfying 4 and 5 for the ⊕ ⊕ ⊕ ⊕ − ⊕ − monomials belonging to this space and let e ::: e be of index i 1: Suppose j > j . Then we set j1 ⊗ ⊗ jn ≥ k k+1 h i σ(e ::: e ) = σ(e ::: e e ::: e ) + σ((e ::: e ;e ::: e ) (6) j1 ⊗ ⊗ jn j1 ⊗ ⊗ jk+1 ⊗ jk ⊗ ⊗ jn j1 ⊗ ⊗ jk jk+1 ⊗ ⊗ jn This makes sense since the two terms on the right are in K1 T1g T1g ::: Tn 1g Tn;i 1g.