Lecture 4 - Lie Algebra Cohomology I

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1 Lecture 4 - Lie Algebra Cohomology I January 25, 2013 Given a differentiable manifold M n and a k-form ω, recall Cartan s formula for the exterior derivative k dω(v 0,..., v k ) = ( 1) i x i (ω(x 0,..., ˆx i,..., x k )) i=0 + (1) i<j( 1) i+j ω([x i, x j ], x 0,..., ˆx i,..., ˆx j,..., x k ) where x 0,..., x k are vectors, extended in any (differentiable) fashion around a given point.. 1 Lie derivatives If M is a differentiable manifold and ϕ τ a 1-parameter family of diffeomorphisms, we define the Lie derivative of the p-form η along ϕ τ by Lη = d dτ ϕ ϕ τ η η τ η = lim τ 0 τ which is a differential operator of order 0. If X is a differentiable vector field and ϕ τ is its integral flow, we define L X η as above. Clearly L X f = X(f) in the case of functions, we have the Leibnitz rule L X (η ω) = (L X η) ω + η L X ω (3) and we have commutativity with the exterior derivative: (2) L X dη = dl X η (4) which follows from the commutativity of d with pullbacks along diffemorphisms. A differential operator D of order D is a map Γ( p T M) Γ( p+ D T M) that obeys the Leibniz rule D(η ω) = (Dη) ω + ( 1) D η η Dω. (5) 1

2 If D 1, D 2 are differential operators, we define their commutator by [D 1, D 2 ] = D 1 D 2 ( 1) D1 D2 D 2 D 1 (6) which is another differential operator, of order D 1 + D 2. The Jacobi identity does not hold in general, although does hold if the orders of the operators all have the same parity. It is easily seen that L X η = [d, i x ] η = (di X + i X d) η (7) as can be verified by noting that the formula holds both for functions and for 1-forms. This is known as Cartan s formula. In particular, L X is a differential operator of order 0. We have the following additional commutator relations: (Jacobi identity) [L X, L Y ] = L [X,Y ] [L X, d] = 0 [L X, i Y ] = i [X,Y ] As a final note, one defined Lie derivatives on sections of arbitrary (p, q)-tensors p T M q T M similary, via a combination of pushforwards and pullbacks along diffeomorphisms. Notably, given vector fields X, and Y, we have L X Y = [X, Y ] (8) and given sections η of p T M q T M and ω of r T M s T M we have the Leibnitz rules: (T ensor P roducts) L X (η ω) = (L X η) ω + η (L X ω) (Contractions) L X (η(y )) = (L X η)(y ) + η (L X Y ). (9) In fact, an axiomatic approach to Lie derivatives is often useful. give a vector fields X and Y, tensor fields T 1 and T 2 of any (p, q) type, constants c 1 and c 2, a 1-form η, and a function f, we have the 5 axioms 1) (Linearity in the second variable) L X (c 1 T 1 + c 2 T 2 ) = c 1 L X T 1 + c 2 L X T 2 2) (Evaluation on functions) L X f = X(f) 3) (Evaluation on vector fields) L X Y = [X, Y ] 4) (Leibniz rule for tensor products) L X (T 1 T 2 ) = (L X T 1 ) T 2 + T 1 (L X T 2 ) 5) (Leibniz rule for contractions) L X (η(y )) = (L X η) (Y ) + η (L X Y ) which uniquely characterize the Lie derivative. 2

3 2 Forms and Invariant Forms Assume G is a compact (and connected) Lie group. We have a left action of G on p-forms given by pullback along left translations. If P G, then L P : G G and L P : p p. Consider a closed p-form η Γ( p T M), an element X g which can be considered a left-invariant vector field on G, and the left-action by L exp(x). We have that d dt L exp(tx) η is a derivative along diffeomorphisms, so is a Lie derivative. Then ) 1 ( ) d (L exp(x) η η p = p 0 dt L exp(tx) η dt p = = ( 1 = d L X ( ϕ t η ) dt di X ϕ tη dt 0 ) i X ϕ tη dt so that a closed p-form and its left translation differ by an exact p-form, and so in particular lie in the same derham class. If the Lie group is compact, we can therefore average the p-form over all left-translations to obtain a left-invariant p-form in the same derham class. This shows that derham classes can be represented by invariant p-forms, although not necessarily uniquely. Also note that if two invariant forms are in the same derham class, their difference, dη for some η, can be taken to be invariant as well, since L X and d commute. Thus to develop derham cohomology, we can restrict to invariant p-forms, to obtain an algebra isomorphism Now notice that on left-invariant forms η we have (10) H Inv DR (M; F) H DR (M; F). (11) dη(x 0,..., x k ) = i<j( 1) i+j η ([x i, x j ], x 0,..., x i,..., x j,..., x k ) (12) so that the exterior differential is given entirely in terms of the Lie algebra structure. Note also that k,invt M k g (13) This gives us a natural exterior differential on the vector space k g itself, and we are led to the subject of Lie algebra cohomology. 3

4 3 Basic Lie Algebra Cohomology 3.1 Definition Define d : k g k+1 g by (dω)(x 0,..., x k ) = i<j( 1) i+j ω([x i, x j ], x 0,..., x i,..., x j,..., x k ) (14) where x 0,..., x k are elements of g. We have that d 2 = 0, which is essentially obvious: it follows directly via the Jacobi identity, or indirectly via the identification of elements of p g with left-invariant sections of k T G where G is an appropriate Lie group. We can therefore form a cohomology theory. We define the p-cochain group C p (g, F) to be the p-cocycles Z p (g, F) to be C p (g, F) p g Z p (g, F) { ω C p (g, F) dω = 0 } (15) (16) and the p-coboundaries to be B p (g, F) dc p 1 (g, F) = { da C p (g, F) A C p 1 (g, F) }. (17) These are all vector spaces, so we have the p-cohomology groups H p (g, F) = Z p (g, F) / B p (g, F). (18) 3.2 Cup product We have a natural wedge product : C p (g, F) C q (g, F) C p+q (g, F) (19) the Leibniz formula d(ω η) = dω η + ( 1) ω ω dη implies that : Z p (g, F) Z q (g, F) Z p+q (g, F) (20) : Z p (g, F) B q (g, F) B p+q (g, F) (21) so that the wedge product passes to cohomology, where it is called the cup product: : H p (g, F) H q (g, F) H p+q (g, F) (22) 4

5 4 Interpretations By left translation we can extend the spaces C p (g, F), Z p (g, F), and B p (g, F) to the leftinvariant vector spaces C p,inv (G, F), Z p,inv (g, F), and B p,inv (G, F), and form the invariant cohomology groups H p,inv dr (G, F) = Zp,Inv(G, F) / Bp,Inv(G, F) (23) dr which we have already seen are isomorphic to the derham cohomology groups H p (G, F). We therefore have isomorphisms dr H p (g, F) H p,inv dr (G, F) Hp dr (G, F). (24) 4.1 H 0 (g, F) We can make a few basic computations. We have C 0 (g, F) F (25) and all sections are closed. Thus there is a natural isomorphism H 0 (g, F) F. (26) 4.2 H 1 (g, F) We have The boundary space is easy to compute: The exterior derivative on 1-forms is C 1 (g, F) = g. (27) B 1 (g, F) = dc 0 (g, F) = {0}. (28) dω(x, y) = ω([x, y]) (29) so that Z 1 (g, F) are those linear operators that evaluate to zero on the derived subalgebra. We therefore have H 1 (g, F) (g/[g, g]). (30) This implies, in particular, that if G is a compact Lie group with a semi-simple Lie algebra, we have that the fundamental group π 1 (G) is finite. Its universal cover, another semi-simple Lie group, is therefore also compact. 5

6 4.3 H 2 (g, F) We have C 2 (g, F) = 2 g (31) The closed 2-forms ω are those with 0 = dω(x, y, z) = ω([x, y], z) + ω([y, z], x) + ω([z, x], y) (32) and the exact 2-forms have the form for any functional η : g F (that is, any η g = C 1 (g, F)). dη(x, y) = η([x, y]) (33) Any closed 2-form leads to a Lie algebra structure on the 1-dimensional extension R g, which makes the R factor central. The Lie bracket on two elements (a, x), (b, y) is We check that the Jacobi identity works: so that [(a, x), (b, y)] = (ω(x, y), [x, y]). (34) [(a, x), [(b, y), (c, z)]] = (ω(x, [y, z]), [x, [y, z]]) (35) [(a, x), [(b, y), (c, z)]] + [(b, y), [(c, z), (a, x)]] + [(c, z), [(a, x), (b, y)]] = (ω(x, [y, z]), [x, [y, z]]) + (ω(y, [z, x]), [y, [z, x]]) + (ω(z, [x, y]), [z, [x, y]]) = (ω(x, [y, z]) + ω(y, [z, x]) + ω(z, [x, y]), [x, [y, z]] + [y, [z, x]] + [z, [x, y]]) = (0, 0) (36) (recalling that the vector space structure is from the cross product, not the tensor product). Now given any exact 2-form dη, we can construct a map η : R g R g given by η(a, x) = (a + η(x), x). (37) This is a Lie algebra isomorphism from the extension ω to the extension ω + dη, as we see from the computation η ([(a, x), (b, y)] ω ) = η(ω(x, y), [x, y]) = (ω(x, y) + η[x, y], [x, y]) (38) [η(a, x), η(b, y)] ω+dη = [(a + η(x), x), (b + η(y), y)] ω+dη = (ω(x, y) + η([x, y], [x,(39) y]) Further, one can see that if two central extensions by F are isomorphic, the Lie algebras differ by such an exact 2-form. Therefore H 2 (g, F) are the isomorphism classes of central extensions of g by F. 6