Lecture 5 - Lie Algebra Cohomology II

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1 Lecture 5 - Lie Algebra Cohomology II January 28, Motivation: Left-invariant modules over a group Given a vector bundle F ξ G over G where G has a representation on F, a left G- action on ξ is a map that commutes with left-multiplication on G itself: for P, Q G and f F Q (the fiber over Q) we have P.f F P Q, so that Id.f = f and P.Q.f = (P Q).f (1) The first standard example is of the tensor bundles over the manifold G. Given P in G, we have the pushforward map L P : T Q M T P Q M and the pullback map L P : T P Q M T Q M which induce maps on sections of all tensor bundles. The second standard example is of a trivial bundle whose fiber is a G-module V, where the action is obvious. If V is a vector space on which G acts, we can consider the case of V -valued p-forms, namely sections of the various bundles p V (where V over G is considered the cross product bundle). We have a combined action induced on homogeneous elements (elements of the form ω v where ω is a section of p and v V can be taken as a section of the trivial bundle) given on homogeneous elements by p (ω v) = L pω p.v (2) Note that we can extend the exterior derivative by taking dv = 0 for constant sections v of V. We get d : p V p+1 V induced by d(ω v) = dω v. (3) Unfortunately this version of the exterior derivative takes no account of the structure of V as a g-module. Consider the derivative terms x i (ω(x 0,..., ˆx i,..., x k )) in the Cartan formula (4), where the vectors x i at a point can be extended in any differentiable way. In particular they can be considered left-invariant fields, and therefore elements of g, which of course has 1

2 its standard derived action on V. We get a new exterior differential k d(ω v)(x 0,..., x k ) = ( 1) i x i (ω(x 0,..., ˆx i,..., x k )) v + i=0 k ( 1) i ω(x 0,..., ˆx i,..., x k ) x i.v i=0 + i<j( 1) i+j ω([x i, x j ], x 0,..., ˆx i,..., ˆx j,..., x k ) v (4) 2 General Lie algebra cohomology 2.1 Definition Let V be a g-module, and define d : k g V k+1 g V by d(ω v)(x 0,..., x k ) = k ( 1) i ω(x 0,..., x i,..., x k ) x i.v i=0 + i<j( 1) i+j ω([x i, x j ], x 0,..., x i,..., x j,..., x k ) v (5) where x 0,..., x k are elements of g. To see that d 2 = 0, note that we have the usual Leibnitz rule so that d(ω v) = dω v + ( 1) ω ω dv (6) d 2 (ω v) = d 2 ω v + ω d 2 v (7) The first term is zero (as can be seen 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). To see that the second term is zero, let {v i } be a basis for V, and given a fixed element v V define elements η i g so that for all x g. Then by definition dv(g) = η i v i. We have ddv(x, y) = d(η i v i )(x, y) η i (x)v i = x.v (8) = η i (x)y.v i + η i (y)x.v i η i ([x, y])v i = y.(η i (x)v i ) + x.(η i (y)v i ) η i ([x, y])v i = y.x.v i + x.y.v i [x, y].v = 0 (9) 2

3 We can therefore form a cohomology theory. We define the p-cochain group C p (g, V ) to be C p (g, V ) p g V (10) the p-cocycles Z p (g, V ) to be Z p (g, V ) { A C p (g, V ) } da = 0 (11) and the p-coboundaries to be B p (g, V ) dc p 1 (g, V ) = { da C p (g, V ) A C p 1 (g, V ) } (12). These are all vector spaces, so we have the p-cohomology groups H p (g, V ) = Z p (g, V ) / B p (g, V ). (13) 2.2 Cup product We have a natural wedge product : C p (g, F) C q (g, V ) C p+q (g, V ) (14) which passes to a cup product on cohomology : H p (g, F) H q (g, V ) H p+q (g, V ) (15) 2.3 Naturality, and Short Exact Sequences Any homomorphism V W of g-modules induces a map on cohomology H p (g, V ) H p (g, W ). Any short exact sequence of g-modules induces a long exact sequence in cohomology: 0 V W K 0 (16)... H p (g, V ) H p (g, W ) H p (g, K) H p+1 (g, V )... (17) 3 Interpretations 3.1 The groups H p (g, F) Consider a trivial representations of G on the base field F, given by P.f = f for all P G, f F. Then the action of g on F is also trivial: x.f = 0 for all x g. By left translation 3

4 we can extend the spaces C p (g, F), Z p (g, F), and B p (g, F) to the left-invariant 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) (18) 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). (19) We can make a few basic computations. We have C 0 (g, F) F (20) and all sections are closed. Thus there is a natural isomorphism Also The exterior derivative is H 0 (g, F) F. (21) C 1 (g, R) = g. (22) dω(x, y) = ω([x, y]) (23) so that Z 1 (g, F) are those linear operators that evaluate to zero on the derived subalgebra. Since B 1 (g, F) = {0} we have H 1 (g, F) (g/[g, g]). (24) 3.2 The groups H p (g, g) In these groups of course the representation of g on g is the adjoint representation The bracket operation on cohomology Recall that we have a bracket operation ( pg ) ( qg ) [, ] : g g p+q g g (25) given by [ω i g i, η j g j ] = ω i η j [g i, g j ] (26) 4

5 where {g i } is a basis of g and ω i p g, η i q g. Note that for g, h, x g we have d([g, h])(x) = ad x [g, h] = [ad x (g), h] + [g, ad x (h)] (27) so that d[g, h] = [dg, h] + [g, dh] and d[ω i g i, η j g j ] = dω i η j [g i, g j ] + ( 1) ωi ω i dη j [g i, g j ] Therefore we have + ( 1) ωi + η j ω i η j [dg i, g j ] + ( 1) ωi + η j ω i η j [g i, dg j ] = [dω i g i, η j g j ] + ( 1) ωi [ω i g i, dη j g j ] + ( 1) ωi [ω i dg i, η j g j ] + ( 1) ωi + η j [ω i g i, η j dg j ] ) ] = [(dω i g i + ( 1) ωi ω i dg i, η j g j [ )] + ( 1) ωi ω i g i, (dη j g j + ( 1) ηj η j dg j = [ d ( ω i ) g i, η j ] g j + ( 1) ω [ i ω i g i, d ( η j )] g j In particular, the bracket passes to maps and d[ω, η] = [dω, η] + ( 1) ω [ω, dη]. (29) [, ] : Z p (g, g) Z q (g, g) Z p+q (g, g) (30) [, ] : Z p (g, g) B q (g, g) B p+q (g, g) (31) so that we also have a bracket operation in cohomology: [, ] : H p (g, g) H q (g, g) H p+q (g, g). (32) (28) Interpretations We can give good interpretations of the groups H 0 (g, g) and H 1 (g, g). The zero-cocycles are those elements x g such that dx = 0. Since dx(y) = ad y (x) = [y, x], these are the central elements. Therefore H 0 (g, g) Z(g). (33) Now consider the elements of 1 g g = C 1 (g, g), which are just the linear operators g g. Now let {g i } be a basis for g, and let ω = ω i g i be a closed 1-form. That means 0 = (dω)(x, y) = ω i (x)y.g i + ω i (y)x.g i ω i ([x, y])g i = [ω(x), y] + [x, ω(y)] ω([x, y]) (34) 5

6 so that 1-cocycles are precisely the derivations of g. The 1-coboundaries are those derivations ω so that ω(x) = (dg)(x) for some g g, or so that ω(x) = (dg)(x) = x.g = [g, x] = ad g (x) (35) which are precisely the inner derivations of g. Thus the group is isomorphic to the vector space of outer derivations of g. H 1 (g, g) Der Out (g) (36) 3.3 General case: H 0 (g, V ) The zero cochain group (vector space) is simply C 0 (g, V ) = V. (37) There are no coboundaries, and the cocycles are the v V such that dv = 0, or that dv(x) = x.v = 0 (38) for all x g. That is, the invariant elements. We have H 0 (g, V ) = { v V g.v = {0} } (39) 3.4 General case: H 1 (g, V ) The cocycle space is C 0 = g V, or linear functionals f : g V. The boundaries are those functionals f(x) = dv(x) = x.v for some v V, and the cocycles are those functionals f so that 0 = df(x, y) = x.f(y) y.f(x) f([x, y]). (40) Such maps are called derivations of g into V, or, intuitively, those maps that behave well with respect to the bracket: Der(g, V ) = { f : g V f([x, y]) = x.f(y) y.f(x) }. (41) The derivations that can be considered inner are the maps f v where f v (x) = x.v. We shall call this vector space IDer(g, V ). We therefore have H 1 (g, V ) = Der(g, V ) / IDer(g, V ) (42) which can be regarder as the space of outer derivations of g into V. 6

7 3.5 General case: H 2 (g, V ) Consider first the case of V = F, with the trivial module structure. The cochains are C 2 (g, R) = 2 g F, or simply 2-forms, and the boundaries are those ω so that ω(x, y) = df(x, y) = f([x, y]). (43) for some 1-form f : g F. The closed elements (the 2-cocycles) are those ω so that 0 = dω(x, y, z) = ω([x, y], z) + ω([y, z], x) + ω([z, x], y) (44) which resembles a Jacobi identity. Given any such ω, we can create a new Lie algebra on the direct sum R g, with bracket [(a, x), (b, y)] ω = (ω(x, y), [x, y]). (45) The bilinearity and anti-symmetry of this bracket are clear, and to check the Jacobi identity we compute [(a, x), [(b, y), (c, z)]] + [(b, y), [(c, z), (a, x)]] + [(c, z), [(a, x), (b, y)]] = [(a, x), (ω(y, z), [y, z])] + [(b, y), (ω(z, x), [z, x])] + [(c, z), (ω(x, y), [x, y])] = (ω(x, [y, z]), [x, [y, z]]) + (ω(y, [x, z]), [y, [x, z]]) + (ω(z, [x, y]), [z, [x, y]]) = (ω(x, [y, z]) + ω(y, [x, z]) + ω(z, [x, y]), [x, [y, z]] + [y, [x, z]] + [z, [x, y]]) = (0, 0). (46) Notice that R is central in R g. To be clear about which central extension is being considered, the bracket given by (45) will be indicated by R ω g. (47) Now how do we interpret the coboundaries? Assume that the algebra is given by (45) where ω = df, and consider the vector space isomorphism M f : R 0 g R df g M f (a, x) = (a + f(x), x). (48) On the one hand we have M f [(a, x), (b, y)] = M f (0, [x, y]) = (f[x, y], [x, y]) (49) and on the other hand we have [M f (a, x), M f (b, y)] df = [(a + f(x), x), (b + f(y), y)] df = (f[x, y], [x, y]). (50) Therefore f is a Lie algebra isomorphism. On the other hand, it can be shown that if R ω g has a Lie algebra isomorphism that is an isomorphism on the restriction to both R 7

8 and g, then ω = df for some f (this is left as an exercise). Thus H 2 (g, F) is the space of isomorphism classes of central extensions of g by F. What about the case of a general g-module V? Given a V -valued 2-form ω we can create a bracket on V ω g by [(a, x), (b, y)] ω = (x.b y.a + ω(x, y), [x, y]). (51) Note that the subalgebra V V ω g is abelian, although the only elements of V that are actually central are the invariant elements. It can be shown that two isomorphic central extensions differ by an exact, so H 2 (g, V ) is the set of isomorphism classes of abelian extensions of g by V. 4 The Whitehead Lemmas The Whitehead lemmas provide basic structure theorems on the cohomology of semisimple Lie algebras. They are Theorem 4.1 If g is a semisimple Lie algebra and V any finite-dimensional g-module, then H 1 (g, V ) = {0} and H 2 (g, V ) = {0}. Theorem 4.2 If g is a semisimple Lie algebra and V a finite-dimensional g-module without invariant elements (ie. there is no v V so that g.v {0}), then H n (g, V ) = {0} for all n. 8