From: Thomas Larsson Subject: Lie algebra cohomology (was Re: This Week's Finds in Mathematical Physics (Week 162)) Date: 6 Feb 2001 18:59:16 GMT Newsgroups: sci.physics.research Summary: Formalisms and interpretations of Lie-algebra cohomology Chris Hillman wrote in message news:Pine.OSF.4.21.0102021846490.272664-100000@goedel1.math.washington.edu... > > > On 30 Jan 2001, Thomas Larsson wrote: > > > I have never worked actively with Lie algebra cohomology myself, but I > > do appreciate the results. At least extensions of algebras are > > classified with cohomology. By definition, an extension L^ of an > > algebra L by a module M is the exact sequence > > > > 0 --> M --> L^ --> L --> 0 > > > > If we regard the L module M as a commutative algebra, L^ is an abelian > > extension. The non-split extensions (i.e. not semi-direct products) > > are classified by the relative cohomology groups H^2(L,M). > > Can anyone clarify this? > Since I started this, I guess that means me. First some literature. Most work on Lie algebra cohomology has been done by Israel Gelfand's school in Moscow, and the standard reference is Fuks' (as in Feigin-Fuks) book: Cohomology of Infinite-Dimensional Lie Algebras (Contemporary Soviet Mathematics) by D.B. Fuks, Hardcover (February 1987) Consultants Bureau; ISBN: 0306109905 Unfortunately, this book is not available at Amazon (and at USD 152, it is too expensive anyway). The basic results can probably be found as applications in books on homological algebra. A concise explanation can be found in Cederwall et al, Nucl. Phys. B 424 (1994) 97, http://www.arxiv.org/abs/hep-th/9401027, section 5. As I stated before, I have never worked actively with this, and chasing diagrams is as intimidating to me as to anybody else. But this is what I read: Let L be a Lie algebra and M an L-module. We denote the linear space of k-linear skew-symmetric maps c: L x ... x L -> M by C^k(L,M) and put C^*(L,M) = direct sum_(k in Z) C^k(L,M); C^k(L,M) = 0 for k < 0. The coboundary operator d: C^k(L,M) -> C^k+1(L,M) is defined by the formula dc(g_1, ... g_k+1) = sum_(1<=s S -> N -> M -> 0 is a sequence of L modules, then N is a right extension of M by S or a left extension of S by M. Two examples: 2a. L = vect(n), S = C (trivial module), M = tensor field of type (1,2). Then N is the module of connections \Gamma^i_jk. 2b. L = map(n,g), S = C, M = adjoint. Then N is the module of gauge connections A^a_i. In physics lingo: the connection transforms inhomogeneously in an interesting way under diffeomorphisms/gauge transformations. 3. H^1(L,L) is the space of exterior derivations of L, i.e. all derivations modulo inner ones. 4. H^2(L,M) is the space of extensions of L by the module M. More about this below. 5. H^2(L,L) is the space of Lie algebra deformations of L 6. The interpretation of H^k(L,M) with k>=3 is not so clear. Since the cocycles involve three elements, they probably label obstructions to associativity/ Jacobi identity. The case H^2(L,M) is probably best illustrated by an example. Set L = vect(1) (vector fields on the circle), M = C (trivial module). In a Fourier basis, there are two cocycles: [L_m, L_n] = (m-n) L_m+n + (1/12) (c m^3 + b m)\delta_m+n. The linear cocycle proportional to b is also a coboundary; it can be eliminated by the redefinition L_m -> L_m + (b/24)\delta_m. Conventionally, one uses this freedom to set b = -c. In other words, this is just the semidirect product L |X M (even direct product since M is the trivial module), and the linear cocycle vanishes in cohomology. In contrast, the cubic cocycle (proportional to c) is not a coboundary. dim H^2(vect(n), C) = 1. In general, one can show that the condition that the extension L^ satisfies the Jacobi identities is exactly equivalent to the cocycle condition dc = 0. This is done e.g. in the article by Cederwall et al. above. Askar Dzhumadildaev has classified central extensions [D1], abelian extensions of vect(1) [D2] and abelian extensions of vect(n) and its subalgebras [D3]. [D1] A S Dzhumadildaev, On the cohomology of modular Lie algebras, Math. USSR-Sb 47 (1984) 127-143 [D2] A S Dzhumadildaev, Cohomology and nonsplit extensions of modular Lie algebras, Contemporary Math. 131 (1992) 31-43 [D3] A S Dzhumadildaev, Virasoro type Lie algebras and deformations, Z. Phys. C 72 (1996) 509 Askar is probably the world leader in this field. What's really impressive is that he works in Kazachstan. Unfortunately, his papers are not easy to read for a physicist (at least not for this one). I spent a ridiculous amount of time trying to understand [D3] (on and off for three years), and when I finally figured his results out, I wrote it up in component formalism and put it on the web: T A Larsson, Extensions of diffeomorphism and current algebras, http://www.arxiv.org/abs/math-ph/0002016 (2000) Abelian extensions are important because any quantum deformation of a classical symmetry reduces to an extension in the hbar->0 limit. This follows immediately by power counting. The only condition is really that the universal enveloping algebra must be associative. Consider e.g. the quantum group U_q(sl(2)), with non-zero brackets [H,E] = E, [H,F] = -F, [E,F] = sin(H). It is a good exercise to work out the corresponding extension (hint: expand sin(H) in a power series and throw away all terms higher than cubic). ============================================================================== From: Thomas Larsson Subject: Re: Lie algebra cohomology Date: 7 Feb 2001 17:18:27 GMT Newsgroups: sci.physics.research Thomas Larsson wrote in message news:95phi4$p52$1@news.state.mn.us... >By definition, an extension L^ of an algebra L by a module M is the exact sequence > >0 --> M --> L^ --> L --> 0 > Let me elaborate a bit on this. Exactness of this sequence means the following: 0 --> M --> L^ : The map from M to L^ is an embedding, i.e. the image of M in L^ is faithful. L^ --> L --> 0 : The projection from L^ to L is onto. M --> L^ --> L : The image of the embedding is the kernel of the projection. As a vector space, L^ = L+M, and M is an ideal in L^. This is the definition of an extension. The sequence above is considered trivial if it *splits*, i.e. the projection is invertible. Then there is not only an arrow from L^ to L, but also in the opposite direction: 0 --> M --> L^ <==> L --> 0 > Let L be a Lie algebra and M an L-module. We denote the linear space of > k-linear skew-symmetric maps c: L x ... x L -> M by C^k(L,M) and put > C^*(L,M) = direct sum_(k in Z) C^k(L,M); C^k(L,M) = 0 for k < 0. > The coboundary operator d: C^k(L,M) -> C^k+1(L,M) is defined by the formula What confused me for a long time is that because the sequence above is exact, it has no cohomology. Non-split extensions are classified by the cohomology of an associated complex, which is not exact. The relevant part is ... --> C^1(L,M) --> C^2(L,M) --> C^3(L,M) --> ... where the maps are given by the coboundary operator d. > to c) is not a coboundary. dim H^2(vect(n), C) = 1. This is a typo. dim H^2(vect(1), C) = 1. If n>1, dim H^2(vect(n), C) = 0, but dim H^2(vect(n), M) = 2, where M is the module of closed dual one-forms, i.e. closed (N-1)-forms. This was actually my main point.