FTUV/98-4 IFIC/98-4 January, 1998 An introduction to some novel applications of Lie algebra cohomology in mathematics and physics∗ J. A. de Azc´arraga†, J. M. Izquierdo‡ and J. C. P´erez Bueno† † Departamento de F´ısica Te´orica, Univ. de Valencia and IFIC, Centro Mixto Univ. de Valencia-CSIC, E–46100 Burjassot (Valencia), Spain. ‡ Departamento de F´ısica Te´orica, Universidad de Valladolid E–47011, Valladolid, Spain Abstract After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents 1 Preliminaries: LX , iX , d ........................... 1 2 Elementary differential geometryon Lie groups . ..... 3 3 Liealgebracohomology: abriefintroduction . ..... 4 4 Symmetric polynomials and higher order cocycles . ...... 7 5 HigherordersimpleandSHLiealgebras. .. 11 6 Higher order generalizedPoissonstructures . ...... 20 7 Relative cohomology, coset spaces and effective WZW actions ....... 23 1 Preliminaries: LX, iX , d Let us briefly recall here some basic definitions and formulae which will be useful later. Consider a uniparametric group of diffeomorphisms of a manifold M, eX : M → M, which takes a point x ∈ M of local coordinates {xi} to x′i ≃ xi + ǫi(x) (= xi + Xi(x)). Scalars and (covariant, say) tensors tq (q = 0, 1, 2,...) transform as follows j j j ′ ′ ′ ′ ∂x ′ ′ ∂x 1 ∂x 2 φ (x )= φ(x) , t i(x )= tj(x) , t i i (x )= tj j (x) ... (1.1) ∂x′i 1 2 1 2 ∂x′i1 ∂x′i2 arXiv:physics/9803046v1 [math-ph] 30 Mar 1998 In physics it is customary to define ‘local’ variations, which compare the transformed and original tensors at the same point x: ′ ′ δφ(x) ≡ φ (x) − φ(x) , δti(x) ≡ t i(x) − ti(x) , ... (1.2) Then, the first order variation defines the Lie derivative: j j j δǫψ = −ǫ (x)∂jψ(x) := −LX ψ , (δǫt)i = −(ǫ ∂jti + (∂iǫ )tj) := −(LX t)i , j j j (1.3) (δǫt)i1i2 = −(ǫ ∂jti1i2 + (∂i1 ǫ )tji2 + (∂i2 ǫ )ti1j) := −(LX t)i1i2 . Eqs. (1.3) motivate the following general definition: ∗ To appear in the Proceedings of the VI Fall Workshop on Geometry and physics (Salamanca, September 1997). 1 Definition 1.1 (Lie derivative) i1 iq k ∂ Let α be a (covariant, say) q tensor on M, α(x)= αi1...iq dx ⊗ . ⊗ dx , and X = X ∂xk a vector field X ∈ X(M). The Lie derivative LX of α with respect to X is locally given by k k k ∂αi1...iq ∂X ∂X (LX α)i ...i = X + αki ...i + . + αi ...i − k . (1.4) 1 q ∂xk 2 q ∂xi1 1 q 1 ∂xiq 1 On a q-form α(x)= α dxi1 ∧ . ∧ dxiq , α ∈∧ (M), L α is defined by q! i1...iq q Y q (L α)(X ,...,X ) := Y · α(X ,...,X ) − α(X ,..., [Y, X ],...,X ) ; (1.5) Y i1 iq i1 iq X i1 i iq i=1 p on vector fields, LX Y = [X,Y ]. The action of LX on tensors of any type tq may be found using that LX is a derivation, ′ ′ ′ LX (t ⊗ t ) = (LX t) ⊗ t + t ⊗ LX t . (1.6) Definition 1.2 (Exterior derivative) The exterior derivative d is a derivation of degree +1, d : ∧q(M) → ∧q+1(M); it satisfies Leibniz’s rule, q d(α ∧ β)= dα ∧ β + (−1) α ∧ dβ , α ∈ Λq , (1.7) and is nilpotent, d2 = 0. On the q-form above, it is locally defined by 1 ∂αi ...i dα = 1 q dxj ∧ dxi1 ∧ . ∧ dxiq . (1.8) q! ∂xj The coordinate-free expression for the action of d is (Palais formula) q+1 (dα)(X ,...,X , X ) := (−1)i+1X · α(X ,..., Xˆ ,...,X ) 1 q q+1 X i 1 i q+1 i=1 (1.9) + (−1)i+jα([X , X ], X ,..., Xˆ ,..., Xˆ ,...,X ) . X i j 1 i j q+1 i<j In particular, when α is a one-form, dα(X1, X2)= X1 · α(X2) − X2 · α(X1) − α([X1, X2]) . (1.10) Definition 1.3 (Inner product) The inner product iX is the derivation of degree −1 defined by (iX α)(X1,...,Xq−1)= α(X, X1,...,Xq−1) . (1.11) On forms (Cartan decomposition of LX ), LX = iX d + diX , (1.12) from which [LX , d] = 0 follows trivially. Other useful identity is [LX , iY ]= i[X,Y ] ; (1.13) from (1.12) and (1.13) it is easy to deduce that [LX ,LY ]= L[X,Y ]. 2 2 Elementary differential geometry on Lie groups ′ ′ ′ Let G be a Lie group and let Lg′ g = g g = Rgg (g , g ∈ G) be the left and right actions G × G → G with obvious notation. The left (right) invariant vector fields LIVF (RIVF) on G reproduce the commutator of the Lie algebra G of G [XL (g), XL (g)] = Ck XL (g) , [XR (g), XR (g)] = −Ck XR (g) , Cρ Cσ = 0 , (i) (j) ij (k) (i) (j) ij (k) [i1i2 ρi3] (2.1) where the square bracket [ ] in the Jacobi identity (JI) means antisymmetrization of the 1 indices i1, i2, i3. In terms of the Lie derivative, the L- (R-) invariance conditions read L R L R L R LXR g X (g) = [X (g), X (g)] = 0 , LXL g X (g) = [X (g), X (g)] = 0 . (2.2) (j)( ) (i) (j) (i) (i)( ) (j) (i) (j) L(i) Let ω (g) ∈ ∧1(G) be the basis of LI one-forms dual to a basis of G given by LIVF L(i) i (ω (g)(XL(j)(g)) = δj). Using (1.10), we get the Maurer-Cartan (MC) equations 1 dωL (i)(g)= − Ci ωL (j)(g) ∧ ωL (k)(g) . (2.3) 2 jk In the language of forms, the JI in (2.1) follows from d2 = 0. If the q-form α is LI dαL(XL,...,XL )= (−1)s+tαL([XL, XL], XL ,..., Xˆ L,..., Xˆ L,...,XL ) , (2.4) i1 iq+1 X is it i1 is it iq+1 s<t L L ˆ L L 2 since α (X1 ,..., Xi ,...,Xq+1) in (1.9) is constant and does not contribute . To facilitate the comparison with the generalized dm to be introduced in Sec. 5, we note here that, with d2 ≡−d, eq. (2.4) is equivalent to e e L L L 1 1 j1...jq+1 L L L L L d2α (X ,...,X )= ε α ([X , X ], X ,...,X ) . (2.5) i1 iq+1 (2 · 2 − 2)! (q − 1)! i1...iq+1 j1 j2 j3 iq+1 e The MC equations may be written in a more compact way by introducing the (canonical) (i) G-valued LI one-form θ on G, θ(g)= ω (g)X(i)(g); then, MC equations read 1 dθ = −θ ∧ θ = − [θ,θ] (2.6) 2 (i) (j) since, for G-valued forms, [α, β] := α ∧ β ⊗ [X(i), X(j)]. The transformation properties of ω(i)(g) follow from (1.5): (j) j (k) LX(i)(g)ω (g)= −Cikω (g) . (2.7) 1 For a general LI q-form α(g)= α ω(i1)(g) ∧ . ∧ ω(iq )(g) on G q! i1...iq q 1 is (i1) \(i ) (k) (iq ) L α(g)= − C αi ...i ω (g) ∧ . ∧ ω s (g) ∧ ω (g) ∧ . ∧ ω (g) . (2.8) X(i)(g) X q! ik 1 q s=1 1The superindex L (R) in the fields refers to the left (right) invariance of them; LIVF (RIVF) generate right (left) translations. 2From now on we shall assume that vector fields and forms are left invariant (i.e., X ∈ XL(G), etc.) and drop the superindex L. Superindices L, R will be used to avoid confusion when both LI and RI vector fields appear. 3 3 Lie algebra cohomology: a brief introduction 3.1 Lie algebra cohomology Definition 3.1 (V -valued n-dimensional cochains on G) Let G be a Lie algebra and V a vector space. A V -valued n-cochain Ωn on G is a skew- symmetric n-linear mapping n 1 Ω : G∧ ···∧G → V , ΩA = ΩA ωi1 ∧ . ∧ ωin , (3.1) n n n! i1...in where {ω(i)} is a basis of G∗ and the superindex A labels the components in V . The (abelian) group of all n-cochains is denoted by Cn(G, V ). Definition 3.2 (Coboundary operator (for the left action ρ of G on V )) A C Let V be a left ρ(G)-module, where ρ is a representation of the Lie algebra G, ρ(Xi).C ρ(Xj).B − A C A n n+1 ρ(Xj).C ρ(Xi).B = ρ([Xi, Xj ]).B. The coboundary operator s : C (G, V ) → C (G, V ) is defined by n+1 (sΩ )A (X , ..., X ) := (−)i+1ρ(X )A (ΩB(X , ..., Xˆ , ..., X )) n 1 n+1 X i .B n 1 i n+1 i=1 n+1 (3.2) + (−)j+kΩA([X , X ], X , ..., Xˆ , ..., Xˆ , ..., X ) . X n j k 1 j k n+1 j,k=1 j<k Proposition 3.1 The Lie algebra cohomology operator s is nilpotent, s2 = 0. Proof. Looking at (1.9), s in (3.2) may be at this stage formally written as A A A i i (s).B = δBd + ρ(Xi).Bω , (s = d + ρ(Xi)ω ) . (3.3) Then, the proposition follows from the fact that 2 i j i j i j 2 s = (ρ(Xi)ω + d)(ρ(Xj )ω + d)= ρ(Xi)ρ(Xj )ω ∧ ω + ρ(Xi)ω d + ρ(Xj)dω + d 1 1 = − ρ(X )Cj ωl ∧ ωk + [ρ(X ), ρ(X )]ωi ∧ ωj = 0 .