Remarks on the Schouten-N jenhuis bracket P. W. MiCHOR Estratto PROCEEDINGS OF THE wINTER' SCHOOL ON GEOMETRY AND PHYSICS SRNI 10-17 JANUARY, 1987 Supplemento ai Rendiconti del Circolo Matematico di Palertno Serie II- numero 16 - 1987 Via Archiraft, 34 - 90123 Palermo (Italia) REMARKS ON THE SCHOUTEN-NIJENHUIS BRACKET Peter W. Miehor In 1940 Schouten introduced the differential invariant of two purely contravariant tensor fields. In 1955 Nijenhuis showed that for skew symmetric contravariant tensor Fields (also called skew multivector Fields) this concomitant satisfies the Jacobi identity and gives a structure of a graded Lie algebra to the space of all multivector Fields. Fhe same is true For the symmetric multivector Fields. in 1974 Tulezyjew gave o coordinate Free treatment of the bracket For skew multi- vector Fields and clarified its relation to certain differential operators on the space of differential forms, which are similar to those of he beter known and more importan FrOlicher-Nijenhuis bracket For angent bundle valued differential Forms. The 5chouten-Njenhuis bracket can be used to express integrablty properties (1.$) and its vanishing is also the condition on a 2-vector field to define a Podsson bracket For-Functions (a coordinate Free proof of this is in .4). Recently Koszul explained some relations of this bracket to tie agebra eohomology. The Sehouten-Nijenhuis bracket For symmetric multivector fields is well known to coincide with the Poisson bracket of the associated functions on the cotangent bundle, which are polynomial along the Fibres. It will no be treated in this paper - similar results as those treated in this paper are true For it. In this paper we introduce the SchoutenNijenhuis brackeL For skew multiveetoc Fields as extension of the Le bracket For vector Fields satisfying certain properties. We sceteh ts uses and we ederive the formulas of Tuleryjew concerning the tie differentials of Forms ending up with the definition lulczyjew sarLed with. Note hat the bracket defined here differs in sign From the usual one. In the second part we show that he Sehouten-Nijenhuis bracke is naura with respect Lo F-dependence of multivector Fields, and Finally, that it is (up o a muItipieatve constant) the unique natural concomitant mapping a p-Field and a q-field to a p+q-1 - field. I want o thank i. Kola, H. Urbantke and K. Wegenkttl For valuable hints. This paper ls in Final form and no version of it will appear elsewhere. 208 MICHOR 1. The Schouten-Nijenhuis bracket for skew multivector fields. 1.1. Let M be a smooth manifold, finite dimensional and paracompact. Then the Lie algebra (M) : r(TH) of vector fields on M is a module over the commutative alge- bra Ca(M) of smooth functions on M, and (M) acts on Ca(M) as Lie algebra of derivations, via Q: (M) + Der(C(H)) ß This is sometimes called a Lie-module. Let us now consider r(ATM), the graded eommutative algebra of (skew) multivector fields on M. It coincides with m(M)r(TM), the space or skew elements in  Ca(M) Y(TM), the C(M)-tensor algebra generated by the C(M)-module F(TM). 1.2. Theorem; The following bracket is well defined on F(ATM) and gives a graded Lie algebra structure with grading ((ATM), [ , ])p -- ?(AP+iTM): y1/%...AYq ] = [XA.. ,^ Xp,  ] Z (-1) p-i+j-1 XiA.../Xi -..AXpA[Xi,Yj]AYiA''' Yj ...AYq ...... j ' q If,U] =-(df)U For f in C(H) and U in (ATH). We also have [U, VAW] = [U,V]AW + (-1) (u-l)v V^[U,W], so that ad: (F(ATM), [ , ]) + Der(F(ATM), A ) is a homomorphism of graded Lie algebras. Proof: Fo vector fields X i and Y. and f in C(M) the following s easily seen to J hold' [XA...AX , YiA...AF.YjA-.q] : F'[Xi^'''AX p' YI^'''AYq ] + ß I p + (_I)P-1 [(dF)(X1A...AXp)AY1A...Ayq , where (df) is the insertion operator, a derivation of degree -1. The formula given in the theorem defines a piori a bilinear mapping AP¾(TM) x AqF(TM) A P+q-IY(TM) If we map it into A p+q-1 F(TM) = F(A P+q-ITM) + ß then by the formula above and by antisymmetry it Factors over 2(APTM) x r(^qTM). So iL is well defined. Then one has to check the graded Jacobi-identit¾. This is an elementary but very tedious calculation. The last property is rather easily qed. checked. Remark: The bracket defined in the theorem is he universal extension of the C(M)-Lie module (M) to a graded version. 1.. Integrability lemma: Let FClM be a 2-dimensional sub vector bundle (a distribution or 2-plane field). Let U in (A2TM) be a (local) "basis" For it ((so X ( F iff X AU : 0). Then F is integrahie if and only x x to,u] = 0. REMARKS ON THE SœHOUTEN-NIJENHUIS BRACKET Proof: Let , Y be local vector Fields spanning F. We may assume that U = XAY. Then [XA'Y,XAY] = 2 X[X,Y]AY by 1.2, which is zero iFF [X,Y] is in F. qed. 1.4. Characterisation of Poisson structures: Let P be in 2(A2TM). Then the skew symmetric product {f,g} ;= on C(M) satisfies the Jacobi identity if and only if [P, P] = 0. Froof: {F,g = = = = [g,[F,?]]'. So F,g,h]) = [[h,[g,P]],[F,P]]. Now a straightforward computation involving gra- ded Jaeobi identity and skew symmetry of [he Schouten-Nijenhus bracket gives: [h,[g,EF,[P,P]]]]= -2({F,{g,h}} + {g,{h,f)} + [h,{F,g}}). Since [h,[g,[F,[P,P]]]] = : (M) x '(ATM) + E(H), given by = det(wi(Xj) ). P For each C(M)-linear (tensorial) mapping F: OP(M) + q(M) we have an adjoint F*:2(AqTM) + (APTH) and conversely. For w in P(H) consider the C(M)-linear mapping (o): k(H)  k+P(M), (w)$ = A . It's adjoirlt willbe denoted by (o) := k(w)*: F(AmTM) + F(A-PTM). Likewise For U in 2(APTM) consider the C(M) - linear mapping (U): (AkTM) + F(Ak+PTM), (U)V = U^V. Its adjoint is denoted by i(U) := p(U): m(M) + om-P(M). Other common notations or these two mappings are Ujw = i(U)w , wU = (w)U. 1.6. Lemma: Let U be in F(APTM). Then i(U): O(M) + (M) is a homogeneous Cø(M)- modu}e homomorphism of degree -p, the graded commutator [i(U) i(V)] = 0 i(U) is a graded derivation of O(M) if and only if the degree of U is 1. For co in l(M) and  in O(M) we have: (U)(mAO) = i((co)U) + (-1)PCOAi(U), hat is [i(U), l(w)] -- i((w)U) in mnd(O(M)). Finaily i(UAV) = i(V)oi(U). Proof: Put U = X1A...AXp, apply i(U) to ml...AW q , evaluate at Z1A...AZ and expand the determinant by the First lne. fhen i(U)(wA½) = i((w)U)kb + (-1)Pco/i(U)½ Follows. lhe rest is easy. qed. 1.7. The Lie differential operator: For U in 2(APTM) we define (U): O(M) + (M) by (U) := [l(U),d] = i(U)d - (-1)Pdi(U). Then (U) is homogeneous of degree 1-p and As called the Lie differential operator along U. It is a derivation if and only if U is a vector Field. Note (hat [©(U), d] = 0 by the graded Jacobi identity. 10 MICHOR 1.8. Lemma: -, For U in l  (Au[M) and V in 1  (VTM) we have 0(UAV) : i(V)O(U) + (-1) u O(V)i(U). 2. O(XiA...^X p) : œ (-1) j-1 i(Xp)...i(Xj+l)O(Xj)i(Xj'_l)-..i(X1 ). 3. O(f) = [i(f),d] : [p(f),d] =- w(df). Proof: 1 follows from 1.6 and the definition. 2 follows by induction on p. 3 is clear. qed. 1.9. Theorem: For U in ?(AUTM) and V in r(AVTM) we have: 1. to(u), i(v)] = (-i) (u-1)(v-l) i([u,v]) .: i(-[v,u]). 2. tO(U), 0(V)] : (-1) (u-1)(v-1) O([U,V]) : O(-[V,U]). Proof: Using the graded Jacobi identity the following Formula is easy to cheek: (3)[e(U), i(V)] : (-1)v-tEl(U), O(V)] :-(-i)(u-l)(v-l)[e(V), i(U)]. Then I can be proved by induction on u+v, using ().and all Lhe results above, in particular 1.8.1. 2 follows also by induction on u+v. qed. 1.10. Let U be in 2(AU[M), V in r(AVTM) and w in ou+v-2(M). [hen we have o(-[v,u]): [e(u),o(v)]: e(u)e(v)- (-z)(u-1)(v-1)e(v)e(u). ©(U)©(V)m : (i(U)d - (-t)Udi( U))(i(V)d - (-1)Vdi(V)) w : i(U)di(V)dm + O. : . Similarly we get O(V)©(U)O)= e(- EV,U ])o) = i(- EV,U ])do) - (-1)u+V-ldi(-[V,U])c0 : ß Putting everything together, we have the following Lemma: For U in F(hUTH), V in F(AVTM) and w in u+v-2(M) we have: = - (-1)(u-1)(v-) . This Formula suffices to compute :[U,V] in coordinates, and i remains valid, if we insert any closed form  in ou+v-l(M) instead of dw. [his formula is the sEarLing poinL of Tulczyjew, who considers [he bracket [U,V] Tul = -EV,U] : (-1)(u-1)(v-1)[U,V]. The coordinate version of it boils down to the definition of Schouten. 2. Naturality of the Schouten-Nijenhuis bracket for skew fields. 2.1. Let F: M + N be a smooth mapping between manifolds. We say that U in I(AUTM) and U'in (AUTN) ae f-relaed or f-dependent, if AuTf. U = U'.F holds. AWTM APTf  AUTN tu f H  N REMARKS ON THE SCHOUTEN-NIJENHUIS BRACKET 2.2. Proposition: 1. U and U' as above are f-related if end only if i(U)o f* : F*o i(U'): O(N) + (M). 2. U and U' are F-related JF and only F 0(U)o F* = F%O(U'): (N) + O(M). ). If U. and U.' are F-related For i = 1,2, then also the Sehouten-Nijenhuis $ 1 brackets [U1, U2] and [U U] are f-related. Proof: 1. Let V in ?(ATH) and w in (M) be such that degU+ degV= degw . Then = <(f'w) x, U x  Vx> )ø1 Txf IJNA Vx> =  x : <F(x  : < U(x)AAT F.V > = <(i(U')w)F(x) , AT F.V > ' x : mr(x)' x x x x > : . : <(F* i(U')m)x, V x x 2. IF U and U' are f-related then e(U) F* = [i(U),d] F* = F* [i(U'),d] by 1. For the converse let w n u-l(N). hen we have ©(U) F* w = i(U) F* d and F* 8(U ) w = F* i(U') dw ß Since the dw , w in u-l(N), generate oU(N) over C(N), we have J(U) F*looU(N) = F* J(U') I Ou(N). This implies, using the proof of i For V = l, that U and U' are F-elated. . This Follows immedialely From 2 and 1.9.2. qed. 2.. For a vector Field X on M let Fl denote the locai Fiow of X. For a multi- vector Feld let he Lie derivative of U along X be given by ©(X)U : --10(F])*U. Lemma: O AP+q-iBn F tA]t x Alt x Alt  ' Alt p+q- t n P ]7n 17 n* q ]7n   © ©  ' .. Since the vertJeal mappings in this diagram (where Alt stands For the appropriate alternator) are all GL2(n,)-equivariant it suffices to determine all GL2(n,)- equivariant mappings in the lower line. The action of  . 8L2(n ) = GL(n ) )L 2 (n,n) on &n is given by the tranformatlon law '  sym 1 "' 81 ...B (2) P : u p 8xgl ''' 8xgp ' P n n. , and the action on © © the space of the first order partial derivatives is qiven by the tranormation law (3) al ...a P = U P 8al Bi...B p (821 jxm 8a2 ap 87al j2p 8x m) + U Bxmax81 a k axB2 ... 8xBp +.-.+ axB1..-BxmBxB p The GL2(n,)-equivariant mppir, gs are in particular GL(n,)-equivarian, sLemming from he embedding GL(n,R) <'* GL. 2(n,) , see (3) with all second partial derivatives O. According o the theory of invarian tensors, as explained in Dieudonn-Carrell, the GL(n,)-equvariant mappings are given by all permutations of the indices, all contractions and tensorizing with the identity. Permutations of indices do not play a role since we take alternator afterwards he identity cannot. appear since the result is purely contravariant. So we may just contract the derivation entry of U into the vector par of U and the same with U and V interchanged. So we have he Following 4-parameter Famlly where U = Uaaa, V = V8%, dU = U m a  dj B(U,V) = BYa , ,j ' We do not indicate alternation in the upper indiees and we wrte , B for any kind of multivector index, so ¾ = (,8) etc. = . U m V B U  V mB U +c +d (a) O  a ,m + b ,m ,n ,n But [e mapping B is also equXvarXanL with respect to the abelJan normal subgroup {Id} x L 2 (n,n) c+ GL2(n, ). The action of an element fid,S) = (Id sym P n P n. on  is the denLLy, on  n  i is gven by (using (3)): (5)  : U  'P ? ., ., + s + ... + . L 2 (n n) So the expression (4) has o be &nvariank unde he on the igh hed side. [his &s equivalent o he Following equaklon: 2]4 MICHOR (6) 0 =  ( U LQS m U mo ... U m S ) V B ... u ) v + b ( IJ La S 1 + + + [n ''' This can be smpliFied after taking alternation: U m S t V 8 U m vn U s S t vn U m S$1 V n8 (7) 0 = a mt + b p S ap + c + d q mn nt mn Now we can compare coefficients and get the Following a = O, p.b + (-1) p-I q.d = O, c = 0 . So there is only one parameter surviving and we get = . U s vmB U n V $ U s vmB un V 8 B Y o ( q + (-1) p p ) = c.( q - p ) ,m ,n ,m ,n' = c.[u,V]. The comparison of coefficients is valid if these tensors ae really independent. If p+q) dm M some expressions are O. Snee they may be viewed as linear mappings in S, they are linearly independen[ as long as they are nonzero. And if  summand in (7) becomes O, the corresponding one in (4) is zero also. qed. REFERENCES Ill M. CAHEN, M. DE WILDE, S. 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