summary:A brief exposition of Lie algebroids, followed by a discussion of vector forms and their brackets in this context - and a formula for these brackets in “deformed” Lie algebroids

Nijenhuis, Albert

Czech digital mathematics library

Vector form brackets in Lie algebroids

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumAlbert NijenhuisVector form brackets in Lie algebroidsArchivum Mathematicum, Vol. 32 (1996), No. 4, 317--323Persistent URL: http://dml.cz/dmlcz/107584Terms of use:© Masaryk University, 1996Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 32 (1996), 317 { 323VECTOR FORM BRACKETS IN LIE ALGEBROIDSAlbert NijenhuisTo Ivan Kolar, on the occasion of his 60th birthday.Abstract. A brief exposition of Lie algebroids, followed by a discussion of vec-tor forms and their brackets in this context - and a formula for these brackets in\deformed" Lie algebroids.1. IntroductionThe sections in the tangent bundle of a (smooth) manifold can be dened as thederivations on the ring of (smooth) functions on the manifold, and thus are seento form a Lie algebra. A Lie algebroid [3] is a direct generalization: it consists ofa triple, say (A; [; ]A; a), where A is a vector bundle over a base manifold, say B,and [; ]A is a Lie algebra product on  (A), the (smooth) section in A. Further,a : A ! TB, the anchor, is a bundle map to the tangent bundle of B, whichestablishes a homomorphism between the Lie algebras  (A) and  (TB) :a([u; v]A) = [au; av]; u; v 2  (A);and satises the product rule[u; fv]A = f [u; v]A + (a(u)  f)vwhere f 2 F (the ring of functions on B).Given a Lie algebroid A and its dual bundle A, all tensor bundles can beconstructed as a straightforward generalization of the structures based on TB andT B. This note shows some less obvious constructions.Part of the material in this note was included in a talk at the Pacic NorthwestGeometry Seminar, Corvallis, OR, U.S.A., on November 9, 1996.2. A handy notation. [4] Denote by Sm;n the set of (m;n)-shues, that is,the permutations  = (i1; : : : ; im+n) of m + n symbols such that i1 <    < imand im+1 <    < im+n. (Any other selection from the cosets Sm+n=Sm  Sn1991 Mathematics Subject Classication : 17B66, 17B70.Key words and phrases: vector valued form, Lie algebroid.318 ALBERT NIJENHUISwill serve equally well.) Let  = (1; 2), where 1 = (i1; : : : ; im) and 2 =(im+1; : : : ; im+n). Let V be a vector space, and ;  multilinear maps of V to V ,say  2 Hom(Va V; V )  2 Hom(Vb V; V ), then dene  Z  2 Hom(Va+b 1; V )by(2:1) ( Z )(v1; : : : ; va+b 1) = X(1;2)2Sb;a 1 sgn(1; 2)((v1 ); v2):It is well known that, for  2 Hom(Vc V; V ) we have(2:2) ( Z ) Z     Z ( Z ) = ( 1)(b 1)(c 1)(( Z ) Z     Z ( Z ));while, if  2 Hom(V; V ) then both sides of (2.2) vanish.Based on Z we dene a commutator bracket;(2:3) [; ]Z =  Z    ( 1)(a 1)(b 1) Z :It is easy to show that [ ; ]Z denes a graded Lie algebra structure with reducedgrading:(2:4) Xcycl( 1)(c 1)(a 1)[[; [; ]Z]Z = 0:Our application is to the case when V =  (A)  F ; i.e., when the entries in; , etc. are sections in a Lie algebroid A or functions on the base space, orformal sums of the two. In the latter case, the linearity permits a decompositionof (v1; : : : ; va) into pure terms, in which each entry is either a section in A or afunction.Each A-(dierential) form or A-vector form ! is identied with an element ofHom(VV; V ), also denoted !, which takes the same values when evaluated on A-sections, and vanishes when any one entry is a function. As a result, all expressionsof the form L Z ! vanish when L is an A-form or a A-vector form and ! an A-form.The structure of Lie algebroid is incorporated in an element  2 Hom(V2 V; V ),the multiplication map, as follows. (It is not a vector form!)(u; v) = [u; v]A for u; v 2  (A);(u; f) =  (f; u) = a(u)  f for u 2  (A); f 2 F ;(2:5)where a is the anchor. Finally, (f; g) = 0 for f; g 2 F .Lemma 1. If  is the multiplication map of a Lie algebroid, then [; ]Z = 0and (u; fv) = f(u; v) + (u; f)v. Conversely, if [ ; ]A is any alternating producton  (A), a : A ! TB any bundle map,  dened by (3.5) and [; ]Z = 0, then(A; [ ; ]A; a) is a Lie algebroid with multiplication .We may write (A; ) instead of (A; [ ; ]A; a).VECTOR FORM BRACKETS IN LIE ALGEBROIDS 319Proof. There are three cases to be considered for the rst formula, dependingon how many of the variables in [; ]Z(:::) are A-sections and how many arefunctions. Note that [; ]Z = 2 Z .( Z )(u; v; w) =Xcycl ((u; v); w) =Xcycl[[u; v]A; w]A = 0;( Z )(u; v; f) = ((u; v); f) + ((v; f); u) + ((f; u); v)= ([u; v]A; f) + (a(v)  f; u) + ( a(u)  f; v)= a([u; v]A)  f   a(u)  a(v)  f + a(v)  a(u)  f= a([u; v]A)  f   [a(u); a(v)]  f = 0;Finally,  Z (:::) is easily seen to vanish when two or more of the variables arefunctions.The second formula is just a re-write of the product rule in a Lie algebroid. Consider an A-form ! 2  (VpA), then for its A-exterior derivative we nd(dA!)(u0; : : : ; up) = pXi=0( 1)ia(ui)  !(u0; : : : ; î; : : : ; up)+Xi<j ( 1)i+j!([ui; uj]A; u0; : : : ; î; : : : ; hatj; : : : ; up)= pXi=0( 1)i(ui; !(u0; : : : ; î; : : : ; up)+Xi<j ( 1)i+j!((ui; uj); u0; : : : ; î; : : : ; ĵ; : : : ; up)= ( 1)p+1( Z !)(u0; : : : ; up)  (! Z )(u0; : : : ; up)=  [!; ]Z(u0; : : : ; up):We have just shownLemma 2. dA! =  [!; ]Z.The Jacobi indentity (2.4) easily implies that (dA)2 = 0.3. Derivations on dierential forms. The classical theory of derivationson the graded ring of dierential forms (i.e., the case of the standard tangent Liealgebroid), see [1], states that every derivation is uniquely a sum of one of type iand one of type d. A derivation is of type i if it vanishes on functions, and is ofthe form ! 7! iL! = ! Z L, where L is a vector form, L 2  (TB VT B), andit is of type d if it commutes with the exterior derivative d; in this case it is ofthe form ! 7! dL! = [iL; d] = (iLd+ ( 1)qdiL)!, where L 2  (TB Vq T B).320 ALBERT NIJENHUISIn a general Lie algebroid A the A-forms also admit derivations of types iA (thesame as i above), and dA (in obvious generalization of the above), but these neednot span all derivations.For example, consider the Lie algebroid A with bundle space TB but trivialbracket and anchor. Then the classical d is a derivation, but is not a sum ofderivations of types iA and dA , because the rst vanish on functions, and thesecond are trivial (zero).The commutator relations for derivations of types i and d for the standardtangent Lie algebroid are (see [1])(3:1) [iL; iM ] = i[M;L]Z ;(3:2) [iL; dM ] = dM Z L + ( 1)mi[L;M ];(3:3) [dL; dM] = d[L;M ]:We generalize these formulas to Lie algebroids, as follows. (3.1) is \the same" (seeabove); (3.2) is seen as a denition of [L;M ], and requires a proof that [il; iM ] dM Z L is indeed of type iA . Then (3.3) easily follows from (3.2) as a consequenceof the Jacobi identity for derivations.In what follows, the A for Lie algebroids has been suppressed in the formulas.{ Dene (L;M ) by(3:4) (L;M )(u1; : : : ; uq+m) = X(1;2)2Sq;m sgn(1; 2)(L(u1 );M (u2)):Note that (L;M ), though not an A-vector form, vanishes if any one entry is afunction; in particular, (L;M ) Z ! = 0 for A-forms !.Lemma 3. We have the following(3:5) (L;M ) = ( 1)q(m 1)(( Z L) Z M    Z (L Z M ));(3:6) dL! = ( 1)q+1! Z [L; ]Z + ( 1)pq+q( Z L) Z !;(3:7) [M Z L; ]Z = M Z [L; ]Z   ( 1)q [M;]Z Z L + ( 1)q+1(L;M );(3:8) [! Z L; ]Z = ! Z [L; ]Z   ( 1)q [!; ]Z Z L + ( 1)qp+1( Z L) Z !:Proof. For (3.5) see page 104 of [4]. (Note that the proof of (2.2) in [1] containstwo canceling errors, and would give an incorrect sign in (3.5).) The other formulasrequire simple calculations using the denitions and (2.2). VECTOR FORM BRACKETS IN LIE ALGEBROIDS 321Theorem 1. The derivations of types i and d in a Lie algebroid satisfy (3.1-3).The bracket [L;M ] is given by(3:9) [L;M ] = (L;M ) + ( 1)m(q 1)L Z [M;]Z + ( 1)q+1M Z [L; ]Z:Proof. In the following calculations, the Z on [ ; ]Z will be suppressed. UseLemma 3.( 1)m[iL; dM ]!   ( 1)mdL Z M!= ( 1)m(dM!) Z L   ( 1)qmdM (! Z L)   ( 1)mdM Z L!= ( 1)m(( 1)m+1! Z [M;] + ( 1)m(p 1)( Z M ) Z !) Z L)  ( 1)qm(( 1)m+1(! Z L) Z [M;] + ( 1)(p+q)m( Z M ) Z (! Z L))  ( 1)m(( 1)q+m! Z [M Z L; ] + ( 1)(p 1)(q+m 1)( Z (M Z L)) Z !):We rst combine the rst terms in each line.  (! Z [M;]) Z L  ( 1)qm+p+q (! Z L) Z [M;]  ( 1)q! Z [M Z L; ]=  (! Z [M;]) Z L+ (( 1)(q 1)m(! Z (L Z [M;] + (! Z [M;]) Z L   ! Z ([M;] Z L))  ( 1)q! Z (M Z [L; ]  ( 1)q [M;] Z L+ ( 1)q+1(L;M ))= ! Z (( 1)(q 1)mL Z [M;] + ( 1)q+1M Z [L; ] + (L;M )):This proves (3.9), after we show the vanishing of the remaining terms:( 1)pm((( Z M ) Z !) Z L   ( Z M ) Z (! Z L)+ ( 1)(p 1)(q 1)( Z (M Z L)) Z !)= ( 1)pm( ( Z M ) Z (L Z !) + ( 1)(p 1)(q 1)((( Z M ) Z L) Z !  ( Z (M Z L)) Z !))) = 0 + ( 1)pm+(p+m 1)(q 1)(M;L) Z ! = 0: 4. The \deformed" Lie algebroid. [2] The operator iL, so far dened asacting on  (VA), is extended to act on Hom(VV; V ) (V as dened in section 2)in the case when L is an A-vector 1-form. In that case we prefer the notation h,k, etc. over L, etc., and set(4:1) ih =  Z h  h Z :According to (2.2) and the line following, ih satises a product rule with respectto Z : ih( Z ) = (ih) Z  +  Z ih:322 ALBERT NIJENHUISIf A is a Lie algebroid with multiplicatiom , and h an A-vector 1-form, a new,deformed multiplication h is given byh(u; v) = (hu; v) + (u; hv)  h(u; v);i.e., by h = ih. (This implies (see (2.5)) that a deformed anchor map ah is givenby ah(u) = a(hu).) In general, h does not dene a Lie algebroid structure on thebundle space A.Lemma 4. If [h; h] = 0 then h denes a Lie algebroid structure (A; h), and his a homomorphism to (A; ).Proof. The product rule h(u; fv) =    follows by a simple calculation, usingjust the F -linearity of h and the fact that h acts trivially on functions.Again, we suppress the Z on [; ]Z below. The formula (3.9) with L = M = h,and the observation that (h; h)(u; v) = 2(hu; hv), yields(hu; hv) =  h Z [h; ](u; v) = hh(u; v);so h gives the homomorphism of h to .Formulas (3.7) and (3.9), with L = M = h, using [h; ] =  ih =  h give riseto [h2; ] =  h Z h   h Z h + (h; h); 0 = (h; h)  2h Z h:Elimination of (h; h) by subtraction yields [h2; ] =  [h; h]. Bracketing with ,combined with the product rule for ih yields[; [h2; ]] =  [; ihh] =  ih[; h] + [ih; h] = ih[; [h; ]]+ [h; h]:Now, [; [; k]] =  12 [k; [; ]] = 0 for any vector 1-form k (Jacobi identity), so wend [h; h] = 0. Hence, h satises the Jacobi identity. Given a Lie algebroid (A; ) and a deformed Lie algebroid (A; h), we dene[L;M ]h, the A-vector form bracket with respect to h by replacing in (3.9) all  byh. (Formulas (3.1-3) will then be valid, after the same substitution, see Lemma2.)Theorem 2. Let [L;M ]h denote the vector form bracket in a deformed Lie alge-broid (A; h), then(4:2) [L;M ]h = ih[L;M ]  [ihL;M ]  [L; ihM ]:Proof. Let P (X;Y; : : : ) be a polynomial, linear in each of X;Y; : : : , withX;Y; : : : 2 Hom(VV; V ), and with the (non-commutative, non-assocative) prod-uct Z , with respect to which ih is a derivation, thenihP (X;Y; : : :) = P (ihX;Y; : : : ) + P (X; ihY; : : : ) + : : : :Apply this to P (L;M; ) = [L;M ] (see (3.9)), and observe that[L;M ]h = P (L;M; ih) :The result follows immediately. VECTOR FORM BRACKETS IN LIE ALGEBROIDS 323ReferencesThe following articles contain suitable introductions to the relevant topics, as well as referencesto further information.[1] Frolicher, A., Nijenhuis, A., Theory of vector-valued dierential forms, Part I., Kon. Nederl.Akad. Wetensch. Proc. A 59 (= Indag. Math. 18), 338-359 (1956).[2] Kosmann-Schwarzbach,Y., Magri, F., Poisson-Nijenhuis structures, l Ann. Inst. Henri Poincare53, 35-81 (1990).[3] Mackenzie, K., Ping Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73, 415-452(1994).[4] Nijenhuis, A., Richardson, R.W., Deformations of Lie algebra structures, J. Math. Mech. 17,89-105 (1967).Department of MathematicsUniversity of WashingtonSeattle WA 98195, U.S.A.E-mail : nijenhuis@math.washington.edu

English

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.2449

Vector form brackets in Lie algebroids

Abstract

Similar works

Full text

Available Versions

Czech digital mathematics library

Institute of Mathematics AS CR, v. v. i.