BULLETIN DE L’ACADEMIE
POLONAISE DES SCIENCES
sérieetdesphys.

sciences— Vol,math.

astr.

XXII, No,9, 1974

:MATHEMATICS|(DIFFERENTIAL GEOMETRY)The Graded Lie Algebra of Multivector Fields andthe Generalized Lie Derivative of Forms
by|W. M. TULCZYJEW|

Presented by A. TRAUTMAN

on November 28, 1973

Summary. The definition of a bracket operation for multivector fields and an extension of thenotion of the Lie derivative are given. With the new bracket operation the exterior algebra ofmultivector fields acquires the structure of a graded Lie algebra.

1. Introduction. The object of this paper is to define for multivector fields
a bracket operation which is a natural extension of the Lie bracket of vector fields.
With this bracket operation the exterior algebra of multivector fields acquires thestructure of a graded Lie algebra. The bracket is a special case of a bilinear differential
concomitantforarbitrary

contravariant

tensorfieldsdefinedby

Schouten[1].

A discussion of this special case of Schouten’s concomitant is included in a paper by
Nijenhuis [2]. Most properties of the bracket are derived there by coordinate methodsof classical differential geometry. ..

|Section 2 is a summary of generally known definitions and results stated without
proofs. The main results of the paper are formulated in Sec. 3. Sec. 4 isdevoted to a discussion of the possibility of further generalizations. It is shown thatthe construction of the bracket operation for multivector fields is formally the samea8 that used by Frolicher and Nijenhuis [3] for the bracket of vector-valued forms.

It is also shown that the construction-does not extend to multivector-valued- forms.

The generalized Lie derivative of forms is defined in the last section.

|2. The graded algebras of multivector fields and forms. Let M beCompact C* manifold and F the ring of differentiable functions on M.

a paraConstantfunctions on M form a subring of F identified with the field R of real numbers.
2.1 DEFINITION. A derivation D of a commutative ring A relative to a subring K
15 a map D: A—A

which satisfies(@Dk=0ifkek,(b)¢)

D(a+b)=Da+Db,
D (ab)=(Da) b+aDb.

[937]

938

W. M. Tulezyjew_———

Derivations of F relative to R form the F-module of vector fields on M, denotg
by V.

Let A?= A? V denote the p-fold exterior power of V. Also let 4° denote F apy
let A?=0 for

negativep<0. Elements of A? are called multivector fields of degree p. The nop.

graded F-module

commutative, associative
product, satisfying

A={4?}
gradedis called the exterior algebra of V. It is 4algebra

witha productA,(a)degree (XA Y)=degree X+ degree(b)(©)(d)

XAY=(—1)P2YAX if degree X=p and degree Y=gq,XA(YAZ)=(XATY)AZ,XA +Z)=XAY+XAZ if degree Y=degree Z.

called theexterip

Y,|Let 4, denote the F-module Hom (A?, F) dual to 47. Elements of 4, are called
forms of degree p. The non-negative graded F-module A*={4,}, dual to A, is
isomorphic to the exterior algebra of V*=Hom (V, F). Formal properties of theexterior product A of forms are the same as those of the exterior product of multivector fields.
Duality between 4 and 4* implies the existence of the evaluation map A x A—=F: (X, f)—=»{X| uy, where {X| w=pu(X) if degree X=degree u and {(X| u)=0
if degree X+degree ut.

Thesuchinterior products

thatdegree_| and |(X _|p)=degree

are operations: AX A*—A*:(X, W)—X _|uu—degree

X and{Y |X _|)={XAY|pu) for

each multivector field Y; andAx A*—A4:(X, pu) X|_u such that degree (X|_ p==degree X —degree u and {X|_u|v>=<{X|unav) for each form v.
2.2 PROPOSITION. Relation (XA Y)|_u=X|_WAY+(—12 XA(Y|_ pn) holds if
pis a form of degree 1, X a multivector field of degree p and Y any multivector
field. Also

X (pan) =(X1_ pu) _v+(~=1? gA(X _Iv)
holds if degree X=p,
2.3 PROPOSITION.

degree p=1
Thereand v is any form.

is a unique differential d in A* satisfying {X|df>=Xf

Jor each vector field Xe V and each function fe F, and(a)degree du=degree u+1,(b)ddu=0,(©

d(utv)=du+dv@d@uav)=dunv+(=1)°
undv if degree p=p.

if degree u=degree v,

3. The graded Lie algebra of multivector fields. Let X

andY be multivector

fields of degree p and g¢, respectively.
3.1. PROPOSITION. There is a unique multivector field [X, Y] of degree p+q—1
such that(IX, Yl dy =(X |d(¥_|du)y+(—Dpe++a(Y|d(X_| dw).

The Graded Lie Algebra of Multivector Fields_YY

proof. To define a multivector field it is sufficient to give its evaluation withforms of the type fdu, where f is a function. Let f; f' be functions andforms such that fdu=f"dy'. ThenAXAdp) +(=1)perrr ea fKY dXx, u”du)=(X|d(Y _J fay + (=D era Y dX _| fawy —(— DP KYA Yd(f du)>=(X|d(Y _1f'du’)y +(=1p*r
Pra Yd X_|f'dp’))— (= 1)" KX AY |d(f dp’)=f (X|d(Y_{du' y+ (—1)Perea
f° Y dX _|du'))Thus a multivector field [X, Y'] of degree p+¢g—1

is correctly defined by

IX, Y1| fdpy=f<X|d(¥ _|dpy+(—1)rrr+af{Y dX _| dp).
Uniqueness||of [X, Y] is obvious.

3.2 PROPOSITION. The multivector field |X, Y] satisfies|

1X, YI df=I[X, Y|_df]-(-1D?[X|_df, Y]
for each function f.
Proof.

CX, YI df |duwy=<IX, Y]ld(f dw)=X |d(Y _{(dfrdw))+(—Drere+al(Y|d(X _|(dfndw)>=(Xd((Y df) Jdu)>—(—DHX|_df|[d(¥Y _ldw)y=(—=1per?*a Y [dX _Ldf) dup — (1YL_df |d(X
=X, YL_df
foreach]ldwy—(— DX,functionfandeach_{dw)Y_df]| dpformwu.

Hence[X, Y]| _df=I[X, Y|_df]—-(-D)XLd, Yl
3.3. PROPOSITION.

The multivector field [X, Y] satisfies[X, Y] _{p=X_1d(Y{p+ (-1re"?*1Y{dX _|u)— (=P? (XA Y) pu) —(—=1PH (XA TY) _ldu
for each form u.

|Proof. The proposition is true for degree u=0, and if the proposition is true fordegree pu=p then

1X, Y]_1@fA =X,YIL_df) _{u+(=DP*1 dfa(lX, Y] _1 p)=X, Y_df] _lp—(—DXLdf, Y]_{p—(—1y* df A(X, Y] _J 1)=X _1d((YL_df)_lw)— (=D(Y df) _dX_|wy— (12 d((XA(Y_df)) lu)+(=1yperr ta (XA (YL df) _ldu—(=DUX_df)_Jd(¥_|@)+(=Dpretr+a

Y_1d((X df) 1g) — (= 12 d(((XL_df)A Y) du) +(— 1p ((XL_df)A Y) dpe~(=DPredfa(X_1d(Y _1p)—(=D dfa(Y _1d(X_i p))+(—Dparagfad(XAY)|p) +(—1P4 2 dfa (XA Y) _] dp)=X_1d(Y _1(dfAw)—(=1petrrey(=p?

d((XAY) _{@d Aw) —(—=1)P* (XAT) _Jddfrp.

Hence proof by induction.
3.4 THEOREM.

satisfies [X,[dX _{(dfrpw)

Thereis|aunique

operation

AX

A—A:(X,Y)—[X,Y]

f1=Xf for each vector field Xe V and each function fe F, andwhick“04()W. M. Tulczyjew

-_—

|(a)

degree[X, Y]=degreeX {degree Y—1,b)[X, Y]+(—DE-D@=-D[Y, X1=0 if degree X=p and degree Y=gq,c)(—~D)-1DE-1) [X, [Y, Z| +(— 1)(a-1 2-1) [Y, [Z, X71]|+(=1De-D@=D[Z [X, Y1]|=0 if degree x=p, degree Y=q anddeg,Z=r,(d)[X, Y+Z]=[X, Y]1+I|X, Z] if degree Y=degree Z,€)[X, YAZ]=YALX, Z]+(—1y®-D[X, YIAZ if degree X=p and degre,=r.

Proof. It follows

fromelementary computation

that(X, Y)—[X, Y], where

TX, Y]is defined in Proposition 3.1, has all the required properties. TO prove uniqye.
ness note that [X,f]=Xf=X|
df for degree X=1, and [Y,f]=Y[_df implyIXAY f=INY=XAY fl=XLd)ANY—=XA(Y|_df)=(XAY)_df by theuse of (b) and (e). Hence [X, f]=X|_df by induction on degree X. This result, together

with (b) and (0), implies [X, Y]L_df=[[X; Y], f1=IX, [¥, f1]—(~1)*[[X, f1, Y]~=[X, Y| _df]-(—D?[X|_df, Y]. An inductive argument similar to that used toprove Proposition 3.3 leads to {[X, YI =<X|d(Y_| p)> +(—1)P4+?+*
LY |d(X_| pn)—(—=1)P2*2{XAY| duy. Hence [X, Y] is unique.
3.5 DermNiTION. The graded algebra of multivector fields with the operation{X, Y)—[X, Y] is called the graded Lie algebra of multivector fields.
4. Multivector forms. Let4]

denotetheF-moduleHom (47, 49).

Elementsof 4% are called multivector forms of degree (p, q). The module AY is identified withA% and 4)=4,. Elements of 4; are the vector forms discussed by Frolicher and‘Nijenhuis [3].
|For each multivector form XK of degree (p, q) and each form x of degree r let
K _{u be a form of degree p+r—g such that K | u=0 for r—g<0 and for r—g=0
K_1p(XAAXarppg)=Sol==)

2,TTI

SEN au (K (Xe, A AXa) A Xo),|A ver A XD)’where X,, ..., X. reg are vector fields and « ranges over the.symmetric group Sp+r—q-Also let K|ux be a multivector form of degree (p, g—r) such that K|:. u)(X)='=K(X)|_ pu for each multivector field X of degree p.

The above definitions extend in a natural way the applicability of interior
products.

Relation

K_1(pav)=E|_p) _v+(=1F"7 pua(K_]v)
for degreeK=(p, q) and degree p=1, is an immediate consequence of the definitions.

To each multivector form K of degree (p, q) there correspond operators ig: A™*——>A*: u—»K_Ju,and di=igd—(—=1)?P"9dig: A* > A*: pK _ dy —(—1)?"12
A(K _|p). The

operator=degree u+p—gq,ix is anoperatorof degree p—gand dg is an operator of degree p—qg-+1

sincedegree (ix 1)=

since degree (dy p)=EThe=u+p—q+1.

GradedLie Algebra of Multivector Fields

041

For two operators a and b of degrees r and s respectively let# p] denote the commutator ab—(—1)" ba. Then dy=[ik, d]. Also [ig, L]=ix| 4fF 1 u is a form of degree 1 and (dk, ir]=lix, iar] =ix, ar if f is a function.

} 4.1 PROPOSITION. Let Xe A? ond Y € A* be multivector fields. There is a unique
Sultivector field |X, Ye A?* 41 such that¥(dx, Iyl=irx,v .

ls;[dy,{:YJdy] =dx,

The proof of this proposition follows directly from Propositions 3.1 and 3.3.

fhe following proposition is due to Frolicher and Nijenhuis.
4.2form

PROPOSITION. Let Ke A: and L € A
thatsuch[K, L] € As,

be vector forms. There is a unique vector[dk, dr]=dg, Ly

Proposition 4.2 implies the existence of a bracket operation for vector forms.
ft has been shown by Frolicher and Nijenhuis that this operation makes the moduleof vector forms a graded Lie algebra over R. The intersection of this algebra withthe graded Lie algebra of multivector fields is the Lie algebra of vector fields.
LetRed),Sed;andKe 4)

bemultivector

forms

such

that

n—m==p+q—r—s-+1. Then | [d, irl; ig =lix, ars i,]=0 for any functions f and g.
However [[[dr, dsl, ir], i) =lir| as» is| agl+[ir|ag» Is) ar] is in general different
from zero. Hence re’ations of the type[dR,ds]=dx

do not exist in general and Propositions 4.1
case of arbitrary multivector forms.

and4.2

donot

generalizetothe

5. The generalized Lie derivative
5.1 DEFINITION.

Let’ X be a multivector field of degree p and ux a form. Then

Eyp=X_|du—(=1)?d(X_|p)is called the generalized Lie derivative of un withrespect to X.

5.2 PROPOSITION. The following relations hold for all multivector fields X and 7Y,
all forms pu and v and each function f:(a)degree £5 yu=degree—degree X+1(b)Lxutv)=Ly u+Lxv, if degree u=degree v©
dD

Lx(fw=fLxu+XL_d) _lu
Lydu=dLyp|€)Lx(¥Y | w=[X,Y]_|u+(=D1"=DY| Ly u, if degree

X=p and degree

Y=q6)

Lxiyu=Lypu+Ly un, if degree X=degree(8)

Lxpyu=flypu—(—DPdf AX _|p), if degree X=p)

Lyru=Y_Lyand degree0)6

p+ (122 X _| Ly u+(—DP [X, Y]_|4, if degree X=pY=gq

Lrxy pZLy Ly
Y=gq.

Yu— (=)Ly Oyu, if degree X=pand degree

942

W. M. Tulczyjew

Eee

The proof of this proposition follows easily from Definition 5.1 and the resyjt,of the preceding sections.
In agreement with a theorem by Frolicher and Nijenhuis [3], the generalized I,
derivative with respect to a .multivector field X is not a derivation of the gradeg
algebra of forms unless degree X=1.
DEPARTMENT(CANADA)OF MATHEMATICS,

UNIVERSITY OF CALGARY,CALGARY,ALBERTA

T2N[nyREFRENCES[11 J. A. Schouten, Uber Differentialkomitanten zweier kontravarianter Gréssen, Proc, Kon,
Ned. Akad. Wet. Amsterdam, 43 (1940), 449—452.
[2] A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensoy
fields, ibid., A58(1955),

390—403.

[3] A. Frolicher, A. Nijenhuis,
338—359.
.

Ba M. Tynpunen,
dopm Jin

Copepacanne.

Theory of vector-valued differential forms, ibid., 59 (1956),

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