doi: 10.17951/a.2022.76.2.1-13
ANNALES
UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA
L U B L I N – P O L O N I A
VOL. LXXVI, NO. 2, 2022 SECTIO A 1–13
MIROSLAV DOUPOVEC, JAN KUREK
and WŁODZIMIERZ M. MIKULSKI
Na u al a ino s and o sion o connec ions
on Weil like unc o s on double ec o bundles
Abs ac . We desc ibe comple ely all na u al a ino s on p oduc p ese ing
gauge bundle unc o s on double ec o bundles. Nex , we s udy o sion o
double-linea connec ions.
1. In oduc ion. We assume ha any mani old conside ed in he pape
is Hausdo , second coun able, ini e dimensional, wi hou bounda y and
smoo h (i.e. o class C∞). All maps be ween mani olds a e assumed o be
smoo h (o class C∞).
The concep o double ec o bundles was in oduced in [12] and u he
s udied in [1, 7, 9], e c. The amewo k o double ec o bundles is con-
enien o many cons uc ions like linea o ms, linea Poisson s uc u es,
linea connec ions, e c. The equi alen concep o double ec o bundles
can be ound in [10]. We ci e i in Sec ion 2 o he p esen no e.
The gene al concep o gauge bundle unc o s can be ound in [5]. The
concep o p oduc p ese ing gauge bundle unc o s (ppgb- unc o s) on he
ca ego y o double ec o bundles can be ound in [10], oo. We ci e i in
Sec ion 3.
2010 Ma hema ics Subjec Classi ica ion. 58A20, 53A55.
Key wo ds and ph ases. Double ec o bundle, p oduc p ese ing gauge bundle unc-
o , na u al a ino , o sion o double-linea connec ion.
The i s au ho was suppo ed by he p ojec no. FSI-S-23-8161.
2 M. Doupo ec, J. Ku ek and W. M. Mikulski
In [10], i is p o ed ha he ppgb- unc o s Fon he ca ego y o double
ec o bundles a e in bijec ion wi h he AF-bilinea maps
⋄F:UF×VF→WF,
whe e AFa e Weil algeb as and UFand VFand WFa e ini e dimensional
(o e R)AF-modules. Mo eo e , gi en a ppgb- unc o Fon double ec o
bundles and a poin c∈AF, in [10] an a ino (i.e. enso ield o ype (1,1))
a (c):TFK →TFK
on FK is cons uc ed o any double ec o bundle K.
The main esul o he p esen no e is he ollowing one ex ending [6].
Theo em 1.1. Le Fbe a ppgb- unc o on double ec o bundles. The
canonical a ino s a (c) o all c∈AFa e all na u al a ino s on F K.
Canonical (la e called na u al) a ino s on some o he bundle unc o s
a e desc ibed in [2, 3, 6, 8], e c. We poin ou ha na u al a ino s play an
impo an ole in di e en ial geome y. Fo example, na u al a ino s a e
use ul in he p olonga ion o ec o ields o p oduc p ese ing bundles, see
e.g. [5]. Na u al a ino s can be also used o de ine he gene al concep o a
o sion o a connec ion, [6].
2. On double ec o bundles.
De ini ion 2.1 ([10]). An almos double ec o bundle is a sys em K=
(K , Kl, E , El)o ec o bundles K = (K, τ , E ),Kl= (K, τl, El),E =
(E , τl, M)and El= (El, τ , M)such ha τl◦τ =τ ◦τl.
I K′= (K′
, K′
l, E′
, E′
l)is ano he almos double ec o bundle, an almos
double ec o bundle map K→K′is a map :K→K′such ha he e
a e maps :E →E′
, l:El→E′
land :M→M′such ha ( , ):
K →K′
,( , l):Kl→K′
l,( , ):E →E′
and ( l, ):El→E′
la e
ec o bundle maps.
We call M he basis o Kand :M→M′ he base map o .
We ha e he i ial almos double ec o bundle
Rm1,m2,n1,n2= (K , Kl, E , El),
whe e Kl= (Rm1×Rm2×Rn1×Rn2, τl,Rm1×Rn1),K = (Rm1×
Rm2×Rn1×Rn2, τ ,Rm1×Rm2),E = (Rm1×Rm2, τl,Rm1)and El=
(Rm1×Rn1, τ ,Rm1), and whe e τ ,τl,τ ,τla e he ob ious p ojec ions.
De ini ion 2.2 ([10]). A double ec o bundle is a locally i ial almos dou-
ble ec o bundle K, ha is, he e a e non-nega i e in ege s m1, m2, n1, n2
such ha o any x∈M he e is an open neighbo hood Ω⊂Mo xsuch
ha K|Ω=Rm1,m2,n1,n2modulo an almos double ec o bundle isomo -
phism.
Na u al a ino s and o sion o connec ions... 3
A e y impo an example o a double ec o bundle is he angen bun-
dle TE = (TE, T E, E, T M)o a ec o bundle E= (E, π, M), whe e
τ :=pT E :TE →E,τl:=Tπ :TE →TM,τ :=pT M :T M →M,
τl:=π:E→M. Ano he such example is he co angen bundle T∗E=
(T∗E, T ∗E, E, E∗)o a ec o bundle E, see [9]. Double ec o bundle s uc-
u es on TTM and TT∗Mmake possible he Lag angian o mula ion o he
dynamics in classical mechanics, see [13].
All double ec o bundles and almos double ec o bundle maps be ween
hem o m a ca ego y which we deno e by DVB. (In [10], he no ion o 2-VB
ins ead o DVB is used.) Any DVB-map :Rm1,m2,n1,n2→Rm′
1,m′
2,n′
1,n′
2is
o he o m
(1)
(x, u, , w)
=a(x),X
j
aj(x)uj,X
k
bk(x) k,X
j,k
cjk(x)uj k+X
l
dl(x)wl
o some maps a:Rm1→Rm′
1, aj:Rm1→Rm′
2,bk:Rm1→Rn′
1,cjk :
Rm1→Rn′
2,dl:Rm1→Rn′
2,j= 1, . . . , m2,k= 1, . . . , n1,l= 1, . . . , n2,
whe e x∈Rm1,u= (u1, . . . , um2)∈Rm2, = ( 1, . . . , n1)∈Rn1,w=
(w1, . . . , wn2)∈Rn2.
By he local desc ip ion (p esen ed in [7]) o double ec o bundles in he
sense o [9], he double ec o bundles in ou sense a e equi alen o he one
o [9].
3. On ppgb- unc o s on double ec o bundles. Le FM deno e
he ca ego y o ib ed mani olds and ib ed maps. The gene al concep o
(gauge) bundle unc o s can be ound in he book [5]. We need he ollowing
pa icula case o i .
De ini ion 3.1 ([10]). A gauge bundle unc o on DVB is a co a ian unc o
F:DVB → FM sending any double ec o bundle Kwi h he basis M
in o ib ed mani old pK:FK →Mo e Mand any double ec o bundle
map :K→K′wi h he base map :M→M′in o ib ed map F :
FK →FK′o e :M→M′and sa is ying he ollowing condi ions:
(i) (Localiza ion condi ion) Fo e e y double ec o bundle Kwi h he
basis Mand any open subse U⊂M, he inclusion map iK|U:K|U→K
induces di eomo phism FiK|U:F(K|U)→p−1
K(U), and
(ii) (Regula i y condi ion)F ans o ms smoo hly pa ame ized amilies
o DVB-maps in o smoo hly pa ame ized amilies o FM-maps.
A gauge bundle unc o Fon DVB is called a Weil like unc o (o ppgb-
unc o ) i F(K1×K2) = F(K1)×F(K2) o any DVB-objec s K1and K2.
An example o a ppgb- unc o on DVB is he angen unc o Tsending
any DVB-objec Kwi h basis Min o he angen bundle TK ( ea ed as
4 M. Doupo ec, J. Ku ek and W. M. Mikulski
he ib ed mani old o e M) and any DVB-map :K→K′in o T :
TK →TK′.
In [10], i is p o ed ha he ppgb- unc o s Fon he ca ego y o double
ec o bundles a e in bijec ion wi h he AF-bilinea maps
⋄F:UF×VF→WF,
whe e AFa e Weil algeb as and UFand VFand WFa e ini e dimensional
(o e R)AF-modules. We ha e
FRm1,m2,n1,n2= (AF)m1×(UF)m2×(VF)n1×(WF)n2,
and i :Rm1,m2,n1,n2→Rm′
1,m′
2,n′
1,n′
2is o he o m (1), hen
F :(AF)m1×(UF)m2×(VF)n1×(WF)n2→
→(AF)m′
1×(UF)m′
2×(VF)n′
1×(WF)n′
2
is o he simila o m
(2)
F (x, u, , w)
=aAF(x),X
j
aAF
j(x)uj,X
k
bAF
k(x) k
,X
j,k
cAF
jk(x)uj⋄F k+X
l
dAF
l(x)wl
,
x∈(AF)m1,u= (u1, . . . , um2)∈(UF)m2, = ( 1, . . . , n1)∈(VF)n1,
w= (w1, . . . , wn2)∈(WF)n2, whe e aAF=TAFa:TAFRm1= (AF)m1→
TAFRm′
1= (AF)m′
1,aAF
j=TAFaj:(AF)m1→(AF)m′
2,bAF
k=TAFbk:
(AF)m1→(AF)n′
1,cAF
ij =TAFcjk :(AF)m1→(AF)n′
2,dAF
l=TAFdl:
(AF)m1→(AF)n′
2a e he alues o a, aj, bk, cjk, dlby he (usual) Weil
unc o TAFo Weil algeb a AF. So, Fhas alues in DVB, i.e. F:DVB →
DVB.
4. Tangen bundle o a ppgb- unc o on double ec o bundles.
I is obse ed ha any ppgb- unc o Fon DVB has alues in DVB. So,
we can compose ppgb- unc o s F1and Fand ob ain ppgb- unc o F1Fon
DVB. In pa icula , he composi ion T F o he angen unc o Tand a
ppgb- unc o Fon DVB is again a ppgb- unc o on DVB. We ha e
AT F =AF×AF, UT F =UF×UF, V T F =VF×VF, WT F =WF×WF
and he algeb a mul iplica ion (o AT F ) and he module mul iplica ions (o
UT F and VT F and WT F ) and he AT F -bilinea map ⋄T F sa is y
(3)
(a1, a2)(b1, b2)=(a1b1, a2b1+a1b2),
(a1, a2)(u1, u2)=(a1u1, a2u1+a1u2),
(a1, a2)( 1, 2)=(a1 1, a2 1+a1 2),
(a1, a2)(w1, w2)=(a1w1, a2w1+a1w2),
(u1, u2)⋄T F ( 1, 2)=(u1⋄F 1, u2⋄F 1+u1⋄F 2)
Na u al a ino s and o sion o connec ions... 5
o any a1, a2, b1, b2∈AF,u1, u2∈UF, 1, 2∈VF,w1, w2∈WF.
In [10], o any c∈AF, i is cons uc ed a DVB-in a ian a ino
a (c):TFK →TFK
on FK o any DVB-objec K. I K=Rm1,m2,n1,n2, hen
a (c)((a1, u1, 1, w1),(a2, u2, 2, w2)) = ((a1, u1, 1, w1), c(a2, u2, 2, w2))
o any a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1,w1, w2∈
(WF)n2, whe e he s anda d iden i ica ion TX =X×X o ec o spaces
Xis applied. The in a iance means ha i :K→K1is a DVB-map,
hen TF ◦a (c) = a (c)◦TF .
5. The na u al a ino s on ppgb- unc o s on double ec o bun-
dles. Le DVBm1,m2,n1,n2be he ca ego y o all DVB-objec s Kbeing lo-
cally isomo phic wi h Rm1,m2,n1,n2wi h local DVB-isomo phisms be ween
hem as mo phisms.
De ini ion 5.1. ADVBm1,m2,n1,n2-na u al a ino on Fis a DVBm1,m2,n1,n2-
in a ian amily B:TF →TF o a ino s
B:TFK →TFK
on FK o any DVBm1,m2,n1,n2-objec K. I means ha TF ◦B=B◦TF
o any DVBm1,m2,n1,n2-map :K→K′.
Theo em 5.2. I m1≥2, hen he na u al a ino s
a (c):TF →TF
o c∈AFa e all DVBm1,m2,n1,n2-na u al a ino s on a ppgb- unc o F.
P oo . Le Bbe a DVBm1,m2,n1,n2-na u al a ino on a ppgb- unc o Fon
DVB. Clea ly, Bis de e mined by he a ino
B:TFRm1,m2,n1,n2→TFRm1,m2,n1,n2
on FRm1,m2,n1,n2= (AF)m1×(UF)m2×(VF)n1×(WF)n2. Then
B:FRm1,m2,n1,n2×FRm1,m2,n1,n2→FRm1,m2,n1,n2×FRm1,m2,n1,n2
modulo he s anda d iden i ica ion. So, we can w i e
B(x, y)=(x, ˜
B(x, y))
o all x, y ∈FRm1,m2,n1,n2, whe e ˜
B(x, y)∈FRm1,m2,n1,n2is linea in y.
Because o he in a iance o Bwi h espec o he homo he ies
·idRm1,m2,n1,n2 o > 0,˜
B( x, y) = ˜
B(x, y), i.e. ˜
B( x, y) = ˜
B(x, y).
Consequen ly, ˜
B(x, y)is independen o x. So, we can w i e
B((a1, u1, 1, w1),(a2, u2, 2, w2))
= ((a1, u1, 1, w1),(α(a2, u2, 2, w2), β(a2, u2, 2, w2),
γ(a2, u2, 2, w2), δ(a2, u2, 2, w2)))

6 M. Doupo ec, J. Ku ek and W. M. Mikulski
o all a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1and w1, w2∈
(WF)n2, whe e α(a2, u2, 2, w2)∈(AF)m1is linea in (a2, u2, 2, w2)and
β(a2, u2, 2, w2)∈(UF)m2is linea in (a2, u2, 2, w2)and γ(a2, u2, 2, w2)∈
(VF)n1is linea in (a2, u2, 2, w2)and δ(a2, u2, 2, w2)∈(WF)n2is linea in
(a2, u2, 2, w2).
Le φ , 1, 2, 3:Rm1,m2,n1,n2→Rm1,m2,n1,n2be gi en by
φ , 1, 2, 3(x, y1, y2, y3)=( x, 1y1, 2y2, 3y3)
o all x∈Rm1and y1∈Rm2and y2∈Rn1and y3∈Rn2, whe e , 1, 2, 3
a e posi i e eal numbe s. I is a DVBm1,m2,n1,n2-map. Then, by he in a i-
ance o Bwi h espec o φ , 1, 2, 3, we de i e
α( a2, 1u2, 2 2, 3w2) = α(a2, u2, 2, w2).
Consequen ly, α(a2, u2, 2, w2)is linea in a2and independen o u2, 2, w2.
Simila ly, β(a2, u2, 2, w2)is linea in u2and independen o a2, 2, w2, and
γ(a2, u2, 2, w2)is linea in 2and independen od a2, u2, w2, and
δ(a2, u2, 2, w2)is linea in w2and independen o b, u2, 2. Hence we can
w i e
B((a1, u1, 1, w1),(a2, u2, 2, w2))
= ((a1, u1, 1, w1),(α(a2), β(u2), γ( 2), δ(w2)))
o all a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1,w1, w2∈(WF)n2,
whe e α(a2)∈(AF)m1is linea in a2and β(u2)∈(UF)m2is linea in u2
and γ( 2)∈(VF)n1is linea in 2and δ(w2)∈(WF)n2is linea in w2.
Le :Rm1,m2,n1,n2→Rm1,m2,n1,n2be gi en by
(x, y1, y2, y3)=(x+x1x, y1+x1y1, y2+x1y2, y3+x1y3)
o all x= (x1, . . . , xm1)∈Rm1and y1∈Rm2and y2∈Rn1and y3∈Rn2.
I is a DVBm1,m2,n1,n2-map on he open and dense subse in Rm1,m2,n1,n2,
x1=−1. Then, by he in a iance o Bwi h espec o and (in pa icula )
o mula (2) o TF ins ead o Fand o mulas (3), we ge
((a1+a1
1a1, u1+a1
1u1, . . .),(α(a2+a1
1a2+a1
2a1), β(u2+a1
1u2+a1
2u1), . . .))
= ((a1+a1
1a1, u1+a1
1u1, . . .),(α(a2) + a1
1α(a2) + α1(a2)a1,
β(u2) + a1
1β(u2) + α1(a2)u1, . . .))
o all a1, a2∈(AF)m1and u1, u2∈(UF)m2and ..., whe e (α1(b), . . . , αm1(b))
=α(b)∈(AF)m1and (b1, . . . , bm1) = b∈(AF)m1. Then
α(a1
1a2) + α(a1
2a1) = a1
1α(a2) + α1(a2)a1,
β(a1
1u2) + β(a1
2u1) = a1
1β(u2) + α1(a2)u1,
γ(a1
1 2) + γ(a1
2 1) = a1
1γ( 2) + α1(a2) 1,
δ(a1
1w2) + δ(a1
2w1) = a1
1δ(w2) + α1(a2)w1.
Na u al a ino s and o sion o connec ions... 7
I a1
1= 1, hen
α(a1
2a1) = α1(a2)a1, β(a1
2u1) = α1(a2)u1,
γ(a1
2 1) = α1(a2) 1, δ(a1
2w1) = α1(a2)w1.
I a2= (1,0,...,0) ∈(AF)m1, we ge
α(a) = c1a , β(u) = c1u , γ( ) = c1 , δ(w) = c1w
o any a= (a1, . . . , am1)∈(AF)m1wi h a1= 1 and u∈(UF)m2and
∈(VF)n1and w∈(WF)n2, whe e c1:=α1(1,0,...,0) ∈AF.
Simila ly, eplacing 1by i∈ {1, . . . , m1}, we de i e
α(a) = cia , β(u) = ciu , γ( ) = ci , δ(w) = ciw
o any a= (a1, . . . , am1)∈(AF)m1wi h ai= 1 and u∈(UF)m2and
∈(VF)n1and w∈(WF)n2, whe e ci:=αi(0,...,1,...,0) ∈AF(1in i- h
posi ion).
F om he linea i y o αand m1≥2we ob ain
α(a) = ca , β(u) = cu , γ( ) = c , δ(w) = cw
o any a= (a1, . . . , am)∈(AF)m1and u∈(UF)m2and ∈(VF)n1and
w∈(WF)n2, whe e c:=c1=. . . =cm∈AF. Tha c1=. . . =cmis a
simple consequence o he in a iance o Bwi h espec o he pe mu a ions
o he base coo dina es.
Then
B((a1, u1, 1, w1),(a2, u2, 2, w2)) = ((a1, u1, 1, w1), c(a2, u2, 2, w2))
o all a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1,w1, w2∈(WF)n2,
whe e c∈AFis as abo e. Then B= a (c), as well and he p oo is comple e.
Q.E.D. □
6. On double-linea ec o ields. Le Kbe a double ec o bundle
wi h basis M. A ec o ield Zon Kis called double-linea i he low o Z
is o med by local DVB-isomo phisms.
Le x1, . . . , xm1,u1, . . . , um2, 1, . . . , n1,w1, . . . , wn2be (local) DVB-
coo dina es on K. A map :K→Kis a DVB-map i and only i i is o
he o m (1). Consequen ly, a ec o ield Zon Kis double linea i and
only i i is o he o m
(4)
Z=
m1
X
i=1
ai(x)∂
∂xi+
m2
X
j,j1=1
bj1
j(x)uj∂
∂uj1+
n1
X
k,k1=1
ck1
k(x) k∂
∂ k1
+
n2
X
l,l1=1
el1
l(x)wl∂
∂wl1+
m2
X
j2=1
n1
X
k2=1
n2
X
l2=1
l2
j2k2(x)uj2 k2∂
∂wl2.
8 M. Doupo ec, J. Ku ek and W. M. Mikulski
So, we ha e:
Lemma 6.1 ([11]). The space o all double-linea ec o ields on Kis he
Lie subalgeb a in he Lie algeb a o ec o ields on K.
Le Fbe a ppgb- unc o on DVB. Then F K is again a DVB-objec (see
Sec ion 3).
Lemma 6.2. Le Zbe a double-linea ec o ield on FK and c∈AFbe a
poin . Then he ec o ield a (c)(Z)on F K is also double-linea .
P oo . We may assume ha K=Rm1,m2,n1,n2. Then F K =Am1×Um2×
Vn1×Wn2. Fixing he bases (o e R) o Am1and Um2and Vn1and Wn2,
we can w i e FK =RM1,M2,N1,N2. Le x1, . . . , xM1, u1, . . . , uM2, 1, . . . , N1,
w1, . . . , wN2be he usual coo dina es on RM1,M2,N1,N2. Then Zis o he
o m
(5)
Z=
M1
X
i=1
ai(x)∂
∂xi+
M2
X
j,j1=1
bj1
j(x)uj∂
∂uj1+
N1
X
k,k1=1
ck1
k(x) k∂
∂ k1
+
N2
X
l,l1=1
el1
l(x)wl∂
∂wl1+
M2
X
j2=1
N1
X
k2=1
N2
X
l2=1
l2
j2k2(x)uj2 k2∂
∂wl2.
To p o e ha a (c)(Z)is double-linea , i is su icien o show ha
a (c)(Z)is o he o m (5), oo. O cou se, i is su icien o show ha
a (c)( ∂
∂xi)is he linea combina ion o ∂
∂x1,..., ∂
∂xM1wi h eal coe icien s
and ha a (c)( ∂
∂uj)is he linea combina ion o ∂
∂u1,..., ∂
∂uM2wi h eal co-
e icien s and ha a (c)( ∂
∂ k)is he linea combina ion o ∂
∂ 1,..., ∂
∂ N1wi h
eal coe icien s and ha a (c)( ∂
∂wl)is he linea combina ion o ∂
∂w1,...,
∂
∂wN2wi h eal coe icien s.
Fo example, we p o e ha a (c)( ∂
∂u1)is he linea combina ion o
∂
∂u1,..., ∂
∂uM2wi h eal coe icien s. Le (x, u, , w)∈Am1×Um2×Vn1×
Wn2. Le e1, . . . , eM2be he usual basis in RM2˜=Um2. We can w i e
∂
∂uj|(x,u, ,w)= ((x, u, , w),(0, ej,0,0)). Then
a (c)∂
∂u1|(x,u, ,w)
= ((x, u, , w),(0, c ·e1,0,0)) .
On he o he hand, c·e1∈Um2(as e1∈Um2), and hen c·e1is he
linea combina ion o e1, . . . , eM2wi h eal coe icien s. The p oo o he
p oposi ion is comple e. Q.E.D. □
7. The F-N-b acke and double-linea (semi-basic) angen al-
ued p- o ms. I K→Mis a ib ed mani old, a p ojec able semi-basic
angen alued p- o m on Kis a sec ion φ:K→ ∧pT∗M⊗T K such
Na u al a ino s and o sion o connec ions... 9
ha φ(X1, . . . , Xp)is a p ojec able ec o ield on K o any ec o ields
X1, . . . , Xpon M.
Gi en a p ojec able semi-basic angen alued p- o m φ:K→ ∧pT∗M⊗
TK we ha e he unde lying angen alued p- o m φ:M→ ∧pT∗M⊗T M
on Msuch ha φ(X1, . . . , Xp)is he unde lying ec o ield o φ(X1, . . . , Xp)
o any ec o ields X1, . . . , Xpon M.
Lemma 7.1. Le K→Mbe a ib ed mani old. Gi en a p ojec able semi-
basic angen alued p- o m φ:K→ ∧pT∗M⊗T K on Kand a p o-
jec able semi-basic angen alued q- o m ψ:K→ ∧qT∗M⊗T K on K, he
F oliche –Nijenhuis b acke (F-N-b acke ) [[φ, ψ]] is (again) a p ojec able
semi-basic angen alued (p+q)- o m [[φ, ψ]] :K→ ∧p+qT∗M⊗T K on
Ksa is ying
(6)
[[φ, ψ]](X1, . . . , Xp+q)
=1
p!q!X
σ
sign σ[φ(Xσ1, . . . , Xσp), ψ(Xσ(p+1), . . . , Xσ(p+q))]
+−1
p!(q−1)! X
σ
sign σψ([φ(Xσ1, . . . , Xσp), Xσ(p+1)], Xσ(p+2), . . .)
+(−1)pq
(p−1)q!X
σ
sign σφ([ψ(Xσ1, . . . , Xσq), Xσ(q+1)], Xσ(q+2), . . .)
+(−1)p−1
(p−1)!(q−1)!2! X
σ
sign σψ(φ([Xσ1, Xσ2], Xσ3, . . .), Xσ(p+2), . . .)
+(−1)(p−1)q
(p−1)!(q−1)!2! X
σ
sign σφ(ψ([Xσ1, Xσ2], Xσ3, . . .), Xσ(q+2), . . .)
o any ec o ields X1, . . . , Xp+qon M, whe e sums a e o e all pe mu a-
ions σ:{1, . . . , p +q}→{1, . . . , p +q}.
P oo . I is well-known ac , see e.g. [4]. Q.E.D. □
Le Fbe a ppgb- unc o on DVB and Kbe a DVB-objec wi h basis M.
Then we ha e he ib ed mani old FK →M. We ha e also he DVB-objec
FK wi h basis FM.
De ini ion 7.2. A double-linea semi-basic angen alued p- o m on F K →
Mis a p ojec able semi-basic angen alued p- o m φ:F K → ∧pT∗M⊗
TFK on ( ibe ed mani old) F K (wi h basis M) such ha (addi ionally)
φ(X1, . . . , Xp)is a double-linea ec o ield on DVB-objec F K (wi h basis
FM) o any ec o ields X1, . . . , Xpon M.