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PureMathematicsnØêÆ,2023,13(2),234-243
PublishedOnlineFebruary2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.132028
pOmni-o“êE-éó
xxxzzzhhh
HʘŒÆêƆ&E‰ÆÆ§ôÜH
ÂvFϵ2023c114F¶¹^Fϵ2023c214F¶uÙFϵ2023c222F
Á‡
T©ïÄpomni-o“êE-éó("Äk§ŠâE-Šéóm½Â§3†ÚmD
n
E⊕
JEþ½ÂD
n−1
E-ŠéÚpDorfman)Ò§Ù¥D
n
EÚJE©O••þmEn-‡
©ŽfmÚjetm"pomni-o“êE-éó("Ùg§ÏL4ÙZ]“êš
é§E†pomni-oV“êƒ'š駿…†²…‚mM×Rƒ'p
omni-oV“êdouble"
'…c
pOmni-o“ê¶E-éó¶pOmni-oV“êDouble
E-ValueDualofHigherOmniLie
Algebroids
XiancongZheng
SchoolofMathematicsandInformation Science, NanchangHangkongUniversity,NanchangJiangxi
Received:Jan.14
th
,2023;accepted:Feb.14
th
,2023;published:Feb.22
nd
,2023
Abstract
Inthispaper,westudyaE-valuedualofhigheromniLiealgebroids.First,wedefine
©ÙÚ^:xzh.pOmni-o“êE-éó[J].nØêÆ,2023,13(2):234-243.
DOI:10.12677/pm.2023.132028
xzh
D
n−1
E-valuepairingandhigherDorfmanbracketonthedirectsumbundleD
n
E⊕JE,
whereD
n
EandJEare,respectively,then-thdifferentialoperatorbundleandthejet
bundleofavectorbundleE,constructaE-valuedualofhigheromniLiealgebroids.
Secondly,throughthematchedpairofLeibnizalgebras,constructamatchedpair
associatedtohigheromniLiebialgebroid,andstudyhigheromniLiebialgebroid
doubleassociatedtoatriviallinebundle.
Keywords
HigherOmniLieAlgebroids,E-ValueDual,HigherOmniLieBialgebroidDouble
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
ChenÚLiu3©z[1]¥Ú\omni-o“êVg,´•þ˜mþomni-o“ê•
þmþAÛí2.•þmEþo“ê(Úomni-o“êDirac(ïá˜˜é
A'X.•ïÄomni-o“êpa,©z[2]Ú\pomni-o“êVg,3†Úm
DE⊕J
n
Eþ½ÂJ
n−1
E-ŠéÚ)ÒŽf,Ù¥DEÚJ
n
E©O´•þmEC‡©Žf
mÚn-jetm.·‚•,éuDE¡,Œ±n)•éómE
∗
þ‚5•þ|,éuJE
þ¡k ØÓ)º.JE¡n)•E
∗
þ~1-/ª,omni-o“êDE⊕JEŒ±
n)•IOCourant“êTE
∗
⊕T
∗
E
∗
Weinstein-‚5z;JE¡n)•E
∗
þ
‚51-/ª,omni-o“êDE⊕JE•IOCourant“êTE
∗
⊕T
∗
E
∗
pseudo-‚5z.
©z[3]¥ïÄCourant“êT
∗
E⊕∧
n
T
∗
E
∗
papseudo-‚5zÚWeinstein-
‚5z,Ù¥n-omni-o“êDE⊕J
n
E•Courant“êpapseudo-‚5z.‡5
¿´,•þmEz••þ˜mV,duJ
n
V=0,Ã{¼¤éAn-o“ê.,˜
•¡,Ã{ïá•þmEþ(n+ 1)-o“ê(Úpomni-o“êDE⊕J
n
Eþ
ŒÈfméA'X.éuCourant“êpaWeinstein-‚5z-omnin-o“ê
DE⊕∧
n
JE[3],rank(E)≥2,Uïá•þmEþ(n+1)-o“ê(Úomni n-o“ê
DE⊕∧
n
JEþŒÈfméA'X.©z[3]0˜„‚5mM×Reomnin-o“ê
(TM×R⊕(∧
n
T
∗
M⊕∧
n−1
T
∗
M),(·,·),{·,·},ρ).
©Äupomni-o“ê(DE⊕J
n
E,(·,·),{·,·},ρ),ÏLDEÚJEE-éóm'X[1],
ïÄpomni-o“êE-éó.©(SüXeµ11!0Ä•£Úƒ'Vg.1
DOI:10.12677/pm.2023.132028235nØêÆ
xzh
2!ÏLE-éóm'X,‰Ñpomni-o“êE-éó(E
n
(E),(−,−)
∗
+
,{−,−}
∗
,ρ
∗
)½Â,
±9Ù¤éAšòzV‚5)Ò(−,−)
∗
+
Ú¡þ2ÂDorfman)Ò{−,−}
∗
.13!Ï
L4ÙZ]“êšé,3†Ú•þmE
n
(E) ⊕E
n
(E)þE˜‡4ÙZ]“ê(
(E
n
(E) ⊕E
n
(E),{−,−}
E
n
⊕E
n
,˜ρ).•þmE•²…‚mž,pomni-oV“êdouble
(∆,(−,−),{−,−}
E
n
⊕E
n
,˜ρ),Ù¥∆ = E
n
(E)⊕E
n
(E).14!é©?1o(Ú8B.
2.ý•£
3!¥,k£©¥9Ä•£Úƒ'Vg.
3Ñt6/(M,π)þ,Ñt)Ò• ˜/(½ÑtÜþπ,§÷v[π,π]= 0.‡ƒ,36/M
þ,‰½˜‡V•þ|π∈X(M),e÷v[π,π]= 0,K••˜(½MþÑt)Ò.3©z[4]
¥,Šö‰Ñ6/Mþ˜‡HÜ-Ñt(½Â.
½Â1.1 [4]6/Mþ˜‡HÜ-Ñt(´˜‡n--‚5N{·,···,·}
C
∞
(M)×···×C
∞
(M) −→C
∞
(M)
÷v±e5Ÿ:
(1)‡é¡:é?¿f
i
∈C
∞
(M),(1 ≤i≤n),σ∈S
n
(S
n
´né¡+),
{f
1
,···,f
n
}= (−1)
ε(σ)
{f
σ(1)
,···,f
σ( n)
}.
(2)4ÙZ[5Ÿ:é?¿f
i
,g∈C
∞
(M),(1 ≤i≤n),
{f
1
g,f
2
,···,f
n
}= f
1
{g,f
2
,···,f
n
}+g{f
1
,f
2
,···,f
n
}.
(3)Äðª:é?¿f
i
,g
j
∈C
∞
(M),(1 ≤i≤n−1,1 ≤j≤n),
{f
1
,···,f
n−1
,{g
1
,···,g
n
}}=
n
X
j=1
{g
1
,···,{f
1
,···,f
n−1
,g
j
},···,g
n
}.
Ïd,‰½˜‡÷v±þ5ŸHÜ-Ñt)Ò{·,···,·},K•˜(½Ñtn-ÜþΛ∈Γ(∧
n
TM)
÷vª
Λ(df
1
∧···∧df
n
) = {f
1
,···,f
n
}.(2.1)
E•6/Mþ•þm,Ùþ1-jetmJE[5]´dΓ(E)þda5½Â.é?
¿u,v∈Γ(E),m∈M,XJu(m)= v(m)…d
m
hu,ξi= d
m
hv,ξi,Kk[u]
m
= [v]
m
.Ïd,é?
¿µ∈(JE)
m
,Ñ•3˜‡L«u∈Γ(E)÷vµ=[u]
m
.©z[1]¥,Šöy²jetmJE´
DEE-éóm.?3©z[6]¥,ŠöÏLE-éóm‰Ñn-‡¡jetm.
J
n
E:= Hom(∧
n
DE,E)
JE
= {µ∈Hom(∧
n
DE,E) |Im(µ
]
) ⊂JE)}n≥2,(2.2)
DOI:10.12677/pm.2023.132028236nØêÆ
xzh
Ù¥µ
]
: ∧
n−1
DE→Hom(DE,E)½Â•:
µ
]
(d
1
,....d
n−1
)(d
n
) = µ(d
1
,....d
n
).∀d
1
,....d
n
∈Γ(JE)
¿…n-‡¡jetméAáÜSXe:
0
//
Hom(∧
n
TM,E)
e
//
J
n
E
p
//
Hom(∧
n−1
TM,E)
//
0 .(2.3)
E•6/Mþ•þm,©z[7]¥,Šö0n-‡©ŽfmD
n
E,=
D
n
E:= Hom(∧
n
JE,E)
DE
= {d∈Hom(∧
n
JE,E) |Im(d
]
) ⊂DE)}n≥2,(2.4)
Ù¥d
]
: ∧
n−1
JE→Hom(JE,E)½Â•:
d
]
(µ
1
,....µ
n−1
)(µ
n
) = d(µ
1
,....µ
n
).∀µ
1
,....µ
n
∈Γ(JE)
rank E≥2,Ù¤éAáÜSXe:
0
//
Hom(∧
n
E,E)
i
//
D
n
E
j
//
Hom(∧
n−1
E,TM)
//
0(2.5)
AO,n=1,D
1
E•IemF(E)þgauge-o“ê,=•þmEþC‡©Žf
m.n=2ž,π∈Γ(D
2
E),é?¿µ
1
,µ
2
∈Γ(JE),kπ(µ
1
,µ
2
)∈Γ(E).Œ±pÑ
π
]
: JE→DE…÷v:
hπ
]
(µ
1
),µ
2
i
E
= π(µ
1
,µ
2
).
¿…,3Γ(JE)þ)Ò[·,·]
]
½Â•:
[µ
1
,µ
2
]
]
= L
π
]
(µ
1
)
µ
2
−ι
π
]
(µ
2
)
dµ
1
(2.6)
Ïd,dπ
]
g,pÑo“ê(JE,[·,·]
]
,j◦π
]
).
E•6/Mþ•þm,·‚©O^X
n
lin
(E)ÚΩ
n
lin
(E)5L«•þmEþ‚5n-•þ|
Ú‚5n-/ª8Ü.d©z[3]Œ•,E
∗
þn-‡©ŽfmD
n
E
∗
ÓuEþ‚5n-•þ|
X
n
lin
(E),E
∗
þn-‡¡jetmJ
n
E
∗
ÓuEþ‚5n-/ªΩ
n
lin
(E).
·K1.1 [3] éuEþ‚5n-•þ|X
n
lin
(E)Ú‚5n-/ªΩ
n
lin
(E),XJrankE≥2,Šâ
áÜS(2.5)Ú(2.3),·‚k:
X
n
lin
(E)
∼
=
Γ(D
n
E
∗
)
∼
=
Γ(∧
n
E⊗E
∗
)⊕Γ(∧
n−1
E⊗TM);
Ω
n
lin
(E)
∼
=
Γ(J
n
E
∗
)
∼
=
Γ(∧
n
T
∗
M⊗E
∗
)⊕Γ(∧
n−1
T
∗
M⊗E
∗
).
DOI:10.12677/pm.2023.132028237nØêÆ
xzh
3.pOmni-o“êE-éó
!lE-éóm*gÑu,3pomni-o“ê(DE⊕J
n
E,(−,−)
+
,{−,−},ρ)Ä:
þ,ïÄpomni-o“êE-éó(D
n
E⊕JE,(−,−)
∗
+
,{−,−}
∗
,ρ
∗
).Šâ©z[1]¥E-éó
m½Â,Œ±‰Ñn-‡¡jetmJ
n
E†n-‡©ŽfmD
n
EE-Šé)Òh−,−i.
½Â2.1E•6/Mþ•þm,J
n
E•Ùn-‡¡jetm,n-•þmD
n
E⊂
Hom(∧
n
J
n
E,E)
DE
¡•n-jetmJ
n
EE-éóm, XJE-Šéh−,−i: Γ(J
n
E)×Γ(D
n
E) →
E´šòz.
©z[8],ŠöïÄE/(Γ(Hom(∧
•
JE,E)
DE
),d
J
)ƒ'Vg,Ù¥d
J
•þ>Žf.
½Â2.2 [8]é?¿d∈Hom(∧
k
JE,E)
DE
,±9µ
1
,....,µ
n
∈Γ(JE),N
d
J
: Hom(∧
k
JE,E)
DE
→Hom(∧
k+1
JE,E)
DE
½Â•
(d
J
d)(µ
1
,..µ
k+1
) =
k+1
X
i=1
(−1)
i+1
(π
]
µ
i
)(d(µ
1
,..ˆµ
i
,..µ
k+1
))
+
X
i<j
(−1)
i+j
d([µ
i
,µ
j
]
]
,µ
1
,..ˆµ
i
,..ˆµ
j
,..µ
k+1
).
AO,‰½˜‡o“ê(E,[−,−]
E
,ρ
E
),Œ±†π
]
:JE→DEƒ'þóE/
(Γ(Hom(∧
•
JE,E)),d
J
)fE/(Γ(Hom(∧
•
JE,E)
DE
),d
J
),Šâ©z[9],Œ±Xe½Â
½Â2.3é?¿d∈Hom(∧
k
JE,E)
DE
,±9µ,µ
1
,....,µ
n
∈Γ(JE),N
L
µ
: Hom(∧
k
JE,E)
DE
→Hom(∧
k
JE,E)
DE
½Â•
(L
µ
d)(µ
1
,..µ
k
) = π
]
µ(d(µ
1
,..µ
k
))−
k
X
i=1
d(µ
1
,...,[µ,µ
i
]
]
,...,µ
k
).
¿Žfι
µ
½Â•:ι
µ
d(µ
1
,..µ
k−1
) = d(µ,µ
1
,..µ
k−1
).
Ïd,é?¿µ,ν∈Γ(JE),¤éAofL
µ
Ú ¿Žfι
µ
÷veª:
L
µ
= d
J
ι
µ
+ι
µ
d
J
;
[L
µ
,ι
µ
] = ι
[µ,ν]
]
, [L
µ
,L
ν
] = L
[µ,ν]
]
;
[ι
µ
,ι
ν
] = [d
J
,L
µ
] = 0.(3.1)
½n2.4 -E
n
(E) := D
n
E⊕JE,Kpomni-o“êE-éó(E
n
(E),(−,−)
∗
+
,{−,−}
∗
,ρ
∗
)
(Xe:
(1)‡¡V‚5D
n−1
E-Šé(−,−)
∗
+
½Â•
(d+µ,r+ν)
∗
+
= (ι
ν
d+ι
µ
r);(3.2)
DOI:10.12677/pm.2023.132028238nØêÆ
xzh
(2)ÙþDorfman)Ò{−,−}
∗
: Γ(E
n
(E))×Γ(E
n
(E)) →Γ(E
n
(E))½Â•
{d+µ,r+ν}
∗
= [d,r]+L
µ
r−ι
ν
d
J
d+[µ,ν]
]
= L
µ
r−ι
ν
d
J
d+[µ,ν]
]
;(3.3)
Ù¥[µ,ν]
]
dª(2.6)‰Ñ.
(3)eNρ
∗
: D
n
E⊕JE→DE½Â•
ρ
∗
(d+µ) = π
]
µ,(3.4)
Ù¥?¿d,r∈Γ(D
n
E),µ,ν∈Γ(JE).
4.pOmni-oV“êDouble
©z[10]¥,Šöy²ÏLÚ\oV“êManintriples,Œ±3Courant-“ênØ
µeeEoV“êdouble.3!¥,ÏL4 ÙZ]“êšé5Epomni-o
V“êdouble.
©z[11]¥,Šö04ÙZ]“êšéƒ'Vg,¿y²6/Mþ˜‡n-
(n>2)HÜ-äŒ'(½Â˜éš4ÙZ]“ê.Äk,£˜e4ÙZ]“êš
é½Â.
½Â3.1 [11](A
1
,{−,−}
1
,ρ
1
),(A
2
,{−,−}
2
,ρ
2
)´Mþü‡4ÙZ]“ê,e¦‚†
ÚmA
1
⊕A
2
•¤4ÙZ]“ê((A
1
⊕A
2
,{−,−}
A
1
⊕A
2
,ρ),K¡(A
1
,A
2
)´4ÙZ]“
êA
1
ÚA
2
šé,…÷vA
1
,A
2
´A
1
⊕A
2
4ÙZ]f“ê.
d©z[4]Ú·K2.5Œ•,(E
n
(E),{−,−},ρ),Ú(E
n
(E),{−,−}
∗
,ρ
∗
)¤ü‡4ÙZ]“ê
.Ïd,Šâ4ÙZ]“êšé½Â,Œ±ÑXe½Â.
½Â3.2(E
n
(E),{−,−},ρ),(E
n
(E),{−,−}
∗
,ρ
∗
)´ü‡4ÙZ]“ê,e¦‚†Úm
E
n
(E) ⊕E
n
(E)•¤4ÙZ]“ê((E
n
(E) ⊕E
n
(E),{−,−}
E
n
⊕E
n
,˜ρ),…E
n
(E),E
n
(E)´
E
n
(E)⊕E
n
(E)4ÙZ]f“ê,K¡(E
n
(E),E
n
(E))´4ÙZ]“êE
n
(E)ÚE
n
(E)š
é.
Šâ©ÙŒ•,˜ρ(α+ x)=ρ(α) +ρ
∗
(x),…{α,β}
E
n
⊕E
n
={α,β},{x,y}
E
n
⊕E
n
={x,y}
∗
.d
uΓ(E
n
(E) ⊕E
n
(E))
∼
=
Γ(E
n
(E)) ⊕Γ(E
n
(E)),·‚±Pr
1
L«Γ(E
n
(E) ⊕E
n
(E))
∼
=
Γ(E
n
(E)) ⊕
Γ(E
n
(E))Γ(E
n
(E))Ý,Pr
2
L«Γ(E
n
(E) ⊕E
n
(E))
∼
=
Γ(E
n
(E)) ⊕Γ(E
n
(E))Γ(E
n
(E))
Ý,dd,¦‚©Opѱe‚5N:
ρ
L
: Γ(E
n
(E))×Γ(E
n
(E)) →Γ(E
n
(E)),
ρ
R
: Γ(E
n
(E))×Γ(E
n
(E)) →Γ(E
n
(E)),
ρ
L
∗
: Γ(E
n
(E))×Γ(E
n
(E)) →Γ(E
n
(E)),
ρ
R
∗
: Γ(E
n
(E))×Γ(E
n
(E)) →Γ(E
n
(E)),
DOI:10.12677/pm.2023.132028239nØêÆ
xzh
Ù¥,é?¿α,β∈Γ(E
n
(E)),x,y∈Γ(E
n
(E))÷v
ρ
L
(α)y= Pr
1
{y,α}
E
n
⊕E
n
,ρ
R
(α)y= Pr
1
{α,y}
E
n
⊕E
n
,
ρ
L
∗
(x)β= Pr
2
{β,x}
E
n
⊕E
n
,ρ
R
∗
(x)β= Pr
2
{x,β}
E
n
⊕E
n
.
·K3.3XJ(E
n
(E),E
n
(E))´4ÙZ]“êE
n
(E)ÚE
n
(E)šé,@o•34ÙZ
]“êE
n
(E)3E
n
(E)þL«(ρ
L
∗
,ρ
R
∗
)Ú4ÙZ]“êE
n
(E)3E
n
(E)þL«(ρ
L
,ρ
R
),
…é?¿α,β∈Γ(E
n
(E)),x,y∈Γ(E
n
(E))÷v±eƒN^‡:
duρ:(Γ(E
n
(E)),{−,−})→(X(M),[−,−])Úρ
∗
:(Γ(E
n
(E)),{−,−}
∗
)→(X(M),[−,−])
´‡4ÙZ]“êÓ,Ïd,k
(1)ρ(ρ
R
(α)y)+ρ
∗
(ρ
L
∗
(y)α) = [ρ(α),ρ
∗
(y)];
(2)ρ
∗
(ρ
R
∗
(β)x)+ρ(ρ
L
(x)β) = [ρ
∗
(β),ρ(x)];
Šâ4ÙZ]“êE
n
(E)ÚE
n
(E)¤÷v4ÙZ]ð{α,{β,γ}}−{{α,β},γ}−{β,{α,γ}}=
0.·‚k
(3)ρ
R
({α,β})y= {α,ρ
R
(β)y}−{β,ρ
R
(α)y}+ρ
R
(α,ρ
L
∗
(y)β)−ρ
R
(β,ρ
L
∗
(y)α);
(4)ρ
R
∗
({x,y}
∗
)α= {x,ρ
R
∗
(y)α}
∗
−{y,ρ
R
∗
(x)α}
∗
+ρ
R
∗
(x,ρ
L
(α)y)−ρ
R
∗
(y,ρ
L
(α)x);
(5)ρ
L
({α,β})y= {α,ρ
L
(β)y}−{ρ
R
(α)y,β}+ρ
R
(α,ρ
R
∗
(y)β)−ρ
L
(ρ
L
∗
(y)α,β);
(6)ρ
L
∗
({x,y}
∗
)α= {x,ρ
L
∗
(y)α}
∗
−{ρ
R
∗
(x)α,y}
∗
+ρ
R
∗
(x,ρ
R
(α)y)−ρ
L
∗
(ρ
L
(α)x,y);
(7)ρ
L
({α,β})y= {ρ
L
(α)y,β}+{ρ
L
(β)y,α}+ρ
R
(α,ρ
R
∗
(y)β)−ρ
L
(ρ
R
∗
(y)β,α);
(8)ρ
L
∗
({x,y}
∗
)α= {ρ
L
∗
(x)α,y}
∗
+{ρ
L
∗
(y)α,x}
∗
+ρ
R
∗
(x,ρ
R
(α)y)−ρ
L
∗
(ρ
R
(α)y,x).
Ïd,(E
n
(E),E
n
(E))´4ÙZ]“êšé,3†Ú•þmE
n
(E) ⊕E
n
(E)þk˜‡4
ÙZ]“ê((E
n
(E)⊕E
n
(E),{−,−}
E
n
⊕E
n
,˜ρ),Ù4ÙZ])Ò•
{α+x,β+y}
E
n
⊕E
n
= {α,β}+ρ
R
∗
(y)α+ρ
L
∗
(x)β+{x,y}
∗
+ρ
L
(α)y+ρ
R
(β)x.(4.1)
E= M×R•²…‚5mž,DE= TM×R,JE= T
∗
M×R.Šâ•þmEþn-õ
fD
n
EÚn-‡¡jetmJ
n
E½Â,· ‚kD
n
E
∼
=
∧
n
(TM×R),J
n
E
∼
=
∧
n
(T
∗
M×R),
ÏdÜS(2.5)ŒL«•:
0
//
∧
n
(M×R)⊗M×R
i
//
D
n
(M×R)
j
//
∧
n−1
(M×R)⊗TM
//
0(4.2)
d©z[2]Œ,ÜS(2.3)ŒL«•∧
n
T
∗
M-áÜS:
0
//
∧
n
T
∗
M⊗M×R
e
//
J
n
(M×R)
p
//
∧
n−1
T
∗
M⊗M×R
//
0 .(4.3)
·K3.4 [3]E= M×R,Ùþ‚5n-•þ|X
n
lin
(E)Ú‚5n-/ªΩ
n
lin
(E)kXe Ó
DOI:10.12677/pm.2023.132028240nØêÆ
xzh
'X:
X
n
lin
(M×R)
∼
=
Γ(D
n
(M×R))
∼
=
X
n
(M)⊕X
n−1
(M);(4.4)
Ω
n
lin
(M×R)
∼
=
Γ(J
n
(M×R))
∼
=
Ω
n
(M)⊕Ω
n−1
(M).(4.5)
KE
n
(E) = TM×R⊕∧
n
(T
∗
M×R),E
n
(E) = ∧
n
(TM×R)⊕T
∗
M×R.
½n3.5E´Mþ²…‚m,-∆= E
n
(E) ⊕E
n
(E),Kpomni-oV“ê-double
(∆,(−,−),{−,−}
E
n
⊕E
n
,˜ρ)(Xe:
(1)E
n
(E)⊕E
n
(E) := (TM×R⊕∧
n
(T
∗
M×R))⊕(∧
n
(TM×R)⊕T
∗
M×R)
(2)šòzV‚5é¡C
∞
(M)-Šé(−,−)½Â•
((X,f)⊕(α
n
,α
n−1
),(Λ,Γ)⊕(ξ,g)) = hι
X
α
n
,ι
ξ
Λi+fghα
n−1
,Γi−hι
X
α
n−1
,ι
ξ
Γi(4.6)
(3) -π
i
∈Γ(TM×R),Φ
i
∈Γ(∧
n
(T
∗
M×R)),Π
i
∈Γ(∧
n
(TM×R)),φ
i
∈Γ(T
∗
M×R),i=1,2,
K¡þCourant)ҽ•
{π
1
⊕Φ
1
+Π
1
⊕φ
1
,π
2
⊕Φ
2
+Π
2
⊕φ
2
}
E
n
⊕E
n
= {π
1
⊕Φ
1
,π
2
⊕Φ
2
}
E
n
+ρ
R
∗
(Π
2
⊕φ
2
)(π
1
⊕Φ
1
)+ρ
L
∗
(Π
1
⊕φ
1
)π
2
⊕Φ
2
+ρ
L
(π
1
⊕Φ
1
)(Π
2
⊕φ
2
)+ρ
R
(π
2
⊕Φ
2
)(Π
1
⊕φ
1
)+{Π
1
⊕φ
1
,Π
2
⊕φ
2
}
E
n
(4)eN½Â•
˜ρ((X,f)⊕(α
n
,α
n−1
),(Λ,Γ)⊕(ξ,g))
= ρ((X,f)⊕(α
n
,α
n−1
))+ρ
∗
((Λ,Γ)⊕(ξ,g))
= (X,f)+(π,χ)
]
(ξ,g).
Ù¥(X,f)∈X(M) ×C
∞
(M,R),(α
n
,α
n−1
)∈Ω
n
(M) ×Ω
n−1
(M),(Λ,Γ)∈X
n
(M) ×X
n−1
(M),
(ξ,g) ∈Ω
1
(M)×C
∞
(M,R).
y²Šâª(3.2),ÏLOŽŒ
((X,f)⊕(α
n
,α
n−1
),(Λ,Γ)⊕(ξ,g))
= ((ι
X
α
n
+fα
n−1
,−ι
X
α
n−1
),(ι
ξ
Λ+gΓ,−ι
ξ
Γ))
= (hι
X
α
n
+fα
n−1
,ι
ξ
Λ+gΓi,−hι
X
α
n−1
,ι
ξ
Γi)
= hι
X
α
n
,ι
ξ
Λi+fghα
n−1
,Γi−hι
X
α
n−1
,ι
ξ
Γi
=Œy(4.6).
dª(4.1)Œ•,Ù¥{π
1
⊕Φ
1
,π
2
⊕Φ
2
}
E
n
d©z[3]¤‰Ñ,{Π
1
⊕φ
1
,Π
2
⊕φ
2
}
E
n
dª(3.3)
¤‰Ñ,¿…,-Pr
1
L«Γ(∆)Γ(E
n
(E))Ý,Pr
2
L«Γ(∆)Γ(E
n
(E))Ý,Šâp
DOI:10.12677/pm.2023.132028241nØêÆ
xzh
‚5N,Œ
ρ
R
∗
(Π
2
⊕φ
2
)(π
1
⊕Φ
1
) = Pr
2
{Π
2
⊕φ
2
,π
1
⊕Φ
1
}
E
n
⊕E
n
,
ρ
L
∗
(Π
1
⊕φ
1
)(π
2
⊕Φ
2
) = Pr
2
{π
2
⊕Φ
2
,Π
1
⊕φ
1
}
E
n
⊕E
n
,
ρ
L
(π
1
⊕Φ
1
)(Π
2
⊕φ
2
) = Pr
1
{Π
2
⊕φ
2
,π
1
⊕Φ
1
}
E
n
⊕E
n
,
ρ
R
(π
2
⊕Φ
2
)(Π
1
⊕φ
1
) = Pr
1
{π
2
⊕Φ
2
,Π
1
⊕φ
1
}
E
n
⊕E
n
.
éueN˜ρ: ∆ →DE½Âdª(3.4)‰Ñ.
5.(Ø
©3c<ïÄÄ:þ$^p*:ÐmïÄ.3†ÚmD
n
E⊕JEþEp
omni-o“êE-éóE
n
(E).Ùg,ÏL4ÙZ]“êšé,E†pomni-oV“ê
ƒ'šé,¿…†²…‚mM×Rƒ'pomni-oV“êdouble.
ë•©z
[1]Chen,Z.andLiu,Z.(2010)Omni-LieAlegebroids.JournalofGeometryandPhysics,60,
799-808.https://doi.org/10.1016/j.geomphys.2010.01.007
[2]Bi, Y., Vitagliano, L. and Zhang, T. (2018)Higher Omni-Lie Algebroids. arXivpreprint arXiv:
1812.09496
[3]Lang,H.andSheng,Y.(2020)LinearizationoftheHigherAnalogueofCourantAlgebroids.
JournalofGeometricMechanics,12,585-606.https://doi.org/10.3934/jgm.2020025
[4]Ibanez,R.,Leon,M.andMarrero,J.(1997)DynamicsofGeneralizedPoissonandNambu-
PoissonBrackets.JournalofMathematicalPhysics,38,2332-2344.
https://doi.org/10.1063/1.531960
[5]Saunders,D.J.,Saunders,D.andSaunders,D.J.(1989)TheGeometryofJetBundles.Cam-
bridgeUniversityPress,Cambridge.https://doi.org/10.1017/CBO9780511526411
[6]Chen,Z.,Liu,Z.andSheng,Y.(2010)E-CourantAlgebroids.InternationalMathematics
ResearchNotices,2010,4334-4376.https://doi.org/10.1093/imrn/rnq053
[7]Crainic,M.andMoerdijk,I.(2008)DeformationsofLieBrackets:CohomologicalAspects.
JournaloftheEuropeanMathematicalSociety,10,1037-1059.
https://doi.org/10.4171/JEMS/139
[8]Sheng,Y.(2012)OnDeformationsofLieAlgebroids.ResultsinMathematics,62,103-120.
https://doi.org/10.1007/s00025-011-0133-x
[9].ý¬,‰÷7,•û³.š†omni-o“ê[J].êÆ?Ð,2022,51(5):879-890.
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xzh
[10]Liu,Z.J.,Weinstein,A.andXu,P.(1997)ManinTriplesforLieBialgebroids.Journalof
DifferentialGeometry,45,547-574.https://doi.org/10.4310/jdg/1214459842
[11]Ibaez, R., Lopez, B., Marrero,J.C., etal. (2001) Matched Pairsof Leibniz Algebroids, Nambu-
JacobiStructuresandModularClass.ComptesRendusdel’Acad´emiedesSciences-SeriesI-
Mathematics,333,861-866.https://doi.org/10.1016/S0764-4442(01)02150-4
[12]Bi,Y.andFan,H.(2021)HigherNonabelianOmni-LieAlgebroids.JournalofGeometryand
Physics,167,ArticleID:104277.https://doi.org/10.1016/j.geomphys.2021.104277
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