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PureMathematics
n
Ø
ê
Æ
,2023,13(2),234-243
PublishedOnlineFebruary2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.132028
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Double
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Algebroids
XiancongZheng
SchoolofMathematicsandInformation Science, NanchangHangkongUniversity,NanchangJiangxi
Received:Jan.14
th
,2023;accepted:Feb.14
th
,2023;published:Feb.22
nd
,2023
Abstract
Inthispaper,westudya
E
-valuedualofhigheromniLiealgebroids.First,wedefine
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n
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,2023,13(2):234-243.
DOI:10.12677/pm.2023.132028
x
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-valuepairingandhigherDorfmanbracketonthedirectsumbundle
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where
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are,respectively,then-thdifferentialoperatorbundleandthejet
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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/
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·
K
1.1
[3]
é
u
E
þ
‚
5
n
-
•
þ
|
X
n
lin
(
E
)
Ú
‚
5
n
-
/
ª
Ω
n
lin
(
E
),
X
J
rankE
≥
2,
Š
â
á
Ü
S
(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.2023.132028237
n
Ø
ê
Æ
x
z
h
3.
p
Omni-
o
“
ê
E
-
é
ó
!
l
E
-
é
ó
m
*
g
Ñ
u
,
3
p
omni-
o
“
ê
(
D
E
⊕
J
n
E,
(
−
,
−
)
+
,
{−
,
−}
,ρ
)
Ä
:
þ
,
ï
Ä
p
omni-
o
“
ê
E
-
é
ó
(
D
n
E
⊕
J
E,
(
−
,
−
)
∗
+
,
{−
,
−}
∗
,ρ
∗
).
Š
â
©
z
[1]
¥
E
-
é
ó
m
½
Â
,
Œ
±
‰
Ñ
n
-
‡
¡
jet
m
J
n
E
†
n
-
‡
©
Ž
f
m
D
n
E
E
-
Š
é
)
Ò
h−
,
−i
.
½
Â
2.1
E
•
6
/
M
þ
•
þ
m
,
J
n
E
•
Ù
n
-
‡
¡
jet
m
,n-
•
þ
m
D
n
E
⊂
Hom
(
∧
n
J
n
E,E
)
D
E
¡
•
n
-
jet
m
J
n
E
E
-
é
ó
m
,
X
J
E
-
Š
é
h−
,
−i
: Γ(
J
n
E
)
×
Γ(
D
n
E
)
→
E
´
š
ò
z
.
©
z
[8],
Š
ö
ï
Ä
E
/
(Γ(
Hom
(
∧
•
J
E,E
)
D
E
)
,
d
J
)
ƒ
'
V
g
,
Ù
¥
d
J
•
þ
>
Ž
f
.
½
Â
2.2
[8]
é
?
¿
d
∈
Hom
(
∧
k
J
E,E
)
D
E
,
±
9
µ
1
,....,µ
n
∈
Γ(
J
E
),
N
d
J
:
Hom
(
∧
k
J
E,E
)
D
E
→
Hom
(
∧
k
+1
J
E,E
)
D
E
½
Â
•
(
d
J
d
)(
µ
1
,..µ
k
+1
) =
k
+1
X
i
=1
(
−
1)
i
+1
(
π
]
µ
i
)(
d
(
µ
1
,..
ˆ
µ
i
,..µ
k
+1
))
+
X
i<j
(
−
1)
i
+
j
d
([
µ
i
,µ
j
]
]
,µ
1
,..
ˆ
µ
i
,..
ˆ
µ
j
,..µ
k
+1
)
.
A
O
,
‰
½
˜
‡
o
“
ê
(
E,
[
−
,
−
]
E
,ρ
E
),
Œ
±
†
π
]
:
J
E
→
D
E
ƒ
'
þ
ó
E
/
(Γ(
Hom
(
∧
•
J
E,E
))
,
d
J
)
f
E
/
(Γ(
Hom
(
∧
•
J
E,E
)
D
E
)
,
d
J
),
Š
â
©
z
[9],
Œ
±
X
e
½
Â
½
Â
2.3
é
?
¿
d
∈
Hom
(
∧
k
J
E,E
)
D
E
,
±
9
µ,µ
1
,....,µ
n
∈
Γ(
J
E
),
N
L
µ
:
Hom
(
∧
k
J
E,E
)
D
E
→
Hom
(
∧
k
J
E,E
)
D
E
½
Â
•
(
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
)
.
¿
Ž
f
ι
µ
½
Â
•
:
ι
µ
d
(
µ
1
,..µ
k
−
1
) =
d
(
µ,µ
1
,..µ
k
−
1
).
Ï
d
,
é
?
¿
µ,ν
∈
Γ(
J
E
),
¤
é
A
o
f
L
µ
Ú
¿
Ž
f
ι
µ
÷
v
e
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)
½
n
2.4
-
E
n
(
E
) :=
D
n
E
⊕
J
E
,
K
p
omni-
o
“
ê
E
-
é
ó
(
E
n
(
E
)
,
(
−
,
−
)
∗
+
,
{−
,
−}
∗
,ρ
∗
)
(
X
e
:
(1)
‡
¡
V
‚
5
D
n
−
1
E
-
Š
é
(
−
,
−
)
∗
+
½
Â
•
(
d
+
µ,
r
+
ν
)
∗
+
= (
ι
ν
d
+
ι
µ
r
);(3.2)
DOI:10.12677/pm.2023.132028238
n
Ø
ê
Æ
x
z
h
(2)
Ù
þ
Dorfman
)
Ò
{−
,
−}
∗
: Γ(
E
n
(
E
))
×
Γ(
E
n
(
E
))
→
Γ(
E
n
(
E
))
½
Â
•
{
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
]
]
d
ª
(2.6)
‰
Ñ
.
(3)
e
N
ρ
∗
:
D
n
E
⊕
J
E
→
D
E
½
Â
•
ρ
∗
(
d
+
µ
) =
π
]
µ,
(3.4)
Ù
¥
?
¿
d
,
r
∈
Γ(
D
n
E
)
,µ,ν
∈
Γ(
J
E
).
4.
p
Omni-
o
V
“
ê
Double
©
z
[10]
¥
,
Š
ö
y
²
Ï
L
Ú
\
o
V
“
ê
Manintriples
,
Œ
±
3
Courant
-
“
ê
n
Ø
µ
e
e
E
o
V
“
ê
double
.
3
!
¥
,
Ï
L
4
Ù
Z
]“
ê
š
é
5
E
p
omni-
o
V
“
ê
double.
©
z
[11]
¥
,
Š
ö
0
4
Ù
Z
]“
ê
š
é
ƒ
'
V
g
,
¿
y
²
6
/
M
þ
˜
‡
n-
(
n>
2)
H
Ü
-
ä
Œ
'
(
½
Â
˜
é
š
4
Ù
Z
]“
ê
.
Ä
k
,
£
˜
e
4
Ù
Z
]“
ê
š
é½
Â
.
½
Â
3.1
[11](
A
1
,
{−
,
−}
1
,ρ
1
)
,
(
A
2
,
{−
,
−}
2
,ρ
2
)
´
M
þ
ü
‡
4
Ù
Z
]“
ê
,
e
¦
‚
†
Ú
m
A
1
⊕
A
2
•
¤
4
Ù
Z
]“
ê
(
(
A
1
⊕
A
2
,
{−
,
−}
A
1
⊕
A
2
,ρ
),
K
¡
(
A
1
,A
2
)
´
4
Ù
Z
]“
ê
A
1
Ú
A
2
š
é
,
…
÷
v
A
1
,A
2
´
A
1
⊕
A
2
4
Ù
Z
]
f
“
ê
.
d
©
z
[4]
Ú
·
K
2.5
Œ
•
,(
E
n
(
E
)
,
{−
,
−}
,ρ
),
Ú
(
E
n
(
E
)
,
{−
,
−}
∗
,ρ
∗
)
¤
ü
‡
4
Ù
Z
]“
ê
.
Ï
d
,
Š
â
4
Ù
Z
]“
ê
š
é½
Â
,
Œ
±
Ñ
X
e
½
Â
.
½
Â
3.2
(
E
n
(
E
)
,
{−
,
−}
,ρ
)
,
(
E
n
(
E
)
,
{−
,
−}
∗
,ρ
∗
)
´
ü
‡
4
Ù
Z
]“
ê
,
e
¦
‚
†
Ú
m
E
n
(
E
)
⊕E
n
(
E
)
•
¤
4
Ù
Z
]“
ê
(
(
E
n
(
E
)
⊕E
n
(
E
)
,
{−
,
−}
E
n
⊕E
n
,
˜
ρ
),
…
E
n
(
E
)
,
E
n
(
E
)
´
E
n
(
E
)
⊕E
n
(
E
)
4
Ù
Z
]
f
“
ê
,
K
¡
(
E
n
(
E
)
,
E
n
(
E
))
´
4
Ù
Z
]“
ê
E
n
(
E
)
Ú
E
n
(
E
)
š
é
.
Š
â
©
Ù
Œ
•
,˜
ρ
(
α
+
x
)=
ρ
(
α
) +
ρ
∗
(
x
),
…
{
α,β
}
E
n
⊕E
n
=
{
α,β
}
,
{
x,y
}
E
n
⊕E
n
=
{
x,y
}
∗
.
d
u
Γ(
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
,
d
d
,
¦
‚
©
O
p
Ñ
±
e‚
5
N
:
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.2023.132028239
n
Ø
ê
Æ
x
z
h
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
.
·
K
3.3
X
J
(
E
n
(
E
)
,
E
n
(
E
))
´
4
Ù
Z
]“
ê
E
n
(
E
)
Ú
E
n
(
E
)
š
é
,
@
o
•
3
4
Ù
Z
]“
ê
E
n
(
E
)
3
E
n
(
E
)
þ
L
«
(
ρ
L
∗
,ρ
R
∗
)
Ú
4
Ù
Z
]“
ê
E
n
(
E
)
3
E
n
(
E
)
þ
L
«
(
ρ
L
,ρ
R
),
…
é
?
¿
α,β
∈
Γ(
E
n
(
E
))
,x,y
∈
Γ(
E
n
(
E
))
÷
v
±
eƒ
N
^
‡
:
du
ρ
:(Γ(
E
n
(
E
))
,
{−
,
−}
)
→
(
X
(
M
)
,
[
−
,
−
])
Ú
ρ
∗
:(Γ(
E
n
(
E
))
,
{−
,
−}
∗
)
→
(
X
(
M
)
,
[
−
,
−
])
´
‡
4
Ù
Z
]“
ê
Ó
,
Ï
d
,
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
)];
Š
â
4
Ù
Z
]“
ê
E
n
(
E
)
Ú
E
n
(
E
)
¤
÷
v
4
Ù
Z
]
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
)
.
Ï
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E
n
(
E
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E
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(
E
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4
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]“
ê
š
é
,
3
†
Ú
•
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m
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n
(
E
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n
(
E
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k
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‡
4
Ù
Z
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ê
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(
E
n
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E
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⊕E
n
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,
{−
,
−}
E
n
⊕E
n
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ρ
),
Ù
4
Ù
Z
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)
Ò
•
{
α
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x,β
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y
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E
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n
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+
ρ
R
∗
(
y
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α
+
ρ
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∗
(
x
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β
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{
x,y
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∗
+
ρ
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(
α
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y
+
ρ
R
(
β
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x.
(4.1)
E
=
M
×
R
•
²
…
‚
5
m
ž
,
D
E
=
TM
×
R
,
J
E
=
T
∗
M
×
R
.
Š
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•
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m
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n
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f
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E
Ú
n
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‡
¡
jet
m
J
n
E
½
Â
,
·
‚
k
D
n
E
∼
=
∧
n
(
TM
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R
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J
n
E
∼
=
∧
n
(
T
∗
M
×
R
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Ï
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Ü
S
(2.5)
Œ
L
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:
0
/
/
∧
n
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M
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R
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M
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n
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1
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d
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[2]
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·
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3.4
[3]
E
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R
,
Ù
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‚
5
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lin
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‚
5
n
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n
lin
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k
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e
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DOI:10.12677/pm.2023.132028240
n
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x
z
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:
X
n
lin
(
M
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R
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Γ(
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∼
=
X
n
(
M
)
⊕
X
n
−
1
(
M
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∼
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(
M
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Ω
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1
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M
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(4.5)
K
E
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(
E
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M
×
R
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E
n
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E
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∧
n
(
TM
×
R
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⊕
T
∗
M
×
R
.
½
n
3.5
E
´
M
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²
…
‚
m
,
-
∆=
E
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E
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⊕E
n
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E
),
K
p
omni-
o
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−
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,
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E
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ρ
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X
e
:
(1)
E
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E
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⊕E
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E
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n
(
T
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M
×
R
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⊕
(
∧
n
(
TM
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R
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⊕
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M
×
R
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(2)
š
ò
z
V
‚
5
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∞
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−
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α
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Γ)
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h
ι
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α
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ξ
Λ
i
+
fg
h
α
n
−
1
,
Γ
i−h
ι
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α
n
−
1
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ξ
Γ
i
(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
)
Ò
½
Â
•
{
π
1
⊕
Φ
1
+Π
1
⊕
φ
1
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2
⊕
Φ
2
+Π
2
⊕
φ
2
}
E
n
⊕E
n
=
{
π
1
⊕
Φ
1
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2
⊕
Φ
2
}
E
n
+
ρ
R
∗
(Π
2
⊕
φ
2
)(
π
1
⊕
Φ
1
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ρ
L
∗
(Π
1
⊕
φ
1
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2
⊕
Φ
2
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L
(
π
1
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Φ
1
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2
⊕
φ
2
)+
ρ
R
(
π
2
⊕
Φ
2
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1
⊕
φ
1
)+
{
Π
1
⊕
φ
1
,
Π
2
⊕
φ
2
}
E
n
(4)
e
N
½
Â
•
˜
ρ
((
X,f
)
⊕
(
α
n
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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
).
y
²
Š
â
ª
(3.2),
Ï
L
O
Ž
Œ
((
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+
g
Γ
,
−
ι
ξ
Γ))
= (
h
ι
X
α
n
+
fα
n
−
1
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ξ
Λ+
g
Γ
i
,
−h
ι
X
α
n
−
1
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ξ
Γ
i
)
=
h
ι
X
α
n
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ξ
Λ
i
+
fg
h
α
n
−
1
,
Γ
i−h
ι
X
α
n
−
1
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ξ
Γ
i
=
Œ
y
(4.6).
d
ª
(4.1)
Œ
•
,
Ù
¥
{
π
1
⊕
Φ
1
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2
⊕
Φ
2
}
E
n
d
©
z
[3]
¤
‰
Ñ
,
{
Π
1
⊕
φ
1
,
Π
2
⊕
φ
2
}
E
n
d
ª
(3.3)
¤
‰
Ñ
,
¿
…
,
-
Pr
1
L
«
Γ(∆)
Γ(
E
n
(
E
))
Ý
,
Pr
2
L
«
Γ(∆)
Γ(
E
n
(
E
))
Ý
,
Š
â
p
DOI:10.12677/pm.2023.132028241
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Ø
ê
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x
z
h
‚
5
N
,
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
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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
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⊕
φ
2
,π
1
⊕
Φ
1
}
E
n
⊕E
n
,
ρ
R
(
π
2
⊕
Φ
2
)(Π
1
⊕
φ
1
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Pr
1
{
π
2
⊕
Φ
2
,
Π
1
⊕
φ
1
}
E
n
⊕E
n
.
é
u
e
N
˜
ρ
: ∆
→
D
E
½
Â
d
ª
(3.4)
‰
Ñ
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5.
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ê
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[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:
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[3]Lang,H.andSheng,Y.(2020)LinearizationoftheHigherAnalogueofCourantAlgebroids.
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[9]
.
ý
¬
,
‰
÷
7
,
•
û
³
.
š
†
omni-
o
“
ê
[J].
ê
Æ
?
Ð
,2022,51(5):879-890.
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Ø
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Æ
x
z
h
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Ø
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