n-李代数与 n-泊松结构 n-Lie Algebra and n-Poisson Structure

doi:10.12677/PM.2023.132017

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PureMathematicsnØêÆ,2023,13(2),149-157

PublishedOnlineFebruary2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.132017

n−o“ê†n−Ñt(

oooZZZ

HÊ˜ŒÆêÆ†&E‰ÆÆ,ôÜH

ÂvFÏµ2022c1231F¶¹^FÏµ2023c130F¶uÙFÏµ2023c26F

Á‡

©lp ÝÑu§ÄkïÄn−o“ê(~ê§§´Š•o“êg,í2§´Ä¦{

$Ž•n−‚5$Ž˜«“êX Ú"ÙgÏL½Â 6/þn−Ñt)ÒÚÑn−Ñt(

½Â95Ÿ§n−o“ê†n−Ñt(˜˜éA'X"•3•þmþïÄ{ƒmþn−o

“ê§‰Ñn−o“ê{†n−ÑtN'X"

'…c

n−Ñt(§n−o“ê§n−o“ê

n−LieAlgebraandn−PoissonStructure

JiaLi

CollegeofMathematicsandInformationScience,NanchangHangkongUniversity,Nanchang

Jiangxi

Received:Dec.31

,2022;accepted:Jan.30

,2023;published:Feb.6

,2023

Abstract

Inthispaper,weﬁrststudythestructuralconstantsofn−Liealgebras,whichisa

naturalgeneralizationofLiealgebrasandanalgebraicsystemwhosebasicmultipli-

©ÙÚ^:oZ.n−o“ê†n−Ñt([J].nØêÆ,2023,13(2):149-157.

DOI:10.12677/pm.2023.132017

cationoperationsarelinearoperationsofn−elements.Secondly,thedeﬁnitionand

propertiesofn−Poissonstructurearederivedbydeﬁningthen−Poissonbracketona

manifoldandtheone-to-onecorrespondencebetween n−Liealgebrasand n−Poisson

structureisobtained.Finally,we studythen−Lie algebrasoncotangentbundles,and

give therelationbetween thecomorphismofn−Lie algebrasandn−Poisson mapping.

Keywords

n−PissonStructure,n−LieAlgebras,n−LieAlgebroids

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

n−o“ê[1,2](q¡•Filippov“ê,Nambu-Poisson“ê)´dc€éêÆ[Filippov

31985cJÑ,dõ“êXÚ@®•@/A^ÔnïÄ+•.3Nambu−åÆXÚ

¥[3,4] [f

,···,f

]=det(

∂f

∂x

)´•;.˜‡Ã•‘n−o“ê~f.CAcn−o“êk•2

•A^˜m,¦n−o“ê(œ„uÐ.3[5]¥k•õ'un−o“êSN.{¤

þ,Poisson(Ñt)•ïÄ;åÆXÚ$Ä•§ÚÅð½ÆÄ¯K,3Œ*ÿþ“êþ

Ú?˜‡V *Ò$Ž,53;åÆïÄ¥å-‡Š^,¡•Ñt)Ò.d[6]Ú\

Courant“ê˜‡š~-‡5Ÿ´§¡˜m´˜‡o2−“ê,¦Ñt(†p(—

ƒƒ'.^ponØ•{ïÄÑtAÛ¥¯K,Ø=Ur?‰ÆuÐ,„U•Ï·‚^

•p*:3•pgþ5*Ú)û¯K.n−o“ê´én−o“ê9ƒmí2,´3•

þmþïÄ.

©d[7]¥Šö‰Ño“ê†•þ˜m¥Ñt(˜˜éA'X,UYïÄn−o“ê

n−Ñt(éA'X.36/MþNM—î•þ|8Ü‘k•þ|)Ò$Ž¤o“

ê,K3C

∞

(M)þ•Œ±Ú\)Ò$Ž,=Ñt)Ò.éA 3p•þ|þn−Ñt)Ò.X

3•þmþÚ\o“ê½Â,UY?Ømþn−Ñt6/{ƒn−o“ê.

2.n−o“ê†n−Ñt(éA'X

½Â1.[

] ´•þ˜mgþ˜|Ä,i=1,···,dimg,K½Â˜‡n−o“ê(~ê,X

e[

,···,

]

k=1

···i



DOI:10.12677/pm.2023.132017150nØêÆ

½Â2.[1]˜‡n−o“ê´˜‡•þ˜mg±9˜‡n−-‚5‡é¡)Ò$Ž[·,···,·]

∧

g→g,¦é?¿

,

∈g,±eÄðª¤á:

[

,···,

n−1

,[

,···,

]]

[

,···,

k−1

,[

,···,

n−1

,

],

k+1

,···,

](1)

½Âad: ∧

n−1

g→gl(g) Xe,



,···,

n−1

: 

→[

,···,

n−1

,

],∀

,

∈g(2)

Keq:n-Liealgebraduad



,···,

n−1

´˜‡f,=



,···,

n−1

[

,···,

]

k=1

[

,···,ad



,···,

n−1



,

]

.(3)

·K1.éu˜‡n−o“ê,Ù(~ê÷vª

···j

···i

n−1

k=1

···i

n−1

···j

k−1

,l,j

k+1

···j

y²

[

,···,

n−1

l=1

···j



] =

k=1

[

,···,

k−1

l=1

···i

n−1



,···,

]

l=1

m=1

···j

···i

n−1



k=1

l=1

m=1

···i

n−1

···j

k−1

,l,j

k+1

···j



···j

···i

n−1

k=1

···i

n−1

···j

k−1

,l,j

k+1

···j

½Â3.[8]6/Mþn−Ñt)Ò´˜‡n−-‚5N{.,...,.}: C

∞

(M)×...×C

∞

(M) −→

∞

(M)÷v:

(1)‡é¡:é?¿f

∈C

∞

(M),(1 ≤i≤n),σ∈S

´né¡+),

,···,f

}= (−1)

ε(σ)

σ(1)

,···,f

σ(n)

(2)4ÙZ[5Ÿ:é?¿f

,g∈C

∞

(M),(1 ≤i≤n),

g,f

,···,f

}= f

{g,f

,···,f

}+g{f

,···,f

(3)Äðª:é?¿f

∈C

∞

(M),(1 ≤i≤n−1,1 ≤j≤n),

,···,f

n−1

,{g

,···,g

}}=

j=1

,···,{f

,···,f

n−1

},···,g

DOI:10.12677/pm.2023.132017151nØêÆ

XJ{.,...,.}´˜‡n−Ñt)Ò,K•3n−•þ|π∈Γ(∧

TM) ÷vπ(df

∧···∧df

,···,f

},¡•n−Ñt•þ|.3Ä[

]e,n−Ñt•þ|Œ±L«•

π=

···i



∂

∂

∧···∧

∂

∂

K‚5n−Ñt(¤(½(~ê{C

···i

}=•n−o“ê((~ê.

ƒ‡géó˜mg

∗

Ä•[µ

],¦<µ

,

>= δ

#Kg

∗

þ¼êf•:

df|

i=1

∂f

∂

d

∈T

∗

∀f∈C

∞

∗

),µ∈g

∗

5¿gÄ•{

,···,

},T

∗

kÄ{d

,···,d

},§‚ƒmŒ±ïáÓ'XT

∗

→g,

d

7→

,∀k= 1,···,n,

,···,f

}(µ) =<µ,[df

,···,df

] >.

Ù¥<,>L«éó˜mƒmé$Ž,Ïd

,···,f

}(µ) =

k=1

∂f

∂

,···,

∂f

∂

···i



(µ).

AOég

∗

þ‹I¼êk

{

,···,

}(µ) =

···i



(µ).

ddg

∗

þn−‚5Ñt(,Œ±±e½n.

½n1•þ˜m†Ùéó˜mþn−o“ê(ƒmkg,˜˜éA'X,¦ü«(k

ƒÓ(~ê.

e¡‰Ñn−Ñt(d^‡.

½n2 e{.,...,.}´˜‡n−Ñt)Ò,KkL

...f

n−1

Π = 0

y²



...f

n−1



(dg

,...,dg

)

= L

...f

n−1

(Π(dg

,...,dg

)−

i=1



,...,L

...f

n−1

,...,dg



= L

...f

n−1

({g

,...,g

})−

i=1

,...,L

...f

n−1

,...,g

}

= {f

,...,f

n−1

,{g

,...,g

}}−

i=1

,...,{f

,...,f

n−1

},...,g

DOI:10.12677/pm.2023.132017152nØêÆ

(M,π) ´n−‘Ñt6/,§)Ò•{f

,···,f

}=π(df

,···,d

).Œ±pÑ˜‡N

]

: ∧

n−1

(M) →X(M),½ÂXe:

,···,f

n−1

,g}= π(df

,···,d

n−1

,dg) =<X

,···,f

n−1

,dg>

Ù¥X

,···,f

n−1

´dπ(½M—î•þ|

3.n−Ñt6/þ{ƒn−o“ê

½Â4[8]M´˜‡6/,f:E→M´•þm,6/Mþ˜‡n−o“ê´˜‡n

|(E,ρ,M),Ù¥ρ:∧

n−1

E→TM´mN,…3Γ(∧

n−1

E)ÚX(M)ƒmpo“êÓ,

=é?¿f∈C

∞

(M),X

,···,X

n−1

,Y∈Γ(E),k

[ρ(X

∧···∧X

n−1

),ρ(Y

∧···∧Y

n−1

)] =

n−1

i=1

ρ(Y

∧···[X

,···,X

n−1

]∧···∧Y

n−1

,···,X

n−1

,fY] = f[X

,···,X

n−1

,Y]+ρ(X

∧···∧X

n−1

)(f)Y.

ρ¡•eN.

(M,π)´˜‡n−Ñt6/,ƒmV=TMþ•3˜‡n−Ñt(π

¦é?¿

,···,f

k{f

,···,f

}

={f

,···,f

}

,ùpπ

´˜‡n−Ñt(,KŒ±3éóm

∧

n−1

∗

Mþ½Â˜‡n−o“ê,Ù¥eN÷v

ρ(df

,···,df

n−1

) = X

,···,f

n−1

= π

]

(df

,···,df

n−1

),ρ= π

]

: ∧

n−1

∗

→TM.

·K2.3n−o“ê¡þ)Ò÷v[df

,···,df

n−1

] = d{f

,···,f

y²φ

,φ

[df

,···,df

]

={φdf

,···,φdf

}={f

,···,f

}={df

,···,df

}

d{f

,···,f

}

α

,···,α

n−1

,α

∈Ω

(M)K½Â˜‡n−{ƒmN:

]

(α

∧···∧

n−1

)(α

) = π(α

,···,α

n−1

,α

½Â)Ò

{α

,···,α

n−1

,α

}= L

]

(α

∧···∧α

n−1

)

−L

]

(α

)

(α

,···,α

n−1

)−dΠ(α

,···,α

n−1

,α

N´

{α

,···,α

n−1

,fα

}= f{α

,···,α

n−1

,α

}+Π

]

(α

∧···∧α

n−1

)(f)α

·‚{ƒmn−o“ê.

DOI:10.12677/pm.2023.132017153nØêÆ

½Â5 ˜‡•þm{÷vXe†ã



: M

→M

´ÄN,‰½˜‡•þm{3¡þŒ±.£N

∗

: Γ(E

) →Γ(E

)(4)

½Â6E

→M

´ü‡n−o“ê,eΦ

: E

´˜‡n−o“ê{,

´˜‡•þm{XJ÷v:

(1).£N(4))Ò5Ÿ,

(2)eN÷v:ρ

(Φ

∗

∧···∧Y

n−1

)) ∼

∧···∧Y

n−1

)Œ±Xe†ã



TΦ

E

¡´Y

,···,Y

n−1

,f´M

þ1w¼ê,KkΦ

∗

,···,Y

n−1

,fY

]

= [Φ

∗

,···,Φ

∗

n−1

,(Φ

∗

f)Φ

∗

],KÏLn−o“ê½Âúª:

((Φ

∗

(ρ

∧···∧Y

n−1

)f))−ρ

(Φ

∗

∧···∧Y

n−1

))(Φ

∗

f))Φ

∗

= 0

ùy²(Φ

∗

(ρ

∧···∧Y

n−1

)f)) = ρ

(Φ

∗

∧···∧Y

n−1

))(Φ

∗

f).

½n3 E

→M

´ü‡n−o“ê,•þm{Φ

: E

´˜‡n−“ê

{…=éóNΦ

∗

: E

∗

→E

∗

´˜‡n−ÑtN.

y²éu{Φ

: E

Ú?¿Y

,···,Y

n−1

∈Γ(E

),k

∗

= Φ

∗

‰½¡Y

,···,Y

n−1

∈Γ(E

),f∈C

∞

),k

∗

,···,Y

n−1

]

= Φ

∗

,···,Y

n−1

]

= Φ

∗

{φ

,···,φ

n−1

,φ

[Φ

∗

,···,Φ

∗

n−1

,Φ

∗

]

= {φ

∗

,···,φ

∗

n−1

,φ

∗

}= {Φ

∗

,···,Φ

∗

n−1

,Φ

∗

(ρ

((Y

∧···∧Y

n−1

))f) = Φ

∗

(ρ

((Y

∧···∧Y

n−1

))f) = Φ

∗

{φ

,···,φ

n−1

∗

f.}

DOI:10.12677/pm.2023.132017154nØêÆ

ùª†>un−o“ê,m>un−Ñt.

½Â7Φ:M

→M

´1wN ,n−•þ|ϑ∈X

) †•þ|ζ∈X

) ¡•´

Φ−ƒ'XJ÷v:

Φ(x)

= (d

Φ)

∗

,∀x∈M

dΦ ‡©Œ±pN(d

Φ)

∗

: ∧

→∧

Φ(x)

,ζ= Φ

∗

ϑ.

·K3.(M

,π

)Ú(M

,π

)´ü‡n−Ñt6/,‰½1wNΦ:M

→M

,ŒX

ed'X:

(1)π

†π

´Φƒ'.

(2)éux∈M

÷vXe†ã

∧

n−1

∗

Φ)

∗

]



∧

n−1

∗

Φ(x)

]



Φ)

∗

Φ(x)

y²

((dΦ)

∗

)(α

,···,α

n−1

,α

) = Π

(α

,α

,···,α

)

⇔((dΦ)

∗

)(Π

]

((dΦ)

∗

(α

,···,α

n−1

))) = α

(Π

]

(α

,···,α

n−1

))

⇔α

(dΦ(Π

]

((dΦ)

∗

(α

,···,α

n−1

)))) = α

(Π

]

(α

,···,α

n−1

)).

½Â8 E→M´n−o“ê,÷XN⊆M•3•þfmF⊆E,XJ÷v:

(1)E¡´X

,···,X

n−1

,Y,•›3NþF¡3)Ò[·,·],

(2)ρ(F) ⊆TN.KF¡•˜‡n−of“ê.

·K4.XJF⊆E´÷XN⊆M˜‡n−of“ê,KF7kn−o“ê(,éA

eN•ρ: E→TN,)Ò$Ž•:

,···,X

n−1

,Y|

] = [X

,···,X

n−1

,Y]|

,···,X

n−1

,Y|

∈Γ(F).

y²y²)Ò´û½Â,Y|

= 0,K[X

,···,X

n−1

,Y]|

= 0.PY= Σ

∈C

∞

(M)

36/Nþu0,K

,···,X

n−1

,Y]

,···,X

n−1

+(ρ(X

∧···∧X

n−1

= 0.

ùpρ(X

∧···∧X

n−1

= 0,Ï•ρ(X

∧···∧X

n−1

)†Nƒƒ.

E→M´˜‡n−o“ê,N⊆M´˜‡f6/.ρ

−1

(TN) ⊆E´˜‡1wfm.K

†Nƒƒ•þ|o)ÒE†Nƒƒ,ρ

−1

(TN)⊆E´÷XN⊆M˜‡n−of“ê

DOI:10.12677/pm.2023.132017155nØêÆ

.

i: N→M´˜‡i\N,Ài

E:= ρ

−1

(TN)´•›3Nþ˜‡n−o“êû½Â,±

eü«AÏœ¹:

(1)XJρ†Nƒƒ(ie.ρ(E|

) ⊆TN),Ki

E= E|

(2)XJρ†Nî(ρ(E)+Φ

∗

(TN) = TM),Ki

E´˜‡û½Â,

rank(i

E) = rank(E)−dim(M)+dim(N),i

TM= TN.

½Â9XJE

→M

´ü‡n−o “ê,˜‡•þmNΦ

→E

´˜‡

n−o“êXJ§ãGr(Φ

) ⊆E

×E

−

´÷XGr(Φ

)˜‡n−of“ê.

˜‡n−Ñt•þ|π??†Nƒƒ,Kf6/N⊆M´˜‡n−Ñtf6/.½Â˜‡•þ

|π

∈∧

TN⊆∧

TM,éuNj: N→M,Kπ

∼

π,éAn−Ñt)ÒXe:

∗

,···j

∗

}

= j

∗

,···f

½n4Xe^‡d:

(1)N´˜‡n−Ñtf6/.

(2)Π

]

(∧

n−1

∗

) ⊆TN.

(3)¤kM—î•þ|X

,···,f

n−1

,(f

,···,f

n−1

) ∈C

∞

(M)†Nƒƒ.

(4)3n−Ñt)Òe,¼êf

,···,f

n−1

= 0´˜‡n−o“ênŽ.

y²(1),(2)w,d,^‡(3)¤áž,K36/N?u0¼ê´nŽ,Kg|

= 0žk

,···,f

n−1

,g}|

,···,f

n−1

(g)|

=0 Ï•X

,···,f

n−1

†Nƒƒ.(4)y,ƒ‡XJ(4)¤á,

Kg|

= 0žk{f

,···,f

n−1

,g}|

= 0,=<dg,X

,···,f

n−1

= X

,···,f

n−1

(g)|

= 0,y..

ë•©z

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