In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation.
Dalam matematika, identitas Jacobi adalah sifat dari operasi biner yang menjelaskan bagaimana urutan evaluasi, penempatan tanda kurung dalam beberapa produk, memengaruhi hasil operasi.
The identity is named after the German mathematician Carl Gustav Jacob Jacobi.
Identitas ini dinamai matentikawan asal Jerman Carl Gustav Jakob Jacobi.
The cross product a × b and the Lie bracket operation [ a , b ] both satisfy the Jacobi identity.
Produk silang a × b dan operasi braket Lie [ a , b ] keduanya memenuhi identitas Jacobi.
In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets.
Dalam mekanika analitik, identitas Jacobi menggunakan tanda kurung Poisson.
In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket.
Dalam mekanika kuantum, digunakan oleh operasi komutator dengan ruang Hilbert dan ekuivalen dalam perumusan ruang fase mekanika kuantum oleh kurung siku Moyal.
0. Notice the pattern in the variables on the left side of this identity.
0 ∀ x , y , z ∈ A {\displaystyle x\times (y\times z)\ +\ z\times (x\times y)\ +\ y\times (z\times x)\
In that notation, the Jacobi identity is:
Identitas Jacobi ditulis sebagai:
[ Z , X ] ] + [ Z , [ X , Y ] ]0 That is easily checked by computation.
[ x , [ y , z ] ] + [ z , [ x , y ] ] + [ y , [ z , x ] ]0 .
[ X , Y ]X Y − Y X belongs to V for all X , Y ∈ V , the Jacobi identity continues to hold on V. Thus, if a binary operation [ X , Y ] satisfies the Jacobi identity, it may be said that it behaves as if it were given by X Y − Y X in some associative algebra even if it is not actually defined that way.
A yang ditutup di bawah operasi kurung siku: [ X , Y ]X Y − Y X sebagai V untuk semua X , Y ∈ V , identitas Jacobi tetap menggunakan V. Maka, jika operasi biner
Using the antisymmetry property [ X , Y ]− [ Y , X ] , the Jacobi identity may be rewritten as a modification of the associative property:
Menggunakan sifat antisimetri [ X , Y ]− [ Y , X ] , identitas Jacobi dapat ditulis ulang sebagai modifikasi dari sifat asosiatif:
If [ X , Z ] is the action of the infinitesimal motion X on Z, that can be stated as: The action of Y followed by X (operator [ X ,
Jika [ X , Z ] adalah tindakan dari gerakan kecil X dengan Z, dapat dinyatakan sebagai: Tindakan Y diikuti oleh X sebagai operasi [ X ,
[ Y , ⋅ ] ] ), minus the action of X followed by Y (operator ( [ Y ,
[ Y , ⋅ ] ] , dikurangi tindakan X diikuti oleh Y sebagai operasi ( [ Y ,
[ X , ⋅ ] ] ), is equal to the action of [ X , Y ] , (operator [ [ X , Y ] , ⋅ ] ).
[ X , ⋅ ] ] , sama dengan tindakan [ X , Y ] sebagai operasi [ [ X , Y ] , ⋅ ] .
There is also a plethora of graded Jacobi identities involving anticommutators { X , Y } } , such as: [ { X , Y } , Z ] + [ { Y , Z } , X ] + [ { Z , X } , Y ]0 ,
Terdapat jumlah identitas Jacobi bertingkat yang melibatkan antikomutator { X , Y } } , contoh:
[ { X , Y } , Z ] + { [ Z , Y ] , X } + { [ Z , X ] , Y }
[ x , [ y , [ z , w ] ] ] + [ y , [ x , [ w , z ] ] ] + [ z ,
0. ,Z]+[\{Y,Z\},X]+[\{Z,X\},Y]=0,\qquad [\{X,Y\},Z]+\{[Z,Y],X\}+\{[Z,X],Y\}=0.} Most common examples of the Jacobi identity come from the bracket multiplication
0. ,Z]+[\{Y,Z\},X]+[\{Z,X\},Y]=0,\qquad [\{X,Y\},Z]+\{[Z,Y],X\}+\{[Z,X],Y\}=0.} Lihat pula Kurung siku Lie bidang vektor dan rumus Baker–Campbell–Hausdorff Contoh umum dari identitas Jacobi berasal dari perkalian kurung siku [ x , y ] dengan aljabar Lie dan gelanggang Lie.
The Jacobi identity is written as:
Identitas Jacobi ditulis sebagai:
[ x , [ y , z ] ] + [ z , [ x , y ] ] + [ y , [ z , x ] ]0.
[ x , [ y , z ] ] + [ z , [ x , y ] ] + [ y , [ z , x ] ]0 .
Defining the adjoint operator ad x : y ↦ [ x , y ] _{x}:y\mapsto [x,y]} , the identity becomes: ad x ⁡ [ y , z ][ ad x ⁡ y , z ] + [ y , ad x ⁡ z ] .
Mendefinisikan operasi adjoin ad x : y ↦ [ x , y ] _{x}:y\mapsto [x,y]} sebagai: ad x ⁡ [ y , z ]
_{x}[y,z]=[\operatorname {ad} _{x}y,z]+[y,\operatorname {ad} _{x}z].}
[ ad x ⁡ y , z ] + [ y , ad x ⁡ z ] _{x}[y,z]=[\operatorname {ad} _{x}y,z]+[y,\operatorname {ad} _{x}z]} .
That form of the Jacobi identity is also used to define the notion of Leibniz algebra.
Bentuk identitas Jacobi digunakan untuk mendefinisikan pengertian aljabar Leibniz.
Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation: ad [ x , y ][ ad x , ad y ] .
Penataan ulang lain menunjukkan bahwa identitas Jacobi setara dengan identitas berikut antara operator wakilan adjoin: ad [ x , y ][ ad x , ad y ] .
[x,y]}=[\operatorname {ad} _{x},\operatorname {ad} _{y}].} There, the bracket on the left side is the operation of the original algebra, the bracket on the right is the commutator of the composition of operators, and the identity states that the a d } map sending each element to its adjoint action is a Lie algebra homomorphism.
[x,y]}=[\operatorname {ad} _{x},\operatorname {ad} _{y}].} Di sana, kurung siku di sisi kiri adalah operasi dari aljabar asli, tanda kurung di sebelah kanan adalah komutator dari komposisi operator, dan identitas menyatakan bahwa a d } peta untuk setiap elemen ke tindakan adjoin adalah aljabar Lie homomorfisme.
Structure constants Super Jacobi identity Three subgroups lemma (Hall–Witt identity) Quintuple product identity Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666.
Konstanta struktur Identitas Jacobi super Tiga subgrup lemma (identitas Hall–Witt) Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (Edisi 2nd), Springer, ISBN 978-3319134666.