Jacobi identity

Jacobi identity is an identity satisfied by a bilinear operation $[\cdot ,\cdot ]\colon V\times V\rightarrow V$ ${\ displaystyle [\ cdot, \ cdot] \ colon V \ times V \ rightarrow V}$ ${\ displaystyle [\ cdot, \ cdot] \ colon V \ times V \ rightarrow V}$ in linear space $V$ ${\ displaystyle V}$ $V$ under the following condition:

\forall \,x,y,z\in V\colon [[x,y],z]+[[y,z],x]+[[z,x],y]=0

{\ displaystyle \ forall \, x, y, z \ in V \ colon [[x, y], z] + [[y, z], x] + [[z, x], y] = 0}

{\ displaystyle \ forall \, x, y, z \ in V \ colon [[x, y], z] + [[y, z], x] + [[z, x], y] = 0}

Named after Carl Gustav Jacobi .

The concept of Jacobi identity is usually associated with Lie algebras .

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Examples

The following operations satisfy the Jacobi identity:

Operator Switch
commutator in Lie algebra
Brackets Lee Vector Fields
Poisson brackets of functions on a symplectic manifold
Vector product of vectors

Value in Lie Algebras

If multiplication $[\cdot ,\cdot ]$ ${\ displaystyle [\ cdot, \ cdot]}$ anticommutative , then the Jacobi identity can be given a slightly different form using the adjoint representation of the Lie algebra :

\mathrm {ad} _{x}\colon y\mapsto [x,y]

{\ displaystyle \ mathrm {ad} _ {x} \ colon y \ mapsto [x, y]}

Having written Jacobi's identity in the form

[x,[y,z]]=[y,[x,z]]+[[x,y],z]

{\ displaystyle [x, [y, z]] = [y, [x, z]] + [[x, y], z]}

we get that it is equivalent to the condition for the Leibniz rule for the operator $\mathrm {ad} _{x}$ ${\ displaystyle \ mathrm {ad} _ {x}}$ :

\mathrm {ad} _{x}\,[y,z]=[\mathrm {ad} _{x}\,y,z]+[y,\mathrm {ad} _{x}\,z]

{\ displaystyle \ mathrm {ad} _ {x} \, [y, z] = [\ mathrm {ad} _ {x} \, y, z] + [y, \ mathrm {ad} _ {x} \ , z]}

In this way, $\mathrm {ad} _{x}$ ${\ displaystyle \ mathrm {ad} _ {x}}$ Is a differentiation in a Lie algebra. Any such differentiation is called internal .

Jacobi’s identity can also be given the form

\mathrm {ad} _{[x,y]}=[\mathrm {ad} _{x},\mathrm {ad} _{y}]=\mathrm {ad} _{x}\mathrm {ad} _{y}-\mathrm {ad} _{y}\mathrm {ad} _{x}

{\ displaystyle \ mathrm {ad} _ {[x, y]} = [\ mathrm {ad} _ {x}, \ mathrm {ad} _ {y}] = \ mathrm {ad} _ {x} \ mathrm {ad} _ {y} - \ mathrm {ad} _ {y} \ mathrm {ad} _ {x}}

This means that the operator $\mathrm {ad}$ ${\ displaystyle \ mathrm {ad}}$ defines a homomorphism of a given Lie algebra into the Lie algebra of its derivations.

Jacobi Graduated Identity

Let be $\Omega =\oplus _{i}\Omega ^{i}$ ${\ displaystyle \ Omega = \ oplus _ {i} \ Omega ^ {i}}$ - graded algebra , $[\cdot ,\cdot ]$ ${\ displaystyle [\ cdot, \ cdot]}$ - multiplication in it. They say that multiplication in $\Omega$ ${\ displaystyle \ Omega}$ satisfies the graded Jacobi identity if for any elements $\omega _{i}\in \Omega ^{i}$ ${\ displaystyle \ omega _ {i} \ in \ Omega ^ {i}}$

[\omega _{m},[\omega _{k},\omega _{l}]=[[\omega _{m},\omega _{k}],\omega _{l}]+(-1)^{mk}[\omega _{k},[\omega _{m},\omega _{l}]

{\ displaystyle [\ omega _ {m}, [\ omega _ {k}, \ omega _ {l}] = [[\ omega _ {m}, \ omega _ {k}], \ omega _ {l} ] + (- 1) ^ {mk} [\ omega _ {k}, [\ omega _ {m}, \ omega _ {l}]}

Examples

algebra of external forms ;
differentiation algebra of differential forms ;
algebra of tangential-valued forms with multiplication defined by FN- brackets or NR- brackets;

Jacobi identity

Content

Examples

Value in Lie Algebras

Jacobi Graduated Identity

Examples

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