Jacobi identity

In mathematics , a bilinear mapping on the vector space fulfills the Jacobi identity (according to Carl Jacobi ) if: ${\displaystyle F\colon V\times V\rightarrow V}$ ${\displaystyle V}$

${\displaystyle F(F(x,y),z)+F(F(y,z),x)+F(F(z,x),y)=0}$

for everyone . ${\displaystyle x,y,z\in V}$

If the bilinear mapping is also antisymmetric, it is a Lie bracket . Important examples are the commutator of linear maps, the vector product, and the Poisson bracket .

Other spellings

Let it be in the following

${\displaystyle [{\cdot },{\cdot }]\colon V\times V\to V,\quad (x,y)\mapsto [x,y]}$

an alternating bilinear map. The Jacobi identity is then equivalent to the fact that this mapping defines the structure of a Lie algebra on . ${\displaystyle V}$

Then the Jacobi identity can be rewritten in the following ways:

• ${\displaystyle [x,[a,b]]=[[x,a],b]+[a,[x,b]]}$

In other words: the picture

${\displaystyle a\mapsto [x,a]}$

is a derivation regarding the product . ${\displaystyle [{\cdot },{\cdot }]}$

• ${\displaystyle [[a,b],x]=[a,[b,x]]-[b,[a,x]]}$

In other words: with the notation

${\displaystyle \mathrm {ad} (a)\colon V\to V,\quad x\mapsto \mathrm {ad} (a)(x)=[a,x]}$

applies

${\displaystyle \mathrm {ad} ([a,b])=[\mathrm {ad} (a),\mathrm {ad} (b)];}$

where the bracket on the right is the commutator in the endomorphism algebra of . In other words: the picture ${\displaystyle V}$

${\displaystyle \mathrm {ad} \colon V\to {\mathfrak {gl}}(V)=\mathrm {End} \,V,\quad a\mapsto \mathrm {ad} (a)}$

is a representation of the Lie algebra on itself. It is called the adjoint representation . ${\displaystyle V}$

swell

• Jacobi identity . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .