In mathematics , a bilinear mapping on the vector space fulfills the Jacobi identity (according to Carl Jacobi ) if:
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![{\ displaystyle F (F (x, y), z) + F (F (y, z), x) + F (F (z, x), y) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/689c1a3a211628996b737c3dafdccb3b702e15b3)
for everyone .
![{\ displaystyle x, y, z \ in V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66794d289d37b67c3857e7370ecdff292a909f34)
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
![[{\ cdot}, {\ cdot}] \ colon V \ times V \ to V, \ quad (x, y) \ mapsto [x, y]](https://wikimedia.org/api/rest_v1/media/math/render/svg/765d313ce6d2cd24a550507b1bbbc2794ec71c1c)
an alternating bilinear map. The Jacobi identity is then equivalent to the fact that this mapping defines the structure of a Lie algebra on .
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
Then the Jacobi identity can be rewritten in the following ways:
![[x, [a, b]] = [[x, a], b] + [a, [x, b]]](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb74cc07ffe0cb15694af74f27ca0befe095c931)
- In other words: the picture
![a \ mapsto [x, a]](https://wikimedia.org/api/rest_v1/media/math/render/svg/66194653c7a1c3a403d90582dc049ad73499c69d)
- is a derivation regarding the product .
![[{\ cdot}, {\ cdot}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/e58dc6e37720306f73f8f451fdf795d655f6a1f6)
![[[a, b], x] = [a, [b, x]] - [b, [a, x]]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6213a8642649de088de0ffdc1f1d34b6d4c4ba39)
- In other words: with the notation
![\ mathrm {ad} (a) \ colon V \ to V, \ quad x \ mapsto \ mathrm {ad} (a) (x) = [a, x]](https://wikimedia.org/api/rest_v1/media/math/render/svg/17386dd8a3e69f1309949df27799a6f11849a815)
- applies
![\ mathrm {ad} ([a, b]) = [\ mathrm {ad} (a), \ mathrm {ad} (b)];](https://wikimedia.org/api/rest_v1/media/math/render/svg/8192799c7c7af42c9e11f3ca58277fbdd2296886)
- where the bracket on the right is the commutator in the endomorphism algebra of . In other words: the picture
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![\ mathrm {ad} \ colon V \ to \ mathfrak {gl} (V) = \ mathrm {End} \, V, \ quad a \ mapsto \ mathrm {ad} (a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/dad776967126b23fcf94882c0af1d62f4c553dfa)
- is a representation of the Lie algebra on itself. It is called the adjoint representation .
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
swell
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Jacobi identity . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 978-3-8274-0439-8 .