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Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a derivation. That form of the Jacobi identity is also used to define the notion of Leibniz algebra. Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation:
Carl Gustav Jacob Jacobi (/ dʒ ə ˈ k oʊ b i /; [2] German:; 10 December 1804 – 18 February 1851) [a] was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory.
In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x {\displaystyle x} and y {\displaystyle y} is denoted [ x , y ] {\displaystyle [x,y]} .
This is a representation on an algebra: the (anti)derivation property is the super Jacobi identity. If a vector space is both an associative algebra and a Lie algebra and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a ...
In Lie algebras, the multiplication satisfies Jacobi identity instead of the associative law; this allows abstracting the algebraic nature of infinitesimal transformations. Other examples are quasigroup, quasifield, non-associative ring, and commutative non-associative magmas.
The Schouten–Nijenhuis bracket makes the multivector fields into a Lie superalgebra if the grading is changed to the one of opposite parity (so that the even and odd subspaces are switched), though with this new grading it is no longer a supercommutative ring. Accordingly, the Jacobi identity may also be expressed in the symmetrical form
Thanks to the Jacobi Identity, the three-dimensional cross product gives the structure of a Lie algebra, which is isomorphic to (), the Lie algebra of the 3d rotation group. Because the Jacobi identity fails in seven dimensions, the seven-dimensional cross product does not give R 7 {\displaystyle \mathbb {R} ^{7}} the structure of a Lie algebra.
Jacobi coordinates, a simplification of coordinates for an n-body system; Jacobi identity for non-associative binary operations; Jacobi's formula for the derivative of the determinant of a matrix; Jacobi triple product, an identity in the theory of theta functions; Jacobi's theorem (disambiguation), several theorems