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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:
The book introduces Jacobi elliptic functions and the Jacobi triple product identity. One of the most exciting moments of my life was when, after computing several of these series, I went down to our mathematical library and found some of them in Jacobi's "Fundamenta nova theoriae..."[3], with the same coefficients down to the last decimal digit!
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]} .
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.
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 vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
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