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Jacobi Identities

"The" Jacobi identity is a relationship

[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0,,
(1)

between three elements A, B, and C, where [A,B] is the commutator. The elements of a Lie algebra satisfy this identity.

Relationships between the Q-functions Q_i are also known as Jacobi identities:

Q_1Q_2Q_3=1,
(2)

equivalent to the Jacobi triple product (Borwein and Borwein 1987, p. 65) and

Q_2^8=16qQ_1^8+Q_3^8,
(3)

where

q=e^(-piK^'(k)/K(k)),
(4)

K=K(k) is the complete elliptic integral of the first kind, and K^'(k)=K(k^')=K(sqrt(1-k^2)). Using Weber functions

f_1=q^(-1/24)Q_3
(5)
f_2=2^(1/2)q^(1/12)Q_1
(6)
f=q^(-1/24)Q_2,
(7)

(5) and (6) become

f_1f_2f=sqrt(2)
(8)
f^8=f_1^8+f_2^8
(9)

(Borwein and Borwein 1987, p. 69).

See also

Commutator, Jacobi Triple Product, Partition Function Q, Q-Function, Weber Functions

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References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 3, 1996.

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Jacobi Identities

Cite this as:

Weisstein, Eric W. "Jacobi Identities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JacobiIdentities.html

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