"The" Jacobi identity is a relationship
![[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0,,](/images/equations/JacobiIdentities/NumberedEquation1.svg) |
(1)
|
between three elements
,
, and
, where
is the commutator. The
elements of a Lie algebra satisfy this identity.
Relationships between the Q-functions
are also known as Jacobi identities:
![Q_1Q_2Q_3=1,](/images/equations/JacobiIdentities/NumberedEquation2.svg) |
(2)
|
equivalent to the Jacobi triple product (Borwein
and Borwein 1987, p. 65) and
![Q_2^8=16qQ_1^8+Q_3^8,](/images/equations/JacobiIdentities/NumberedEquation3.svg) |
(3)
|
where
![q=e^(-piK^'(k)/K(k)),](/images/equations/JacobiIdentities/NumberedEquation4.svg) |
(4)
|
is the complete
elliptic integral of the first kind, and
. Using Weber
functions
(5) and (6) become
![f_1f_2f=sqrt(2)](/images/equations/JacobiIdentities/NumberedEquation5.svg) |
(8)
|
![f^8=f_1^8+f_2^8](/images/equations/JacobiIdentities/NumberedEquation6.svg) |
(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.Referenced
on Wolfram|Alpha
Jacobi Identities
Cite this as:
Weisstein, Eric W. "Jacobi Identities."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiIdentities.html
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