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 Group satisfy this identity.

Relationships between the Q-Function $Q_i$ 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

$${Q_2}^8=16q{Q_1}^8+{Q_3}^8,$$ (3)

where

$$q\equiv e^{-\pi K'(k)/K(k)},$$ (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, Q-Function, Weber Functions

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.