So far, I've seen two ways of deriving the Jacobi Triple Product (JTP) formula $$\sum_{m \in \mathbb{Z}} (-u)^m q^{\frac{m(m-1)}{2}}= \prod_{j=0}^{\infty} (1-uq^{j})(1-q^{j+1})(1-u^{-1} q^{j+1})$$ using infinite-dimensional representation theory. On the one hand, it is the Weyl-Kac denominator formula for $\widehat{\mathfrak{sl}_2}$. On the other hand, it is the character $$\text{Tr}(q^{L_0} u^{\alpha_0})$$ computed in two ways using the boson-fermion correspondence. Here $$\alpha_0 = \sum_{i \in \mathbb{Z}'} : \psi_i \psi_i^* :$$ is the zero mode of the Heisenberg algebra which computes the charge of the Dirac sea of free fermions, and $$L_0 = \sum_{i \in \mathbb{Z}'} i : \psi_i \psi_i^* :$$ is the energy operator for a particular representation of the Virasoro algebra on this semi-infinite wedge space.

- Is there a structural relationship between these two appearances of JTP?

Infinite-Dimensional Lie Algebrasby M. Wakimoto, in an AMS translation series (2001). The language tends to be quirky and informal. The Jacobi identity arises from affine Lie algebra representations in Chapter 3; then in Chapter 6 some connections are worked out with the Virasoro algebra. One quote, "In the mid-1980's, The Virasoro algebra fell in passionate love with affine Lie algebras." But I don't see an explicit connection made with Jacobi identity there. $\endgroup$ – Jim Humphreys Feb 27 '13 at 18:48