EXTENSIONS OF THE JACOBI IDENTITY 5

standard A[ -modules in [10] — [12]. In the third subsection we exhibit the

Jacobi identity for relative twisted vertex operators (cf. Theorem 2.1) in the

case of the two operators Y^(va^ z

x

), Y^(vai z2), and we observe that one term

of this identity involves the "generalized powers" of vertex operators used

by A. Meurman and M. Prime in their approach to the construction of the

standard A[ -modules ([13], Section 5). Then in Subsection 3.4 we apply

a multi-operator extension of the Jacobi identity to determine the general

form in which a combination of products of Z-algebra operators and suitable

^-functions can be expressed in terms of a combination of products of the

Meurman-Primc operators and suitable ^-functions. This is the content of

Theorem 3.7. In Subsection 3.5, applying Theorem 3.7, we recover the gen-

erating function identities of [10] (see also [12], Theorems 12.10 and 12.13),

which give the Z-algebra relations for the standard A[ -modules and which

form the main part of the Lepowsky-Wilson interpretation of the generalized

Rogers-Jtamanujan identities (see [10] - [12]). In particular, comparing the

method of [12] and our method, we present a natural (in the sense of the

Jacobi identity) alternate interpretation of the numerical coefficients of the

Lepowsky-Wilson identities (Theorem 3.1).

The equivalence — hinted by A. B. Zamolodchikov and V. A. Fateev in

[14] and clarified by C. Y. Dong and J. Lepowsky in [2] - [4] — between the

notions of untwisted (homogeneous) Z-operator algebra (see [7] - [8]) and

of parafermion operator algebra (nonlocal current algebra) (see [14]) allows

us to establish an analogous equivalence between the twisted Z-operator

algebras of [10] — [12] and the representations of the parafermion algebra

constructed in another paper [15] by A. B. Zamolodchikov and V. A. Fateev.

Our algebraic point of view shows the structure of these representations as a

natural aspect of the Jacobi identity for (relative twisted) vertex operators.

The twisted Z-algebras (and the representations of the parafermion al-

gebra in [15]) are built from the twisted affine Lie algebra A[' (recalled in

Subsection 3.1 here) just as the untwisted Z-algebras of [7] - [8] are built from

the untwisted affine Lie algebra A\ . The positive integer k of the level k

(twisted) standard Ai -modules (see Subsection 3.2 here) corresponds to the

positive integer p of the [Zp]-symmetry (statistical mechanics) model in [15].

(For example, the levels 2 and 3 of the representations correspond, respec-

tively, to the Ising model and to the Potts model in [15].) The identities in

Corollaries 3.2 and 3.6 of the present paper correspond, respectively, to the