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
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