By Cristiano Husu
This booklet extends the Jacobi id, the most axiom for a vertex operator algebra, to multi-operator identities. in keeping with buildings of Dong and Lepowsky, relative ${\mathbf Z}_2$-twisted vertex operators are then brought, and a Jacobi identification for those operators is verified. Husu makes use of those principles to interpret and get well the twisted Z -operators and corresponding producing functionality identities constructed by means of Lepowsky and Wilson for the development of the traditional $A^{(1)}_1$-modules. the perspective of the Jacobi identification additionally indicates the equivalence among those twisted Z-operator algebras and the (twisted) parafermion algebras built by way of Zamolodchikov and Fadeev. The Lepowsky-Wilson producing functionality identities correspond to the identities fascinated by the development of a foundation for the gap of C-disorder fields of such parafermion algebras.
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70) equals Y(Y(. 69) exist. 69) (with n replaced by m). 68) is seen by means of the coefficient of *m+l,m-l*m+l,m D Z H t,m+1> t=m+2 <*, 6, Cj £ Z . 72) •••V2,22lH,Zl)- where both sides of this identity are existing expressions. 73) the multi-operator Jacobi identity. 9. 9 and the following identities, which also show the equality of the two expressions. 1A) x2n_12 • • • • ] ] ^ i ) n ^ ( j=2 \ • • • [Y(v3,z31) £ L Z\ ^ ) J xZ32 Y(v2,z2i)] ••• = = r(y(r(n,,j nJ )y(vi,vy) • • • *>3, z32K *2iK *i)- (n^(^))(n^(^e)).
The next step of the proof will be to compute exp(Az*)A, and Y+(A, z) (cf. 48)). --Yz(ak,wk)t(l) JJ l EXTENSIONS OF THE JACOBI IDENTITY 23 Proof. We first show the existence of Y(Y(- • • Y(Y(vn, 2 n , n - l K - l , Z n - l . 67)). 8). 70) equals Y(Y(. 69) exist. 69) (with n replaced by m). 68) is seen by means of the coefficient of *m+l,m-l*m+l,m D Z H t,m+1> t=m+2 <*, 6, Cj £ Z . 72) •••V2,22lH,Zl)- where both sides of this identity are existing expressions. 73) the multi-operator Jacobi identity. 9. 9 and the following identities, which also show the equality of the two expressions. 1A) x2n_12 • • • • ] ] ^ i ) n ^ ( j=2 \ • • • [Y(v3,z31) £ L Z\ ^ ) J xZ32 Y(v2,z2i)] ••• = = r(y(r(n,,j nJ )y(vi,vy) • • • *>3, z32K *2iK *i)- (n^(^))(n^(^e)).