Generalized vertex algebras and relative vertex operators
Generalized vertex algebras and relative vertex operators
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DOI：
10.1007/978-1-4612-0353-7
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发表时间：
1993
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影响因子：
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通讯作者：
C. Dong;J. Lepowsky
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文献类型：
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作者：
C. Dong;J. Lepowsky
现在我们回到第五章的背景。广义顶点算子代数的公理并不完全适用于Ω*的结构，真空空间(3.29)(对于Heisenberg代数(ĥ*)z)配备相对顶点算子Y*。事实上，有限维公理(6.4)和有界性公理(6.5)一般不成立，因为L1不一定是正定的。此外，雅可比恒等式(5.11)(定理5.1)与雅可比恒等式(6.12)的不同之处在于插入了因子(ā，b)。当然，还必须定义一个合适的groupG。我们想要扩展我们的公理，以便包括这个结构。然后，第7章的对偶论点的相应推广将提供Jacobi恒等式(5.11)(回想备注7.18)的另一种基于对偶的证明，特别是[FLM3]的定理8.6.1和8.8.23。(上面第7章、[FLM3]附录和[FHL]中的论点不够普遍，不足以给出[FLM3]的定理8.8.23的对偶证明-其中h*=0和<；ā，b>；？∈Z的特例(5.11)-因为该定理中的有理格不假定为偶数或整数，并且具有交替形式c(·，·)；在命题7.16的证明中，向量v3是任意的。)
Now we return to the setting of Chapter 5. The axioms for a generalized vertex operator algebra do not quite apply to the structure of Ω*, the vacuum space (3.29) (for the Heisenberg algebra (ĥ*)z) equipped with the relative vertex operatorsY*. In fact, the finite-dimensionality axiom (6.4) and the boundedness axiom (6.5) fail to hold in general, sinceLis not necessarily positive definite. Moreover, the Jacobi identity (5.11) (Theorem 5.1) differs from the Jacobi identity (6.12) by the insertion of the factorc(ā,b). Of course, a suitable groupGwould also have to be defined. We would like to extend our axioms so as to include this structure. Then the corresponding generalization of the duality arguments of Chapter 7 would provide an alternate duality-based proof of the Jacobi identity (5.11) (recall Remark 7.18), and in particular, of Theorems 8.6.1 and 8.8.23 of [FLM3]. (The arguments in Chapter 7 above, in the Appendix of [FLM3] and in [FHL] are not sufficiently general to give a duality-based proof of Theorem 8.8.23 of [FLM3] — the special case of (5.11) in whichh*= 0 and <ā,b> ? ∈Z— because the rational lattice in that theorem is not assumed even or integral, and has an alternating formc(·,·); in the proof of Proposition 7.16, the vectorv3is arbitrary.)