Associative Algebras and Intertwining Operators

• Published in: Communications in mathematical physics Vol. 396; no. 1; pp. 1 - 44
• Main Author: Huang, Yi-Zhi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2022; Springer Nature B.V
• Subjects: Algebra; Classical and Quantum Gravitation; Complex Systems; Logarithms; Mathematical and Computational Physics; Mathematical Physics; Modules; Operators (mathematics); Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-022-04457-z

Let V be a vertex operator algebra and A ∞ ( V ) and A N ( V ) for N ∈ N the associative algebras introduced by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras. arXiv:2009.00262 ). For a lower-bounded generalized V -module W , we give W a structure of graded A ∞ ( V ) -module and we introduce an A ∞ ( V ) -bimodule A ∞ ( W ) and an A N ( V ) -bimodule A N ( W ) . We prove that the space of (logarithmic) intertwining operators of type W 3 W 1 W 2 for lower-bounded generalized V -modules W 1 , W 2 and W 3 is isomorphic to the space Hom A ∞ ( V ) ( A ∞ ( W 1 ) ⊗ A ∞ ( V ) W 2 , W 3 ) . Assuming that W 2 and W 3 ′ are equivalent to certain universal lower-bounded generalized V -modules generated by their A N ( V ) -submodules consisting of elements of levels less than or equal to N ∈ N , we also prove that the space of (logarithmic) intertwining operators of type W 3 W 1 W 2 is isomorphic to the space of Hom A N ( V ) ( A N ( W 1 ) ⊗ A N ( V ) Ω N 0 ( W 2 ) , Ω N 0 ( W 3 ) ) .