Associative algebras and intertwining operators

Publication date: 12 November 2021

Abstract: Let

V

be a vertex operator algebra and

A^{i n f t y} (V)

and

A^{N} (V)

for

N i n m a t h b b N

the associative algebras introduced by the author in [H5]. For a lower-bounded generalized

V

-module

W

, we give

W

a structure of graded

A^{i n f t y} (V)

-module and we introduce an

A^{i n f t y} (V)

-bimodule

A^{i n f t y} (W)

and an

A^{N} (V)

-bimodule

A^{N} (W)

. We prove that the space of (logarithmic) intertwining operators of type

for lower-bounded generalized

V

-modules

W_{1}

,

W_{2}

and

W_{3}

is isomorphic to the space

h o m_{A^{i n f t y} (V)} (A^{i n f t y} (W_{1}) o t i m e s_{A^{i n f t y} (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 i n m a t h b b N

, we also prove that the space of (logarithmic) intertwining operators of type

is isomorphic to the space of

h o m_{A^{N} (V)} (A^{N} (W_{1}) o t i m e s_{A^{N} (V)} O m e g a_{N}^{0} (W_{2}), O m e g a_{N}^{0} (W_{3}))

.

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