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P(z)-tensor product in a module category
Statement
Let A be an abelian group, let à contain A as a subgroup, and let V be a strongly A-graded Möbius or conformal vertex algebra. Let C be a specified full category of strongly Ã-graded generalized V-modules, with the grading-preserving morphisms commuting with the Möbius operators specified by the P(z)-map supplier. For z∈C×, write for the vector space of P(z)-intertwining maps of type , as defined by the P(z)-intertwining-map convention. A P(z)-tensor product of W1,W2∈C is an object T∈C and a P(z)-intertwining map ι:W1⊗W2→T̄ such that for every X∈C, composition with ι gives a natural bijection Hom_C(T,X)→. Existence is an additional theorem or hypothesis, not part of the definition.
Statement references
Facts & Assumptions
Given: Work over C with the formal-variable, grading, and branch conventions in the linked definitions.
P(z)-intertwining map: Let A be abelian, Ã contain A as a subgroup, V be a strongly A-graded Möbius or conformal vertex algebra, and W1,W2,W3 be strongly Ã-graded generalized V-modules with the specified generalized-weight decompositions and Möbius operators. Fix . A P(z)-intertwining map of type (W3;W1,W2) is a linear map whose Ã-degrees add, for which for all sufficiently large integers m for each fixed h,w1,w2, and which satisfies the P(z)-Jacobi identity and, for . There is no L(-1)-derivative axiom for I. Formal variables x0,x1 are independent; z is a fixed complex number.
Proof
The universal property represents the functor assigning to X the space of P(z)-intertwining maps W1,W2→X.
If (T,ι) and (T',ι') both represent this functor, universality gives unique maps each way carrying the universal maps to one another; uniqueness forces their composites to be identities. Thus the product, when it exists, is unique up to unique isomorphism.