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Compatibility subspace yields the P(z)-tensor product

Statement

Let C be a full category of grading-restricted generalized V-modules and W1,W2∈C. If $S = W 1$ P(z) W2 with its induced P(z)-dual module action and its contragredient S' are both objects of C, then its contragredient (W1 P(z) W2)' together with the dual of the natural inclusion W1 P(z) W2↪(W1⊗W2)* is the P(z)-tensor product of W1,W2 in C.

Facts & Assumptions

Given: Work over C with the formal-variable, grading, and branch conventions in the linked definitions.

[F1]

P(z)-tensor product in a module category: For a strongly A-graded Möbius or conformal vertex algebra V, an abelian group Ã containing A, z∈C×, and a specified full category C of strongly Ã-graded generalized V-modules with grading-preserving Möbius-module morphisms, 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 $H o m_{C} (T, X) \to M [P (z)]_{X}^{W 1, W 2}$ . Existence is an additional theorem or hypothesis, not part of the definition.

[F2]

P(z)-compatibility subspace and local grading restriction defines the transported field $Y_{P (z)}^{'}$ by the displayed residue formula and defines ${C O M P}_{P (z)}$ by lower truncation of this field together with the transported compatibility identity. Its P(z)-local grading restriction requires λ to be a finite sum of generalized $L_{P (z)}^{'} (0)$ -eigenvectors, each homogeneous for $\tilde{A}$ ; the generated doubly graded subspace must have finite-dimensional $(β)_{[h]}$ components and be lower truncated in each fixed integer weight string. For each P(z)-map $I$ with target $W \in C$ , the transpose $I^{'} (w^{'}) (w_{1} \otimes w_{2}) = ⟨ w^{'}, I (w_{1} \otimes w_{2}) ⟩$ defines an image; the definition of $W_{1} P (z) W_{2}$ sums these images and states that transposed Jacobi and grading put each one in the stated subspace.

[F3]

Strongly graded contragredient duality for Möbius and conformal modules proves the opposite-field restricted dual, componentwise evaluation $M ≅ M^{''}$ , and functorial restricted transposition for strongly graded generalized modules over strongly graded Möbius or conformal V, preserving the auxiliary and generalized-weight gradings and all three Möbius operators. It imposes no CFT-type hypothesis. Use the ambient strong gradings and morphisms stipulated in F1 and F2. Each global generalized-weight space here is finite dimensional, so only finitely many auxiliary degrees occur at a fixed weight; the doubly restricted dual therefore agrees with $⨁_{h} (M_{[h]})^{*}$ used below. Its weight support and global grading restriction are the same as those of M.

[F4]

P(z)-intertwining map: Let V be strongly A-graded Möbius or conformal, let Ã contain A, and let W1,W2,W3 be strongly Ã-graded generalized V-modules; fix z∈C×. Retain the ambient strong gradings from F1 and F2. The morphisms preserve those gradings and commute with the three Möbius operators; no conformal vector, CFT-type condition or trivial auxiliary-group specialization is required. A P(z)-intertwining map of type (W3;W1,W2) is a linear map $I : W 1 \otimes W 2 \to \overset{ˉ}{W} 3 = \prod_{h \in C} W 3_{[} h]$ whose Ã-degrees add, for which $π_{h - m} I (w 1 \otimes w 2) = 0$ for all sufficiently large integers m for each fixed h,w1,w2, and which satisfies the P(z)-Jacobi identity and, for $j = - 1, 0, 1$ , $L (j) I (w 1 \otimes w 2) = I (w 1 \otimes L (j) w 2) + \sum_{i = 0}^{j + 1} (\binom{(}{j}) + 1, i) z^{i}$ I((L(j-i)w1)⊗w2). There is no L(-1)-derivative axiom for I. Formal variables x0,x1 are independent; z is a fixed complex number.

Proof

1.1givenF1F2F3F4algebra

Write $S = W_{1} P (z) W_{2}$ and $T = S^{'}$ . For $w_{1}, w_{2}$ define $ι (w_{1} \otimes w_{2})$ in $\overline{T} = \prod_{h} (S_{[h]})^{*}$ by evaluation on each finite-dimensional $S_{[h]}$ : its pairing with $s \in S$ is $s (w_{1} \otimes w_{2})$ . The value generally belongs to the completion, not to $T$ . Each $s$ is a finite sum of vectors in images $I_{j}^{'} (X_{j}^{'})$ by the definition of $S$ . Grading compatibility follows from the grading of each $I_{j}$ . Lower truncation follows uniformly because $T$ is grading restricted: if its real weights are bounded below by $B$ , then $T_{[h - m]} = 0$ for every integer $m > R e (h) - B$ , independently of the vector used to test the component. The compatibility identity for $s$ is exactly the transpose of P(z)-Jacobi for $ι$ , and the transported $L (j)$ identities transpose to its three required bracket identities. Thus $ι$ is a P(z)-map. The assumption $T \in C$ is needed here; fullness alone does not imply it.

2.1givenF1F2F3F4step 1.1algebra

For a P(z)-map $I : W_{1} \otimes W_{2} \to \overline{X}$ with $X \in C$ , the map $I^{'} : X^{'} \to S$ , $I^{'} (x^{'}) (w_{1} \otimes w_{2}) = ⟨ x^{'}, I (w_{1} \otimes w_{2}) ⟩$ , has image in $S$ by its defining sum. Transposed P(z)-Jacobi makes it a module homomorphism for the induced action on $S$ . The transposed three Möbius identities also make it intertwine the transported Möbius operators, and the degree-addition condition preserves auxiliary degree; in particular it preserves generalized weights. Its restricted transpose is therefore a grading-preserving Möbius-module map $S^{'} \to X^{''}$ by F3. Identify $X^{''}$ with $X$ using F3; since $C$ is full and $S^{'}, X \in C$ , the resulting map $η : T \to X$ is a morphism of $C$ . Evaluation of pairings gives $I = \overline{η} \circ ι$ .

3.1F1F2F3step 2.1algebra∎

Conversely, if $η : T \to X$ has this property, transpose it and use the evaluation identifications $T^{'} = S^{''} ≅ S$ , $X^{''} ≅ X$ . For every $x^{'} \in X^{'}$ , its transpose must be the vector in $S$ with values $⟨ x^{'}, I (w_{1} \otimes w_{2}) ⟩$ for all $w_{1}, w_{2}$ . The inclusion $S ↪ (W_{1} \otimes W_{2})^{*}$ is injective, so this vector is uniquely $I^{'} (x^{'})$ . Thus $η$ is unique. The pairing formula commutes with postcomposition, which gives naturality and the representing property.

Depends on

Dependency tree · 2 levels

Sources

Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part IV