Vertex Coalgebras, Coassociator, and Cocommutator Formulas

Abstract

Based on the definition of vertex coalgebra introduced by Hubbard, 2009, we prove that this notion can be reformulated using coskew symmetry, coassociator and cocommutator formulas without restrictions on the grading. We also prove that a vertex coalgebra can be defined in terms of dual versions of the axioms of Lie conformal algebra and differential algebra.

1. Introduction

The notion of vertex coalgebra was introduced by Hubbard in [1] as a generalization of the notion of vertex operator coalgebra previously studied in [2–4]. Many of the properties described in these works are in some sense dual to the properties satisfied by vertex algebras. Since the original definition of vertex algebra was introduced by Borcherds [5] in the 1980s, several reformulations have been studied. A vertex algebra can be defined using Lie algebra-type axioms or it can be seen as a generalization of a commutative and associative algebra with unit, focusing on the commutator or associator formulas. These formulations are introduced and thoroughly studied in [6–8]. A vertex algebra can also be defined as the deformation of a Poisson vertex algebra, namely, a Lie conformal algebra and a left symmetric differential algebra with unit satisfying certain compatibilities, as described in [9]. These different approaches engender several equivalent definitions of vertex algebra based on different axioms. Our goal is to prove that these approaches can be, in some sense, dualized to obtain equivalent definitions of vertex coalgebra.

The study of these approaches is not automatic in the case of vertex coalgebras as there might be axioms in the definition of vertex algebra that do not make sense in their dual version. For instance, axioms such as “weak commutativity” and “weak associativity” do not make sense unless we require the use of grading on a vertex coalgebra (see [1]). In [10], the authors show that with a coefficient approach, the Jacobi identity can be proven to follow from the commutator and associator formulas. Based on that idea, we first obtain a reformulation of the original definition of vertex coalgebra analogous to the original definition of vertex algebra introduced by Borcherds [5]. Then, we prove that the original definition of vertex coalgebra can be reformulated in three equivalent definitions: the first based on the coassociator and coskew symmetry formulas, the second based on the cocommutator formula, and the third based on dual versions of the axioms of Lie conformal algebra and differential algebra, following the ideas developed in [9].

In the next section we introduce basic definitions and results, as well as the original definition of vertex coalgebra and its basic properties. In Section 3, following the idea presented in [10], we prove that the coefficient version of the co-Jacobi identity (called co-Borcherds formula) can be deduced from weaker versions of this formula. In Section 4 we prove that the definition of vertex coalgebra can be reformulated based on coskew symmetry, coassociator, cocommutator, and 𝐷^⋆ formulas. Finally, in Section 5, we define a vertex coalgebra in terms of axioms regarding the 𝑛-coproducts for 𝑛≥−1.

2. Basic Properties and the Original Definition

We begin by introducing some basic definitions and results from the calculus of formal variables. A more thorough description of the concepts defined in this section, as well as the proofs of the following results, can be found in [6, 7, 11].

We will consider 𝑥, 𝑦, and 𝑧 commuting formal variables and define the formal 𝛿-function to be 𝛿𝑥,𝑦=_𝑛∈ℤ𝑥^𝑛𝑦^−𝑛−1. (1) Given an integer 𝑛, we define 𝑥±𝑦^𝑛=_𝑘≥0𝑛𝑘𝑥^𝑛−𝑘±𝑦^𝑘, (2) where (^𝑛_𝑘)=(𝑛(𝑛−1)⋯(𝑛−𝑘+1))/𝑘! for 𝑛∈ℤ and 𝑘≥0. Note that 𝛿(𝑥−𝑦,𝑧) is a formal power series in nonnegative powers of 𝑦. The formal residue “Res_𝑥,” of a series 𝑣(𝑥)=∑_𝑛∈ℤ𝑣_𝑛𝑥^𝑛 refers to the coefficient of the negative first power; that is, Res_𝑥𝑣(𝑥)=𝑣₋₁. (3) We will also use Taylor’s Theorem, which states that given a ℂ-vector space 𝑉 and a formal series 𝑓(𝑥)∈𝑉[[𝑥,𝑥⁻¹]], we have 𝑒^{𝑦(𝑑/𝑑𝑥)}𝑓(𝑥)=𝑓𝑥+𝑦. (4) We define the linear map 𝑇∶𝑉⊗𝑉→𝑉⊗𝑉 to be the transposition operator described by 𝑇(𝑣⊗𝑢)=𝑢⊗𝑣 for all 𝑢,𝑣∈𝑉.

Next, we enumerate basic properties of the formal series 𝛿 that we will need throughout this work: (a)Res_𝑦𝛿𝑥,𝑦=Res_𝑦𝛿(b)𝛿(c)𝛿.(d)𝛿.𝑦,𝑥=1.𝑥−𝑦𝑥,𝑦=0.𝑥−𝑦,𝑧=𝛿𝑧+𝑦,𝑥𝑥−𝑦,𝑧−𝛿−𝑦+𝑥,𝑧=𝛿𝑥−𝑧,𝑦 (5)(6)(7)(8) We will also make use of the following result (cf. Proposition 2.1.1 [11]).

Given a formal Laurent series 𝑋(𝑥,𝑦)∈(Hom(𝑉,𝑊))[[𝑥,𝑥⁻¹,𝑦,𝑦⁻¹]] with coefficients which are homomorphisms from a vector space 𝑉 to a vector space 𝑊, if lim_𝑥→𝑦𝑋(𝑥,𝑦) exists (i.e., when 𝑋(𝑥,𝑦) applied to any element of 𝑉 setting 𝑥=𝑦 leads to only finite sums in 𝑊), we have 𝛿𝑥,𝑦𝑋𝑥,𝑦=𝛿𝑥,𝑦𝑋𝑦,𝑦. (9) In particular, we have Res_𝑥𝛿𝑥,𝑦𝑋𝑥,𝑦=𝑋𝑦,𝑦. (10)

Now that we have listed the results that we will need, let us recall the original definition of vertex coalgebra introduced in [1].

Definition 1. A vertex coalgebra consists of a ℂ-vector space 𝑉, together with linear maps ⅄(𝑥)∶𝑉⟶(𝑉⊗𝑉)𝑥,𝑥⁻¹,𝑐∶𝑉⟶ℂ (11) called the coproduct and covacuum map, respectively, satisfying the following axioms for all 𝑣∈𝑉.(i)Left counit: (𝑐⊗Id)⅄(𝑥)𝑣=𝑣. (12)(ii)Cocreation: (Id⊗𝑐)⅄(𝑥)𝑣∈𝑉[[𝑥]],(Id⊗𝑐)⅄(𝑥)𝑣|_𝑥=0=𝑣. (13)(iii)Truncation: ⅄(𝑥)𝑣∈(𝑉⊗𝑉)𝑥⁻¹. (14)(iv)Co-Jacobi identity: 𝛿(⅄(𝑧))⅄𝑦𝑦⅄(𝑥)((𝑥⅄𝑦.𝑥−𝑧,𝑦⊗Id=𝛿𝑥−𝑦,𝑧Id⊗⅄−𝛿−𝑦+𝑥,𝑧𝑇⊗Id)(Id⊗⅄)) (15)

Remarks 1. (a) The operator ⅄(𝑥) is linear so that, for example, (Id⊗⅄(𝑦)) acting on the coefficients of ⅄(𝑥)𝑣∈(𝑉⊗𝑉)[[𝑥,𝑥⁻¹]] is well defined.
(b) Note that when each expression is applied to any element of 𝑉, the coefficient of each monomial in the formal variables is a finite sum due to the truncation condition (14).

As an immediate result, we have the following.

Proposition 2. Let (𝑉,⅄(𝑥),𝑐) be a vertex coalgebra as in Definition 1. Then, the following formulas hold.
(1) Cocommutator formula:𝑅𝑒𝑠_𝑧𝛿(⅄(𝑧))⅄𝑦=𝑦⅄(𝑥)−()×((𝑥⅄𝑦.𝑥−𝑧,𝑦⊗IdId⊗⅄𝑇⊗IdId⊗⅄)) (16)
(2) Coassociator formula:(⅄(𝑧)⊗Id)⅄𝑦=Id⊗⅄𝑦⅄𝑧+𝑦−𝑅𝑒𝑠_𝑥𝛿)×((𝑥⅄𝑦.−𝑦+𝑥,𝑧(𝑇⊗IdId⊗⅄)) (17)

Proof. (1) Taking Res_𝑧 to the co-Jacobi identity (15), we obtain Res_𝑧𝛿𝑥−𝑧,𝑦(⅄(𝑧)⊗Id)⅄𝑦=Res_𝑧𝛿𝑥−𝑦,𝑧Id⊗⅄𝑦⅄(𝑥)−Res_𝑧𝛿−𝑦+𝑥,𝑧(𝑇⊗Id)(Id⊗⅄(𝑥))⅄𝑦. (18) Using (5) twice on the right hand side of (18) we obtain (16).
(2) Taking Res_𝑥 to the Co-Jacobi identity (15) we obtain Res_𝑥𝛿𝑥−𝑧,𝑦(⅄(𝑧)⊗Id)⅄𝑦=Res_𝑥𝛿𝑥−𝑦,𝑧Id⊗⅄𝑦⅄(𝑥)−Res_𝑥𝛿−𝑦+𝑥,𝑧(𝑇⊗Id)(Id⊗⅄(𝑥))⅄𝑦. (19)
Using (5) and (7) on the left hand side of (19) we get (⅄(𝑧)⊗Id)⅄𝑦=Res_𝑥𝛿𝑥−𝑦,𝑧Id⊗⅄𝑦⅄(𝑥)−Res_𝑥𝛿)×((𝑥⅄𝑦.−𝑦+𝑥,𝑧(𝑇⊗IdId⊗⅄)) (20) Finally, using (7) and (10) on the first term of the right hand side of (20) we obtain (17).

As pointed in [1], it is natural to question the effect of applying a formal derivative, 𝑑/𝑑𝑥, to the comultiplication operator. In order to study that effect we need to introduce a map. Given a vertex coalgebra 𝑉, we define the linear map 𝐷^⋆∶𝑉→𝑉 as 𝐷^⋆=Res_𝑥𝑥⁻²(Id⊗𝑐)⅄(𝑥). (21) The following proposition obtained in [1] describes the main properties of the operator 𝐷^⋆ and its relationship to the formal derivative.

Proposition 3. The map 𝐷^⋆ satisfies the following properties:
(1)𝐷^⋆⅄(𝑥)=𝑑⊗Id⅄(𝑥)𝑑𝑥 (22)
(2)𝑒^𝑧𝐷⋆⊗Id⅄(𝑥)=⅄(𝑥+𝑧) (23)
(3)𝑒^𝑧𝐷⋆=(Id⊗𝑐)⅄(𝑧) (24)
(4) coskew symmetry:𝑇⅄(𝑥)=⅄(−𝑥)𝑒^𝑥𝐷⋆ (25)
(5)𝑑⅄(𝑥)(𝑥)𝐷𝑑𝑥=⅄^⋆−Id⊗𝐷^⋆⅄(𝑥) (26)
(6)Id⊗𝑒^{−𝑧𝐷⋆}⅄(𝑥)𝑒^𝑧𝐷⋆=⅄(𝑥+𝑧). (27)

We consider the following expansion of the map ⅄(𝑥): ⅄(𝑥)𝑣=_𝑛∈ℤΔ_𝑛(𝑣)𝑥^−𝑛−1, (28) where Δ_𝑛∶𝑉→𝑉⊗𝑉 is the coefficient of 𝑥^−𝑛−1 in the series ⅄(𝑥). If we look for the coefficient of 𝑥^−𝑝−1𝑦^−𝑞−1𝑧^−𝑟−1 in the co-Jacobi identity (15) we obtain what we will call the co-Borcherds identity: _𝑖≥0𝑝𝑖Δ_𝑟+𝑖⊗IdΔ_{𝑝+𝑞−𝑖}=_𝑖≥0(−1)^𝑖𝑟𝑖Id⊗Δ_𝑞+𝑖Δ_{𝑝+𝑟−𝑖}−(−1)^𝑟(𝑇⊗Id)Id⊗Δ_𝑝+𝑖Δ_{𝑞+𝑟−𝑖}. (29) We can rewrite Definition 1 considering coefficients. With this approach we obtain the following definition of a vertex coalgebra, evidently equivalent to the one previously introduced.

Definition 4. A vertex coalgebra is a ℂ-vector space 𝑉 endowed with a family of linear coproducts indexed by 𝑛∈ℤ and a linear map 𝑐: Δ_𝑛∶𝑉⟶𝑉⊗𝑉,𝑐∶𝑉⟶ℂ (30)
satisfying the following axioms. (i)Left counit: (𝑐⊗Id)Δ_𝑛()Δ=Idif𝑛=−1,𝑐⊗Id_𝑛=0if𝑛≠−1. (31)(ii)Cocreation: (Id⊗𝑐)Δ_𝑛()Δ=0if𝑛≥0,Id⊗𝑐₋₁=Id. (32)(iii)Truncation: for each 𝑣∈𝑉 there exists 𝑁 such that Δ_𝑛𝑣=0∀𝑛≤𝑁. (33)(iv)Co-Borcherds identity: for all 𝑝,𝑞,𝑟∈ℤ, _𝑖≥0𝑝𝑖Δ_𝑟+𝑖⊗IdΔ_{𝑝+𝑞−𝑖}=_𝑖≥0(−1)^𝑖𝑟𝑖Id⊗Δ_𝑞+𝑖Δ_{𝑝+𝑟−𝑖}−(−1)^𝑟(𝑇⊗Id)Id⊗Δ_𝑝+𝑖Δ_{𝑞+𝑟−𝑖}. (34)

Under this coefficient approach the definition of 𝐷^⋆ corresponds to the linear map 𝐷^⋆𝐷∶𝑉⟶𝑉,_⋆=(Id⊗𝑐)Δ₋₂. (35)

Proposition 5. Let 𝑉 be a vertex coalgebra as in Definition 4 and 𝐷^⋆ defined as in (35). Then, the following properties hold.  (1) For all 𝑞∈ℤ, 𝐷^⋆⊗IdΔ_𝑞=−𝑞Δ_𝑞−1. (36) (2) If we denote 𝐷^⋆(𝑖)=(1/𝑖!)(𝐷^⋆)^𝑖 for 𝑖≥0, then𝐷^⋆(𝑖)=(Id⊗𝑐)Δ_−1−𝑖. (37) (3) The operator 𝐷^⋆ is locally nilpotent. Namely, for each 𝑣∈𝑉 there exists 𝑁 such that  (𝐷^⋆)^𝑛(𝑣)=0   for  all  𝑛≥𝑁.

Proof. (1) If we take the generating function in 𝑥 of (36) we obtain 𝐷^⋆⅄(𝑥)=𝑑⊗Id⅄(𝑥).𝑑𝑥 (38) Therefore, (36) is just the coefficient formulation of (22).
(2) It is easy to see that (37) is merely the coefficient formulation of (24).
(3) The proof follows from (37) together with (33).

Remarks 2. Note that for the proof of (1) and (2) in the previous proposition we only used the definition of 𝐷^⋆ together with axioms (22) and (24).

Manipulating expressions that involve coefficients we will get coefficient versions of the cocommutator formula (16), the coassociator formula (17), and the coskew symmetry formula (25) based on Definition 4. These formulas will be axioms in the two definitions of vertex coalgebra that we will introduce in Section 4.

Proposition 6. Let 𝑉 be a vertex coalgebra as in Definition 4. Then, the following formulas hold for every 𝑝,𝑞,𝑟∈ℤ.
(1) Coefficient cocommutator formula: _𝑖≥0𝑝𝑖Δ_𝑖⊗IdΔ_{𝑝+𝑞−𝑖}=Id⊗Δ_𝑞Δ_𝑝−(𝑇⊗Id)Id⊗Δ_𝑝Δ_𝑞. (39)
(2) Coefficient coassociator formula: Δ_𝑟⊗IdΔ_𝑞=_𝑖≥0(−1)^𝑖𝑟𝑖×Id⊗Δ_𝑞+𝑖Δ_𝑟−𝑖−(−1)^𝑟×(𝑇⊗Id)Id⊗Δ_𝑖Δ_{𝑞+𝑟−𝑖}. (40)
(3) Coefficient coskew symmetry: 𝑇Δ_𝑟=_𝑖≥0(−1)^𝑟+1+𝑖Δ_𝑟+𝑖𝐷^⋆(𝑖). (41)

Proof. The proof follows from the fact that Definitions 1 and 4 are equivalent. Formulas (39), (40), and (41) are merely the coefficient formulation of formulas (16), (17), and (25) respectively.

3. Structure of the Co-Borcherds Identity

Following the idea presented in [10], we want to prove that the Co-Borcherds identity can be deduced from the cocommutator and the coassociator formulas. For that reason, we introduce a few auxiliary formulas: CB₁𝑝,𝑞,𝑟∶=_𝑖≥0𝑝𝑖Δ_𝑟+𝑖⊗IdΔ_{𝑝+𝑞−𝑖},CB₂𝑝,𝑞,𝑟∶=_𝑖≥0(−1)^𝑖𝑟𝑖Id⊗Δ_𝑞+𝑖Δ_{𝑝+𝑟−𝑖},CB₃𝑝,𝑞,𝑟∶=_𝑖≥0(−1)^𝑖+𝑟𝑟𝑖×(𝑇⊗Id)Id⊗Δ_𝑝+𝑖Δ_{𝑞+𝑟−𝑖}. (42) Note that with the notation introduced above, the co-Borcherds identity (34) for the indices 𝑝,𝑞,𝑟∈ℤ corresponds to CB₁𝑝,𝑞,𝑟=CB₂𝑝,𝑞,𝑟−CB₃𝑝,𝑞,𝑟. (43) We have the following formulas which are analogous to the results obtained in Section 3.2 [10].

Proposition 7. For all 𝑝,𝑞,𝑟∈ℤ, CB_𝑗𝑝+1,𝑞,𝑟=CB_𝑗𝑝,𝑞+1,𝑟+CB_𝑗𝑝,𝑞,𝑟+1, (44) for 𝑗=1,2,3.

Proof. Note that the co-Borcherds identity (34) is the coefficient version of the co-Jacobi identity (15). Therefore, it is clear that if we consider 𝐺₁(⅄(𝑧))⅄𝑦,𝐺𝑥,𝑦,𝑧∶=𝛿𝑥−𝑧,𝑦⊗Id₂𝑦⅄(𝑥),𝐺𝑥,𝑦,𝑧∶=𝛿𝑥−𝑦,𝑧Id⊗⅄₃𝑥,𝑦,𝑧∶=𝛿−𝑦+𝑥,𝑧(𝑇⊗Id)(Id⊗⅄(𝑥))⅄𝑦, (45) then, for 𝑗=1,2,3, CB_𝑗𝑝,𝑞,𝑟=Res_{𝑥,𝑦,𝑧}𝑥^𝑝𝑦^𝑞𝑧^𝑟𝐺_𝑗𝑥,𝑦,𝑧. (46) We want to prove that for 𝑗=1,2,3, CB_𝑗𝑝+1,𝑞,𝑟−CB_𝑗𝑝,𝑞+1,𝑟−CB_𝑗𝑝,𝑞,𝑟+1=0. (47) Note that (47) is equivalent to Res_{𝑥,𝑦,𝑧}𝑥^𝑝+1𝑦^𝑞𝑧^𝑟𝐺_𝑗𝑥,𝑦,𝑧−𝑥^𝑝𝑦^𝑞+1𝑧^𝑟𝐺_𝑗𝑥,𝑦,𝑧−𝑥^𝑝𝑦^𝑞𝑧^𝑟+1𝐺_𝑗𝑥,𝑦,𝑧=Res_{𝑥,𝑦,𝑧}𝑥^𝑝𝑦^𝑞𝑧^𝑟𝑥−𝑦−𝑧𝐺_𝑗𝑥,𝑦,𝑧=0, (48) for 𝑗=1,2,3, which holds due to (6).

Using (43) and Proposition 7, it is easy to prove the following result.

Proposition 8. The co-Borcherds identities for two of the indices (𝑝+1,𝑞,𝑟),  (𝑝,𝑞+1,𝑟), and (𝑝,𝑞,𝑟+1) imply the co-Borcherds identity for the other index.

Now, we can prove the following statement.

Theorem 9. The co-Borcherds identities for all 𝑝 and 𝑞 with fixed 𝑟 and for all 𝑞 and 𝑟 with fixed 𝑝 imply the co-Borcherds identity for all 𝑝, 𝑞, and 𝑟.

Proof. We assume the co-Borcherds identity for all 𝑝 and 𝑞 with 𝑟₀ fixed and for all 𝑞 and 𝑟 with 𝑝₀ fixed. Using Proposition 8 we have that the co-Borcherds identity also holds for (𝑝,𝑞,𝑟₀+1) for all 𝑝,𝑞∈ℤ. Using that the co-Borcherds identity holds for indices (𝑝,𝑞+1,𝑟₀+1) and (𝑝+1,𝑞,𝑟₀+1) together with Proposition 8, we obtain that the co-Borcherds identity also holds for the indices (𝑝,𝑞,𝑟₀+2) for all 𝑝,𝑞∈ℤ. Inductively we obtain that the co-Borcherds identity holds for (𝑝,𝑞,𝑟₀+𝑛) for all 𝑛∈ℕ for every 𝑝,𝑞∈ℤ.
Analogously, we can prove that the co-Borcherds identity also holds for (𝑝₀+1,𝑞,𝑟) for all 𝑞,𝑟,∈ℤ. Inductively we obtain that the co-Borcherds identity holds for the indices (𝑝₀+𝑛,𝑞,𝑟) for all 𝑛∈ℕ for every 𝑞,𝑟∈ℤ. Thus, it is clear that the co-Borcherds identity holds for (𝑝,𝑞,𝑟) as long as 𝑝≥𝑝₀ or 𝑟≥𝑟₀.
Now, we need to analyze the case in which 𝑝<𝑝₀ and 𝑟<𝑟₀. First, we note that the co-Borcherds identity holds for indices (𝑝₀,𝑠,𝑟₀−1) and (𝑝₀−1,𝑠,𝑟₀) for all 𝑠∈ℤ. Using Proposition 8 those identities imply the co-Borcherds identity for the index (𝑝₀−1,𝑠+1,𝑟₀−1) for all 𝑠∈ℤ. Again, using that the co-Borcherds identity holds for indices (𝑝₀−1,𝑠+1,𝑟₀−1) and (𝑝₀−2,𝑠+1,𝑟₀) we obtain that co-Borcherds identity holds for the index (𝑝₀−2,𝑠+2,𝑟₀−1). Inductively, we obtain that the co-Borcherds identity holds for indices (𝑝₀−𝑛,𝑠+𝑛,𝑟₀−1) for all 𝑛≥0. As this identity holds for all 𝑠∈ℤ, we obtain that the co-Borcherds identity holds for indices (𝑝₀−𝑛,𝑠,𝑟₀−1) for all 𝑛≥0 and 𝑠∈ℤ.
Next, the fact that the co-Borcherds identity holds for indices (𝑝₀−1,𝑙,𝑟₀−1) and (𝑝₀,𝑙,𝑟₀−2) for all 𝑙∈ℤ implies that the co-Borcherds identity holds for the index (𝑝₀−1,𝑙,𝑟₀−2) for all 𝑙∈ℤ. Inductively, we obtain that the co-Borcherds identity holds for indices (𝑝₀−1,𝑠,𝑟₀−𝑛) for all 𝑛≥0 for every 𝑠∈ℤ. Thus, we have proved that if the co-Borcherds identity holds for indices (𝑝,𝑞,𝑟) as long as 𝑝≥𝑝₀ or 𝑟≥𝑟₀, then it also holds for (𝑝,𝑞,𝑟) as long as 𝑝≥𝑝₀−1 or 𝑟≥𝑟₀−1. Continuing with this procedure, we inductively obtain that the co-Borcherds identity holds for every index (𝑝,𝑞,𝑟), finishing the proof.

4. Two Equivalent Definitions

After studying the structure of the co-Borcherds identity, we will prove that we can reformulate the definition of a vertex coalgebra focusing on either the coassociator formula or the cocommutator formula.

Definition 10. A vertex coalgebra consists of a ℂ-vector space 𝑉, together with linear maps ⅄(𝑥)∶𝑉⟶(𝑉⊗𝑉)𝑥⁻¹,𝐷𝑐∶𝑉⟶ℂ,_⋆∶𝑉⟶𝑉 (49) satisfying the following axioms for all 𝑣∈𝑉.(i)Left counit: (𝑐⊗Id)⅄(𝑥)𝑣=𝑣. (50)(ii)Cocreation: 𝑒^𝑥𝐷⋆=(Id⊗𝑐)⅄(𝑥). (51)(iii)𝐷^⋆ formula: 𝑑⅄(𝑥)=𝐷𝑑𝑥^⋆⊗Id⅄(𝑥). (52)(iv)Coskew symmetry: 𝑇⅄(𝑥)=⅄(−𝑥)𝑒^𝑥𝐷⋆. (53)(v)Coassociator formula: (⅄(𝑧))⅄𝑦=𝑦⅄⊗IdId⊗⅄𝑧+𝑦−Res_𝑥𝛿×((𝑥⅄𝑦.−𝑦+𝑥,𝑧𝑇⊗Id)(Id⊗⅄)) (54)

Remarks 3. (a) Note that (51) trivially implies (13). That is, Definition 10 implies that for all 𝑣∈𝑉, (Id⊗𝑐)⅄(𝑥)𝑣∈𝑉[[𝑥]],(Id⊗𝑐)⅄(𝑥)𝑣|_𝑥=0=𝑣. (55)
(b) Note that (51) implies that the map 𝐷^⋆ is the map defined in (21). In fact, if we take the coefficient of 𝑥¹ in (51), we get 𝐷^⋆=(Id⊗𝑐)Δ₋₂.
(c) Using Proposition 3(3), it is clear that (51) is also satisfied under the conditions of Definition 1.

Hence, 𝐷^⋆ is the map we introduced earlier and the conditions regarding the map 𝑐 do not essentially differ from the properties described in the first definitions. We can now state one of our main results.

Theorem 11. Definition 10 is equivalent to Definition 1.

In [1] (cf. Proposition 3), it is proved that Definition 1 implies cocreation (24), coskew symmetry (25), and the 𝐷^⋆ formula (22). In Proposition 2(2) we proved that the coassociator formula (54) follows from Definition 1. Therefore, it is clear that the conditions of Definition 10 are satisfied under the axioms of Definition 1. In order to prove the other implication, we first need to prove some results.

Proposition 12. The coskew symmetry formula (53) and the 𝐷^⋆ formula (52) imply that the map 𝐷^⋆ satisfies the following properties.
(1) 𝐷^⋆-bracket formula:𝑑⅄(𝑥)(𝑥)𝐷𝑑𝑥=⅄^⋆−Id⊗𝐷^⋆⅄(𝑥). (56)
(2) Conjugation formula:Id⊗𝑒^{−𝑧𝐷⋆}⅄(𝑥)𝑒^𝑧𝐷⋆=⅄(𝑥+𝑧). (57)

Proof. (1) Using (53), the product rule, (52), and the fact that 𝑇(𝐷^⋆⊗Id)=(Id⊗𝐷^⋆)𝑇 and reapplying (53), we have 𝑑⅄(𝑥)=𝑑𝑑𝑥()𝑒𝑑𝑥𝑇⅄−𝑥^𝑥𝐷⋆=−𝑇𝐷^⋆⊗Id⅄(−𝑥)𝑒^𝑥𝐷⋆+𝑇⅄(−𝑥)𝑒^𝑥𝐷⋆𝐷^⋆=−Id⊗𝐷^⋆⅄(𝑥)+⅄(𝑥)𝐷^⋆. (58)
(2) Exponentiating the 𝐷^⋆-bracket formula (56) and applying Taylor's Theorem (4) we obtain (57).

Our goal is to prove that the axioms of Definition 10 imply the co-Jacobi identity (15). We begin by proving that the cocommutator formula (16) holds.

Lemma 13. In the presence of the 𝐷^⋆ formula (52) and coskew symmetry (53), the coassociator formula (54) is equivalent to the cocommutator formula (16).

Proof. Assuming that the coassociator formula (54) holds, we multiply it by 𝑒^𝑦𝐷⋆ after replacing 𝑦 by −𝑦 to obtain (⅄(𝑧)⊗Id)⅄−𝑦𝑒^𝑦𝐷⋆=Id⊗⅄−𝑦⅄𝑧−𝑦𝑒^𝑦𝐷⋆−Res_𝑥𝛿()×((𝑥⅄𝑒𝑦+𝑥,𝑧𝑇⊗IdId⊗⅄))−𝑦^𝑦𝐷⋆, (59) which due to (53) and (57) implies (⅄(𝑧))𝑦=𝑦⅄(𝑧)⊗Id𝑇⅄Id⊗𝑇⅄−Res_𝑥𝛿×((𝑥𝑦.𝑦+𝑥,𝑧𝑇⊗Id)(Id⊗⅄))𝑇⅄ (60) Thus, we have that Id⊗𝑇⅄𝑦⅄(𝑧)−(⅄(𝑧)⊗Id)𝑇⅄𝑦=Res_𝑥𝛿)×((𝑥𝑦.𝑦+𝑥,𝑧(𝑇⊗IdId⊗⅄))𝑇⅄ (61) Using that 𝛿(𝑦+𝑥,𝑧)=𝛿(𝑧−𝑥,𝑦) and applying (Id⊗𝑇) to (61), we obtain (Id⊗𝑇)Id⊗𝑇⅄𝑦⅄(𝑧)−(Id⊗𝑇)(⅄(𝑧)⊗Id)𝑇⅄𝑦=Res_𝑥𝛿𝑧−𝑥,𝑦(Id⊗𝑇)(𝑇⊗Id)(Id⊗⅄(𝑥))𝑇⅄𝑦. (62) Using that ()𝑦⅄(𝑧)=𝑦⅄(𝑧),(⅄(𝑧))𝑦=((𝑧⅄𝑦,((𝑥𝑦=(⅄(𝑥))⅄𝑦,Id⊗𝑇Id⊗𝑇⅄Id⊗⅄Id⊗𝑇)(⊗Id𝑇⅄𝑇⊗Id)(Id⊗⅄))Id⊗𝑇)(𝑇⊗Id)(Id⊗⅄))𝑇⅄⊗Id (63) we have that (62) is equivalent to Id⊗⅄𝑦⅄(𝑧)−(𝑇⊗Id)(Id⊗⅄(𝑧))⅄𝑦=Res_𝑥𝛿𝑧−𝑥,𝑦(⅄(𝑥)⊗Id)⅄𝑦, (64) which is the cocommutator formula. Using the same argument in reverse order, it is clear that, in the presence of (52) and (53), the cocommutator and the coassociator formulas are equivalent.

Now, we can finish the proof of Theorem 11. By Remarks 3, it remains to show that Definition 10 implies the co-Jacobi identity (15). On the one hand, the coefficient version of the coassociator formula (54) implies the co-Borcherds identity for all 𝑞,𝑟∈ℤ with fixed 𝑝₀=0. On the other hand, the cocommutator formula, proven to hold under the axioms of Definition 10 in Lemma 13, implies the co-Borcherds identity for all 𝑝,𝑞∈ℤ with fixed 𝑟₀=0. Therefore, using Theorem 9, we have that the axioms of Definition 10 imply the co-Borcherds identity for all 𝑝,𝑞,𝑟∈ℤ. Now, it is clear that the co-Jacobi identity holds due to the equivalence between Definitions 4 and 1.

Next, we will introduce a definition mainly based on the cocommutator formula. In order to prove its equivalence with the definitions previously introduced, we will show that the coskew symmetry can be deduced from the cocommutator formula in the presence of certain axioms of 𝐷^⋆ and 𝑐.

Definition 14. A vertex coalgebra consists of a ℂ-vector space 𝑉, together with linear maps ⅄(𝑥)∶𝑉⟶(𝑉⊗𝑉)𝑥⁻¹,𝐷𝑐∶𝑉⟶ℂ,_⋆∶𝑉⟶𝑉 (65) satisfying the following axioms.(i)Left counit: (𝑐⊗Id)⅄(𝑥)𝑣=𝑣,∀𝑣∈𝑉. (66)(ii)Cocreation: 𝑒^𝑥𝐷⋆=(Id⊗𝑐)⅄(𝑥). (67)(iii)𝐷^⋆ formula: 𝑑⅄(𝑥)=𝐷𝑑𝑥^⋆⊗Id⅄(𝑥). (68)(iv)𝐷^⋆-bracket formula: 𝑑⅄(𝑥)(𝑥)𝐷𝑑𝑥=⅄^⋆−Id⊗𝐷^⋆⅄(𝑥). (69)(v)Cocommutator formula: Res_𝑧𝛿(⅄(𝑧))⅄𝑦=𝑦⅄(𝑥)−((𝑥⅄𝑦.𝑥−𝑧,𝑦⊗IdId⊗⅄𝑇⊗Id)(Id⊗⅄)) (70)

Our goal is to prove the following statement.

Theorem 15. Definition 14 is equivalent to Definition 10.

Due to Proposition 12 and Lemma 13, it is clear that Definition 10 implies the axioms of Definition 14. In light of Lemma 13, in order to prove the other implication it is enough to show that coskew symmetry (53) follows from the axioms of Definition 14. We will use the cocommutator formula (70) and the coefficient cocommutator formula (39) interchangeably because, as we mentioned before, (39) is merely the coefficient version of (70).

First, we will need the following result, which is a dual version of the proposition presented in Section 3.2.9 [12] for the case of vertex algebras.

Lemma 16. In the presence of (68) and (69), the coefficient coskew symmetry (41) for indices 𝑟≥−1 imply the coefficient coskew symmetry for every 𝑟∈ℤ.

Proof. Note that it is enough to show that for fixed 𝑟≠0, the coefficient coskew symmetry for the index 𝑟 implies the coefficient coskew symmetry for the index 𝑟−1. We will denote the coefficient coskew symmetry for fixed 𝑟∈ℤ by coskew(𝑟); that is, coskew(𝑟)∶𝑇Δ_𝑟=_𝑖≥0(−1)^𝑟+1+𝑖Δ_𝑟+𝑖𝐷^⋆(𝑖). (71) We assume coskew(𝑟) for 𝑟≠0 and we want to prove coskew(𝑟−1). First, we write 𝑟⋅_𝑖≥0(−1)^𝑟+𝑖Δ_{𝑟−1+𝑖}𝐷^⋆(𝑖)=Res_𝑥𝑑𝑥𝑑𝑥^𝑟⅄(−𝑥)𝑒^𝑥𝐷⋆=−Res_𝑥𝑥^𝑟𝑑⅄()𝑒𝑑𝑥−𝑥^𝑥𝐷⋆=−Res_𝑥𝑥^𝑟−𝐷^⋆⊗Id⅄(−𝑥)𝑒^𝑥𝐷⋆+⅄(−𝑥)𝑒^𝑥𝐷⋆𝐷^⋆ (72) due to (68). Using coskew(𝑟) we can rewrite the right hand side of (72) to obtain 𝑟⋅_𝑖≥0(−1)^𝑟+𝑖Δ_{𝑟−1+𝑖}𝐷^⋆(𝑖)=−Res_𝑥𝑥^𝑟−𝐷^⋆⊗Id𝑇⅄(𝑥)+𝑇⅄(𝑥)𝐷^⋆. (73) Now, we apply 𝑇 to (69) and use that 𝑇(Id⊗𝐷^⋆⅄(𝑥)=(𝐷^⋆⊗Id)𝑇⅄(𝑥) to get 𝑟⋅_𝑖≥0(−1)^𝑟+𝑖Δ_{𝑟−1+𝑖}𝐷^⋆(𝑖)=−Res_𝑥𝑥^𝑟𝑑(𝑥)𝑑𝑥𝑇⅄=𝑟𝑇Δ_𝑟−1. (74) Dividing by 𝑟, which we have assumed to be nonzero, we obtain from (74) coskew(𝑟−1), finishing the proof of the lemma.

For the proof of Theorem 15 we will also need the following result.

Proposition 17. Let (𝑉⅄(𝑥),𝑐,𝐷^⋆) be a vertex coalgebra as in Definition 14. Then, the coefficient coskew symmetry, denoted coskew(𝑟) in (71), holds for 𝑟≥−1.

Proof. First, let us recall that the cocommutator formula (70) is equivalent to the coefficient cocommutator formula (39) for every 𝑝,𝑞∈ℤ. Second, note that the coefficient cocommutator formula for fixed indices 𝑝,𝑞∈ℤ is exactly the co-Borcherds identity (34) for indices 𝑝,𝑞∈ℤ and 𝑟=0. Now, due to Proposition 8, it is clear that the co-Borcherds identity for every 𝑝,𝑞∈ℤ and 𝑟=0 implies the co-Borcherds identity for every 𝑝,𝑞∈ℤ and 𝑟≥0. In particular, the cocommutator formula implies the co-Borcherds identity for 𝑝=0,  𝑞=−1, and 𝑟≥0: Δ_𝑟⊗IdΔ₋₁=_𝑖≥0(−1)^𝑖𝑟𝑖Id⊗Δ_−1+𝑖Δ_𝑟−𝑖−(−1)^𝑟(𝑇⊗Id)Id⊗Δ_𝑖Δ_{−1+𝑟−𝑖}. (75) Applying (𝑇⊗Id) to (75) and reordering the terms, we obtain 𝑇Δ_𝑟⊗IdΔ₋₁=_𝑖≥0(−1)^𝑖𝑟𝑖(−1)^𝑟+1Id⊗Δ_𝑖Δ_{−1+𝑟−𝑖}−(−1)^𝑟×(𝑇⊗Id)Id⊗Δ_−1+𝑖Δ_𝑟−𝑖. (76) As we mentioned before, the co-Borcherds identity (34) holds for 𝑝,𝑞∈ℤ and 𝑟≥0. Therefore, evaluating 𝑝=−1, 𝑞=0, and 𝑟≥0 in (34), the right hand side of (76) can be rewritten to get 𝑇Δ_𝑟⊗IdΔ₋₁=(−1)^𝑟+1_𝑖≥0𝑖Δ−1_𝑟+𝑖⊗IdΔ_−1−𝑖. (77) Next, using that _𝑖⁻¹=(−1)^𝑖 for 𝑖≥0, we apply (Id⊗Id⊗𝑐) to (77) and reorder the terms to obtain 𝑇Δ_𝑟⊗Id(Id⊗𝑐)Δ₋₁=_𝑖≥0(−1)^𝑟+1+𝑖Δ_𝑟+𝑖(Id⊗𝑐)Δ_−1−𝑖. (78) Using Remarks 2 and 3(b), together with (67) and (68), it is clear that for 𝑖≥0, 𝐷^⋆(𝑖)=(Id⊗𝑐)Δ_−1−𝑖. (79) Replacing (79) in (78) we have 𝑇Δ_𝑟⊗Id(Id⊗𝑐)Δ₋₁=_𝑖≥0(−1)^𝑟+1+𝑖Δ_𝑟+𝑖𝐷^⋆(𝑖), (80) which due to (67) implies 𝑇Δ_𝑟=_𝑖≥0(−1)^𝑟+1+𝑖Δ_𝑟+𝑖𝐷^⋆(𝑖). (81) Therefore, we have proved that the cocommutator formula (70) implies coskew(𝑟) for 𝑟≥0, in the presence of (67) and (68).
In order to finish the proof, we need to show that coskew(−1) also follows from the axioms of Definition 14. To prove that, we use the coefficient cocommutator formula (39) for 𝑝=−1, 𝑞=−1: ^𝑖≥0𝑖Δ−1_𝑖⊗IdΔ_−2−𝑖=Id⊗Δ₋₁Δ₋₁−(𝑇⊗Id)Id⊗Δ₋₁Δ₋₁. (82) Applying (Id⊗Id⊗𝑐) to (82) and using (79) together with _𝑖⁻¹=(−1)^𝑖 for 𝑖≥0, we obtain _𝑖≥0(−1)^𝑖Δ_𝑖𝐷^⋆(𝑖+1)=(Id⊗Id⊗𝑐)Id⊗Δ₋₁Δ₋₁−(Id⊗Id⊗𝑐)(𝑇⊗Id)Id⊗Δ₋₁Δ₋₁. (83) Due to (67), it follows that (83) implies _𝑖≥0(−1)^𝑖Δ_𝑖𝐷^⋆(𝑖+1)=Δ₋₁−𝑇Δ₋₁, (84) which is equivalent to coskew(−1). Therefore, we have seen that under the axioms of Definition 14, the cocommutator formula implies coskew(𝑟) for every 𝑟≥−1.

Now we can complete the proof of Theorem 15. Using Proposition 17 together with Lemma 16 it is clear that Definition 14 implies the coskew symmetry (53). With the coskew symmetry proved, the coassociator formula (54) follows from Lemma 13 together with (67), (68), and (70), finishing the proof of the theorem.

5. Bakalov-Kac Dual Definition

In this section we will introduce the last definition, which is based on the work of Bakalov and Kac [9] for the case of vertex algebras. They present a vertex algebra as a sort of deformation of a Poisson vertex algebra, namely, a ℂ[𝐷]-Lie conformal algebra and a left-symmetric ℂ[𝐷]-differential algebra with certain compatibilities. In light of (36), it is clear that the 𝑛-coproducts Δ_𝑛 for 𝑛<0 can be described using the −1-coproduct and the map 𝐷^⋆. Therefore, the coalgebra structure is fully determined by the coproducts Δ_𝑛 for 𝑛≥−1. That is the motivation for the definition that we present in this section, where the axioms are reformulated to consider properties regarding the 𝑛-coproducts only for 𝑛≥−1.

Let (𝑉,⅄(𝑥),𝑐,𝐷^⋆) be a vertex coalgebra, where ⅄(𝑥)=_𝑛∈ℤΔ_𝑛𝑥^−𝑛−1. (85) In order to consider the −1-coproduct and the 𝑛-coproducts for 𝑛≥0, we define two linear maps: Λ∶𝑉⟶𝑉⊗𝑉𝑎⟼Res_𝑥𝑥⁻¹⅄(𝑥)𝑎=Δ₋₁(𝑎),Δ_𝜆∶𝑉⟶(𝑉⊗𝑉)[[𝜆]]𝑎⟼Res_𝑥𝑒^𝑥𝜆⅄(𝑥)𝑎=_𝑛≥0Δ_𝑛(𝑎)𝜆^𝑛.𝑛! (86)(87)

Remarks 4. Note that the map Δ_𝜆 defined in (87) may contain infinitely many powers of 𝜆, unlike the case of vertex algebras (cf. Section 4 in [9]), where the 𝜆-bracket that encodes the 𝑛-products for 𝑛≥0 is truncated.

We want to prove several properties of the maps defined above that will be axioms in the next definition of vertex coalgebra. First, we need to introduce the (𝑛-)coproduct identity:Δ_𝑛⊗Id⅄𝑦=Res_𝑥𝑥−𝑦^𝑛𝑦⅄(𝑥)−Id⊗⅄−𝑦+𝑥^𝑛(𝑇⊗Id)(Id⊗⅄(𝑥))⅄𝑦. (88) We have the following result.

Proposition 18. The coassociator formula (54) holds if and only if the (𝑛-)coproduct identity (88) holds for all 𝑛∈ℤ.

Proof. Using that (Id⅄(𝑦))⅄(𝑧+𝑦)=Res_𝑥𝛿(𝑥−𝑦,𝑧)(Id⊗⅄(𝑦))⅄(𝑥) and taking the coefficient of 𝑧^−𝑛−1 for each 𝑛∈ℤ in (54), the equivalence is clear.

Now we can prove the list of properties that we will need later.

Proposition 19. Let (𝑉,⅄(𝑥),𝑐,𝐷^⋆) be a vertex coalgebra and let Λ, Δ_𝜆 be the maps defined in (86) and (87), respectively. Then, the following conditions are satisfied.
(1) Counit:(𝑐⊗Id)Λ=Id=(Id⊗𝑐)Λ. (89)
(2) Coderivation with respect to  Λ:Λ𝐷^⋆=𝐷^⋆⊗Id+Id⊗𝐷^⋆Λ. (90)
(3) Counit holomorphic condition:(Id⊗𝑐)Δ_𝜆=0. (91)
(4) 𝐷^⋆-sesquilinearity:𝐷^⋆⊗IdΔ_𝜆=Δ_𝜆(−𝜆),Id⊗𝐷^⋆Δ_𝜆=Δ_𝜆𝜆+𝐷^⋆. (92)
(5) Conformal coskew symmetry:𝑇Δ_𝜆=−Δ_{−𝜆−𝐷}^⋆. (93)
(6) Conformal co-Jacobi formula:Δ_𝜆⊗IdΔ_𝜆+𝜇=Id⊗Δ_𝜇Δ_𝜆−(𝑇⊗Id)Id⊗Δ_𝜆Δ_𝜇. (94)
(7) Co-Wick formula:Δ_𝜆⊗IdΛ=(Id⊗Λ)Δ_𝜆−(𝑇⊗Id)Id⊗Δ_𝜆Λ−^𝜆₀Δ_𝜆⊗IdΔ_𝜇𝑑𝜇. (95)
(8) Cocompatibility:Λ=𝑇Λ+⁰_−𝐷⋆Δ_𝜆𝑑𝜆. (96)
(9) Coleft symmetry:()()Λ=()()Id⊗ΛΛ−𝑇⊗Id)(Id⊗ΛΛ⊗IdΛ−𝑇⊗Id)(Λ⊗IdΛ. (97)

Remarks 5. (a) The expression Δ_{−𝜆−𝐷}^⋆ is interpreted as follows: Δ_{−𝜆−𝐷}^⋆(𝑣)=_𝑛≥0Δ_𝑛𝑛!−𝜆−𝐷^⋆𝑛(𝑣), (98) with the powers of −𝜆−𝐷^⋆ on the right, unlike the case of ℂ[𝐷]-Lie conformal algebras in [9]. Moreover, using that 𝐷^⋆ is locally nilpotent (see Proposition 5(3)), it is clear that Δ_{−𝜆−𝐷}^⋆ makes sense in Hom(𝑉,(𝑉⊗𝑉)[[𝜆]]).
(b) Note that the expression (Δ_𝜆⊗Id)Δ_𝜆+𝜇 is well defined in Hom(𝑉,(𝑉⊗𝑉)[[𝜆,𝜇]]).
(c) The integral notation in (7) and (8) is interpreted as follows: ^𝜆₀Δ_𝜆⊗IdΔ_𝜇𝑑𝜇=_{𝑛,𝑚≥0}Δ_𝑛⊗IdΔ_𝑚𝜆^𝑛+𝑚+1()!,𝑛!𝑚+1_0−𝐷⋆Δ_𝜆𝑑𝜆=−_𝑛≥0Δ_𝑛−𝐷^⋆(𝑛+1), (99) unlike the case of Lie conformal algebras, where the parameter and its evaluation on 𝐷 are always placed on the left. Indeed, from now on expansions inside definite integrals will be interpreted with all the powers of the parameters 𝜆 and 𝜇 placed on the right.
(d) The names of properties (1)--(9) in Proposition 19 are motivated by the Definition of vertex algebra presented in Section 6 [9]. Formulas (1), (2), and (9) are the dual version of the axioms of a left symmetric ℂ[𝐷]-differential algebra with unit (cf. Section 4 [9]). Formula (3) is related to the fact that (Id⊗𝑐)⅄(𝑥) is holomorphic. On the other hand, (4), (5), and (6) are dual to the axioms of a ℂ[𝐷]-Lie conformal algebra. Finally, (7) and (8) relate the two structures in a dual version of the Wick formula and the compatibility described in Theorem 6.9 [9].

Proof of Proposition 19. (1) The proof follows from (12) and (13).
(2) It is clear if we take the coefficient of 𝑥⁰ in (22) and (26).
(3) In light of (13), it is clear that (Id⊗𝑐)Δ_𝜆=Res_𝑥𝑒^𝜆𝑥(Id⊗𝑐)⅄(𝑥)=0.
(4) Using (22), we have 𝐷^⋆⊗IdΔ_𝜆=Res_𝑥𝑒^𝜆𝑥𝐷^⋆⊗Id⅄(𝑥)=−Res_𝑥𝑑𝑒𝑑𝑥^𝜆𝑥⅄(𝑥)=Δ_𝜆(−𝜆). (100) Using (26) and part (4) proved above we have Id⊗𝐷^⋆Δ_𝜆=Res_𝑥𝑒^𝜆𝑥−𝑑⅄(𝑥)(𝑥)𝐷𝑑𝑥+⅄^⋆=Δ_𝜆𝜆+𝐷^⋆. (101)
(5) Using (25) we have 𝑇Δ_𝜆=Res_𝑥𝑒^𝜆𝑥𝑇⅄(𝑥)=Res_𝑥⅄(−𝑥)𝑒^{𝑥(𝜆+𝐷⋆)}=−Δ_{−𝜆−𝐷}^⋆. (102)
(6) Note that for 𝑛≥0 the (𝑛-)coproduct formula (88) can be rewritten as follows: Δ_𝑛⊗Id⅄𝑦=Res_𝑥𝑥−𝑦^𝑛𝑦⅄(𝑥)−((𝑥⅄𝑦.Id⊗⅄𝑇⊗Id)(Id⊗⅄)) (103) Changing the variables 𝑥 for 𝑧 and 𝑦 for 𝑥 in (103) we obtain that the collection of (𝑛-)coproduct identities (88) for 𝑛≥0 is equivalent to Δ_𝜆⊗Id⅄(𝑥)=_𝑛≥0Δ_𝑛⅄(𝑥)𝜆⊗Id^𝑛𝑛!=_𝑛≥0Res_𝑧(𝑧−𝑥)^𝑛×(𝑥⅄(𝑧)−()×((𝑧⅄(𝑥𝜆((Id⊗⅄))𝑇⊗IdId⊗⅄))))^𝑛𝑛!=Res_𝑧𝑒^{𝜆(𝑧−𝑥)}×(𝑥⅄(𝑧)−()×((𝑧⅄(𝑥((Id⊗⅄))𝑇⊗IdId⊗⅄))))=𝑒^−𝜆𝑥(Id⊗⅄(𝑥))Δ_𝜆−(𝑇⊗Id)Id⊗Δ_𝜆⅄(𝑥). (104) Multiplying (104) by 𝑒^{(𝜆+𝜇)𝑥} and taking Res_𝑥 we obtain Δ_𝜆⊗IdΔ_𝜆+𝜇=Id⊗Δ_𝜇Δ_𝜆−(𝑇⊗Id)Id⊗Δ_𝜆Δ_𝜇. (105)
(7) Multiplying (104) by 𝑥⁻¹, using that 𝑒^−𝜆𝑥𝑥⁻¹=𝑥⁻¹+∫₀^−𝜆𝑒^𝜇𝑥𝑑𝜇 and taking Res_𝑥 we have Δ_𝜆⊗IdΛ=(Id⊗Λ)Δ_𝜆−(𝑇⊗Id)Id⊗Δ_𝜆Λ+₀^−𝜆Id⊗Δ_𝜇Δ_𝜆−(𝑇⊗Id)Id⊗Δ_𝜆Δ_𝜇𝑑𝜇. (106) Using the conformal co-Jacobi formula (6) and making a change of variables we can rewrite (106) as follows: Δ_𝜆⊗IdΛ=(Id⊗Λ)Δ_𝜆−(𝑇⊗Id)Id⊗Δ_𝜆Λ−∫^𝜆₀Δ_𝜆⊗IdΔ_𝜇𝑑𝜇. (107)
(8) Using that 𝑥⁻¹𝑒^𝑥𝐷⋆=𝑥⁻¹−∫^−𝐷⋆₀𝑒^−𝜆𝑥𝑑𝜆 and coskew symmetry (25) we have Λ=Res_𝑥𝑥⁻¹⅄(𝑥)=Res_𝑥𝑇⅄(−𝑥)𝑥⁻¹𝑒^𝑥𝐷⋆=Res_𝑥𝑇⅄(−𝑥)𝑥⁻¹−^−𝐷⋆₀Res_𝑥𝑇⅄(−𝑥)𝑒^−𝜆𝑥𝑑𝜆=𝑇Λ+^−𝐷⋆₀𝑇Δ_𝜆𝑑𝜆. (108) Using conformal coskew symmetry proved in (5) and making a change of variables we can rewrite (108) as follows: Λ=𝑇Λ+⁰_−𝐷⋆Δ_𝜆𝑑𝜆. (109)
(9) In order to prove coleft symmetry we will follow the argument developed by Bakalov and Kac for the case of vertex algebras in [9]. We start by proving the following formula: ()()Λ=Λ⊗IdΛ−Id⊗Λ_{𝐷⋆⊗Id0}Id⊗Δ_𝜆Λ+(𝑑𝜆𝑇⊗Id)^𝐷⋆⊗Id₀Id⊗Δ_𝜆𝑑𝜆Λ, (110) where the integral notation is interpreted as follows: first we expand the 𝜆 coproduct and put the powers of 𝜆 on the right inside the integral and then we take the definite integral by the usual rules inside the parenthesis. Namely, ^𝐷⋆⊗Id₀Id⊗Δ_𝜆Λ=𝑑𝜆_{𝐷⋆⊗Id0𝑛≥0}Id⊗Δ_𝑛𝜆^𝑛Λ𝑛!𝑑𝜆=_𝑛≥0𝐷^⋆(𝑛+1)⊗Δ_𝑛Λ. (111) In order to prove (110) we will use the (−1)-coproduct formula proved in Proposition 18: Δ₋₁⊗Id⅄𝑦=Res_𝑥𝑥−𝑦⁻¹Id⊗⅄𝑦⅄(𝑥)−−𝑦+𝑥⁻¹×(𝑇⊗Id)(Id⊗⅄(𝑥))⅄𝑦. (112) We multiply (112) by 𝑦⁻¹, take Res_𝑦, and use that _𝑘⁻¹=(−1)^𝑘 for 𝑘≥0 to obtain (Λ⊗Id)Λ=Res_𝑦𝑦⁻¹Res_𝑥𝑥−𝑦⁻¹𝑦⅄(𝑥)−Id⊗⅄−𝑦+𝑥⁻¹()×((𝑥⅄𝑦𝑇⊗IdId⊗⅄))=Res_𝑦𝑦⁻¹_𝑘≥0(−1)^𝑘−𝑦^𝑘Id⊗⅄𝑦Δ_−𝑘−1−−𝑦^−𝑘−1()×𝑇⊗IdId⊗Δ_𝑘⅄𝑦=Res_𝑦𝑦⁻¹_{𝑛∈ℤ𝑘≥0}𝑦^{−𝑛−1+𝑘}Id⊗Δ_𝑛Δ_−𝑘−1+𝑦^{−𝑛−2−𝑘}(𝑇⊗Id)Id⊗Δ_𝑘Δ_𝑛=_𝑘≥0Id⊗Δ_𝑘−1Δ_−𝑘−1+(𝑇⊗Id)Id⊗Δ_𝑘Δ_−𝑘−2=Id⊗Δ₋₁Δ₋₁+_𝑘≥1Id⊗Δ_𝑘−1Δ_−𝑘−1+(𝑇⊗Id)Id⊗Δ_𝑘−1Δ_−𝑘−1. (113) Using that ((𝐷^⋆)^(𝑗)⊗Id)Δ₋₁=Δ_−1−𝑗 for 𝑗≥0, which follows from (36), we can rewrite the right hand side of (113) to get (Λ⊗Id)Λ=(Id⊗Λ)Λ+_𝑘≥1Id⊗Δ_𝑘−1𝐷^⋆(𝑘)+()⊗Id𝑇⊗IdId⊗Δ_𝑘−1×𝐷^⋆(𝑘)⊗IdΛ. (114) Using (111) it is easy to check that the right hand side of (114) equals (Id⊗Λ)Λ+^𝐷⋆⊗Id₀Id⊗Δ_𝜆Λ+()∫𝑑𝜆𝑇⊗Id^𝐷⋆⊗Id₀Id⊗Δ_𝜆𝑑𝜆Λ, (115) finishing the proof of (110).
In order to complete the proof of (9), we note that the right hand side of (110) is invariant under the operator (𝑇⊗Id). Therefore, its left hand side is invariant under (𝑇⊗Id) as well. This clearly implies coleft symmetry (97), finishing the proof of Proposition 19.

Let us suppose that we start with a ℂ-vector space 𝑉 and four linear maps: 𝐷𝑐∶𝑉⟶ℂ,_⋆Δ∶𝑉⟶𝑉,Λ∶𝑉⟶𝑉⊗𝑉,_𝜆∶𝑉⟶(𝑉⊗𝑉)[[𝜆]], (116) such that 𝐷^⋆ is locally nilpotent and properties (1)--(9) of Proposition 19 hold. Then, we define a new linear map ⅄(𝑥)∶𝑉⟶(𝑉⊗𝑉)𝑥,𝑥⁻¹𝑣⟼Δ_{−(𝑑/𝑑𝑥)}(𝑣)𝑥⁻¹+𝑒^𝑥𝐷⋆⊗IdΛ(𝑣), (117) where the expression Δ_{−(𝑑/𝑑𝑥)}(𝑣)(𝑥⁻¹) is interpreted as Δ_𝜆(𝑣) evaluated in 𝜆=−𝑑/𝑑𝑥 acting on 𝑥⁻¹. Namely, if we write Δ_𝜆(𝑣)=∑_𝑛≥0Δ_𝑛(𝑣)(𝜆^𝑛/𝑛!), then Δ_{−(𝑑/𝑑𝑥)}(𝑣)𝑥⁻¹=_𝑛≥0Δ_𝑛(𝑣)−𝑑𝑑𝑥^(𝑛)𝑥⁻¹=∑_𝑛≥0Δ_𝑛(𝑣)𝑥^−𝑛−1. (118) We want to prove that (𝑉,⅄(𝑥),𝑐,𝐷^⋆) with ⅄(𝑥) defined in (117) is a vertex coalgebra. Indeed, we will show that providing a ℂ-vector space with the structure described above is equivalent to providing 𝑉 with a coalgebra structure.

Remarks 6. (a) If we consider the expansion ⅄(𝑥)=∑_𝑛∈ℤΔ_𝑛𝑥^−𝑛−1 for the map ⅄(𝑥) defined in (117), then we are defining Δ_−𝑛−1∶=((𝐷^⋆)^(𝑛)⊗Id)Λ for 𝑛≥0.
(b) In this context, we can recover the maps Λ and Δ_𝜆 using the formulas Λ=Res_𝑥𝑥⁻¹⅄(𝑥), Δ_𝜆=Res_𝑥𝑒^𝜆𝑥⅄(𝑥) with ⅄(𝑥) defined as in (117).
(c) If we have a vertex coalgebra (𝑉,⅄(𝑥),𝑐,𝐷^⋆) and define maps Λ and Δ_𝜆 as we did in (86), (87), then due to (36) we can reobtain ⅄(𝑥) as in (117).

We introduce the last definition.

Definition 20. A vertex coalgebra consists of a ℂ-vector space 𝑉 together with linear maps 𝐷𝑐∶𝑉⟶ℂ,_⋆Δ∶𝑉⟶𝑉,Λ∶𝑉⟶𝑉⊗𝑉,_𝜆∶𝑉⟶(𝑉⊗𝑉)[[𝜆]], (119) such that 𝐷^⋆ is locally nilpotent and conditions (1)--(9) of Proposition 19 hold.

As we mentioned before, we want to prove the following result.

Theorem 21. Definition 20 is equivalent to Definition 10.

In light of the equivalence between Definitions 10 and 1, Propositions 5(3) and 19 imply that the axioms of Definition 20 are satisfied under the conditions of Definition 10 if we define Λ and Δ_𝜆 as in (86)-(87).

In order to prove the other implication let us first define ⅄(𝑥) as in (117). Using that 𝐷^⋆ is locally nilpotent, it is clear that the image of ⅄(𝑥) actually lies in (𝑉⊗𝑉)((𝑥⁻¹)). Second, note that using (89) and (90) it is clear that 𝑐𝐷^⋆=0. Indeed, (89) and (90) imply 𝑐𝐷^⋆=𝑐(𝑐⊗Id)Λ𝐷^⋆=(𝑐⊗𝑐)Λ𝐷^⋆=(𝑐⊗Id)(Id⊗𝑐)Λ𝐷^⋆=(𝑐⊗Id)(Id⊗𝑐)𝐷^⋆Λ+()⊗Id𝑐⊗Id)(Id⊗𝑐Id⊗𝐷^⋆Λ=(𝑐⊗Id)𝐷^⋆()Λ+(⊗IdId⊗𝑐Id⊗𝑐)Id⊗𝐷^⋆(𝑐⊗Id)Λ=𝑐𝐷^⋆+𝑐𝐷^⋆. (120) Using that 𝑐𝐷^⋆=0 together with (91) and (92), we have (𝑐⊗Id)Δ_𝜆=0=(Id⊗𝑐)Δ_𝜆. The fact that 𝑐𝐷^⋆=0 together with (𝑐⊗Id)Δ_𝜆=0 and (89) implies (50). On the other hand, (91) and (89) imply (51). Next, note that (52) follows from (92). Therefore, in order to finish the proof we need to show that coskew symmetry (53) and the coassociator formula (54) follow from Definition 20.

We start with the proof of coskew symmetry. Note that (90) and (92), together with (52), proven to hold above, imply that ⅄(𝑥) satisfies the 𝐷^⋆-bracket formula ⅄(𝑥)𝐷^⋆=Id⊗𝐷^⋆⅄(𝑥)+𝑑⅄(𝑥).𝑑𝑥 (121) If we define the linear map ⅄^𝑜𝑝(𝑥)∶𝑉⟶(𝑉⊗𝑉)𝑥⁻¹𝑣⟶𝑇⅄(−𝑥)𝑒^𝑥𝐷⋆, (122) it follows that ⅄^𝑜𝑝(𝑥) also satisfies 𝑑⅄𝑑𝑥^𝑜𝑝(𝑥)=𝐷^⋆⊗Id⅄^𝑜𝑝(𝑥). (123) Next, we define Δ_𝜆^𝑜𝑝∶=Res_𝑥𝑒^𝑥𝜆⅄^𝑜𝑝(𝑥),Λ^𝑜𝑝∶=Res_𝑥𝑥⁻¹⅄^𝑜𝑝(𝑥). (124) Using the coefficient version of (123) (cf. proof of Proposition 5(1)) and Remarks 6 it is clear that we can reconstruct ⅄^𝑜𝑝(𝑥)=Δ^𝑜𝑝_{−𝑑/𝑑𝑥}𝑥⁻¹+𝑒^𝑥𝐷⋆⊗IdΛ^𝑜𝑝. (125) Now, conformal coskew symmetry (93) implies Δ_𝜆^𝑜𝑝=Res_𝑥𝑒^𝜆𝑥𝑇⅄(−𝑥)𝑒^𝑥𝐷⋆=−𝑇Δ_{−𝜆−𝐷}^⋆=Δ_𝜆, (126) whereas the fact that 𝑥⁻¹𝑒^𝑥𝐷⋆=𝑥⁻¹−∫^−𝐷⋆₀𝑒^−𝜆𝑥𝑑𝜆 together with conformal coskew symmetry (93) and cocompatibility (96) implies Λ^𝑜𝑝=Res_𝑥𝑥⁻¹𝑇⅄(−𝑥)𝑒^𝑥𝐷⋆=𝑇Λ+^−𝐷⋆₀𝑇Δ_𝜆𝑑𝜆=𝑇Λ−^−𝐷⋆₀Δ_{−𝜆−𝐷}^⋆𝑑𝜆=𝑇Λ+⁰_−𝐷⋆Δ_𝜆𝑑𝜆=Λ. (127) Therefore, using (126) and (127) in (125) together with the definition of ⅄(𝑥) in (117) it follows that ⅄^𝑜𝑝(𝑥)=⅄(𝑥), which implies coskew symmetry (53).

Finally, we want to prove that the coassociator formula holds. Note that due to Proposition 18(2), it is equivalent to the (𝑛-)coproduct identity (88) for all 𝑛∈ℤ. If we denote cop(𝑛)[⅄(𝑧)]∶=Res_𝑥(𝑥−𝑧)^𝑛((𝑧⅄(𝑥)−()Id⊗⅄))−𝑧+𝑥^𝑛()×((𝑥⅄(𝑧𝑇⊗IdId⊗⅄))) (128) then the (𝑛-)coproduct identity is equivalent to Δ_𝑛⊗Id⅄(𝑧)−cop(𝑛)[⅄(𝑧)]=0. (129) In order to prove that it holds for every 𝑛∈ℤ we introduce the following result.

Lemma 22. Let 𝐷^⋆∈𝐸𝑛𝑑𝑉 be a locally nilpotent map and let ⅄(𝑥)∶𝑉⟶(𝑉⊗𝑉)𝑥⁻¹, (130) be a linear map satisfying ⅄(𝑥)𝐷^⋆=Id⊗𝐷^⋆⅄(𝑥)+𝐷^⋆⅄(𝑥),𝑑⊗Id⅄(𝑥)=𝐷𝑑𝑥^⋆⊗Id⅄(𝑥). (131)(132) Then, ⅄(𝑥) satisfies the (𝑛-)coproduct identity (88) for all 𝑛∈ℤ if and only if 𝑅𝑒𝑠_𝑧Δ_𝑛⊗Id⅄(𝑧)−𝑐𝑜𝑝(𝑛)[⅄(𝑧)]𝐹=0 (133) for all 𝑛≥−1 with 𝐹=𝑧⁻¹ and 𝐹=𝑒^𝜆𝑧.

Proof. If we assume that the (𝑛-)coproduct formula (88) holds for every 𝑛∈ℤ, then it is clear that (133) holds for 𝑛≥−1. Conversely, if we assume that (133) holds for 𝑛≥−1 we will show that it holds with 𝐹 replaced by (𝑑/𝑑𝑧)𝐹. In fact, Res_𝑧Δ_𝑛⅄(𝑧)(𝑛⅄(𝑧𝑑⊗Id−cop)[)]𝐹𝑑𝑧=−Res_𝑧𝑑Δ𝑑𝑧^𝑛⊗Id⅄(𝑧)−cop(𝑛)[⅄(𝑧)]𝐹. (134) Using (131) and the coefficient version of (132) we have that 𝑑Δ𝑑𝑧^𝑛⅄(𝑧)=Δ⊗Id_𝑛⊗Id𝐷^⋆⅄(𝑧)=𝐷⊗Id_⋆⊗Id⊗IdΔ_𝑛⅄(𝑧)+⊗IdId⊗𝐷^⋆⊗IdΔ_𝑛⊗Id⅄(𝑧), (135) whereas, (132) implies 𝑑(𝑛⅄(𝑧(⅄(𝑧+𝑑𝑧cop)[)]=−𝑛cop𝑛−1)[)]Id⊗𝐷^⋆⊗Idcop(𝑛)[⅄(𝑧)]. (136) Using (132) and integration by parts we have −𝑛cop(𝑛−1)[⅄(𝑧)]=−Rex_𝑥𝑑()𝑑𝑥𝑥−𝑧^𝑛((𝑧⅄(𝑥)−𝑑Id⊗⅄))()𝑑𝑥−𝑧+𝑥^𝑛×((𝑥⅄(𝑧=𝐷𝑇⊗Id)(Id⊗⅄)))_⋆⊗Id⊗Idcop(𝑛)[⅄(𝑧)]. (137) Replacing (135) and (137) in (134) we obtain Res_𝑧Δ_𝑛⅄(𝑧)(𝑛⅄(𝑧𝑑⊗Id−cop)[)]𝐹𝐷𝑑𝑧=−^⋆⊗Id⊗Id×Res_𝑧Δ_𝑛⅄(𝑧)(𝑛⅄(𝑧𝐹−⊗Id−cop)[)]Id⊗𝐷^⋆⊗Id×Res_𝑧Δ_𝑛⊗Id⅄(𝑧)−cop(𝑛)[⅄(𝑧)]𝐹=0. (138) Hence, it follows that (133) also holds for all 𝐹=𝑧^𝑘, 𝑘≤0. Taking coefficients at powers of 𝜆 for 𝐹=𝑒^𝜆𝑧, we have that (133) also holds for all 𝐹=𝑧^𝑘, 𝑘≥0. This implies the (𝑛-)coproduct identity (88) for 𝑛≥−1. Finally, it follows from the coefficient version of (132) together with (137) that for 𝑛≠0 the (𝑛-)coproduct identity implies the (𝑛−1)-coproduct identity. Therefore, the (𝑛-)coproduct identity holds for all 𝑛∈ℤ.

In light of Lemma 22, in order to finish the proof of Theorem 21 it is enough to prove that (133) holds for all 𝑛≥−1 with 𝐹=𝑧⁻¹ and 𝐹=𝑒^𝜆𝑧. In the proof of Proposition 19(6), we proved that the conformal co-Jacobi formula (94) is equivalent to (133) for all 𝑛≥0 and 𝐹=𝑒^𝜆𝑧. In the proof of (7), we proved that the co-Wick formula (95) together with the conformal co-Jacobi identity (94) implies (133) for all 𝑛≥0 and 𝐹=𝑧⁻¹. In order to prove the remaining two cases we will need the following result.

Proposition 23. Let (𝑉,⅄(𝑥),𝑐,𝐷^⋆) be a vertex coalgebra as in Definition 20. Then, (1)(Λ⊗Id)Δ_𝜆=𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆Λ+(𝑇⊗Id)𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆Λ+^𝜆₀(𝑇⊗Id)Id⊗Δ_𝜆−𝜇Δ_𝜇(2)()Λ=𝑑𝜇.)(Λ⊗IdΛ−Id⊗Λ_{𝐷⋆⊗Id0}Id⊗Δ_𝜆Λ+()𝑑𝜆𝑇⊗Id^𝐷⋆⊗Id₀Id⊗Δ_𝜆𝑑𝜆Λ. (139)(140)

Proof. (1) First, using cocompatibility (96) we have Δ_𝜆⊗IdΛ=Δ_𝜆⊗Id𝑇Λ+⁰_−𝐷⋆Δ_𝜆⊗IdΔ_𝜇𝑑𝜇, (141) which, due to the co-Wick formula (95), implies (Id⊗Λ)Δ_𝜆=Δ_𝜆⊗Id𝑇Λ+(𝑇⊗Id)Id⊗Δ_𝜆Λ+^𝜆_−𝐷⋆Δ_𝜆⊗IdΔ_𝜇𝑑𝜇. (142) Observe that the integral notation in (142) is interpreted as follows: first we expand Δ_𝜆 and Δ_𝜇 inside the integral sign putting the powers of 𝜆 and 𝜇 on the right of the sum. Then, we take the definite integral by the usual rules. Namely, ^𝜆_−𝐷⋆Δ_𝜆⊗IdΔ_𝜇𝑑𝜇=^𝜆_{−𝐷⋆𝑛,𝑚≥0}Δ_𝑛⊗IdΔ_𝑚𝜆^𝑛𝜇𝑛!𝑚^𝑚!𝑑𝜇=_{𝑛,𝑚≥0}Δ_𝑛⊗IdΔ_𝑚𝜆^𝑛+𝑚+1𝑛!(𝑚+1)!−_𝑚≥0Δ_𝜆⊗IdΔ_𝑚−𝐷^⋆(𝑚+1). (143)
After replacing 𝜆 by −𝜆, we compose (142) with 𝑒^{𝐷⋆(𝑑/𝑑𝜆)} to obtain (Id⊗Λ)Δ_−𝜆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=Δ_−𝜆⊗Id𝑇Λ𝑒^{𝐷⋆(𝑑/𝑑𝜆)}+(𝑇⊗Id)Id⊗Δ_−𝜆Λ𝑒^{𝐷⋆(𝑑/𝑑𝜆)}+^−𝜆_−𝐷⋆Δ_−𝜆⊗IdΔ_𝜇𝑑𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}, (144) where 𝑒^{𝐷⋆(𝑑/𝑑𝜆)} on the right indicates the position of the powers of 𝐷^⋆ in the expansion. The powers of (𝑑/𝑑𝜆) act on 𝜆 as if they were placed on the left of each term, namely, as the derivation map. For instance, the expression Δ_−𝜆𝑒^{𝐷⋆(𝑑/𝑑𝜆)} is interpreted as follows: Δ_−𝜆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=^𝑛_{𝑛≥0𝑘=0}Δ_𝑛𝐷^⋆(𝑘)𝑑𝑑𝜆^𝑘(−𝜆)^(𝑛). (145) Using (90) and (92) it follows that Λ𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗𝑒^{𝐷⋆(𝑑/𝑑𝜆)}ΔΛ,_𝜆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗𝑒^{𝐷⋆(𝑑/𝑑𝜆)}Δ_𝜆. (146)(147) Applying (146) and Taylor’s Theorem, we obtain Δ_−𝜆⊗Id𝑇Λ𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=Δ_−𝜆𝑇×𝑒⊗Id_{𝐷⋆(𝑑/𝑑𝜆)}⊗𝑒^{𝐷⋆(𝑑/𝑑𝜆)}Λ=Δ_{−𝜆−𝐷}^⋆⊗𝑒^{𝐷⋆(𝑑/𝑑𝜆)}()𝑇Λ,𝑇⊗IdId⊗Δ_−𝜆Λ𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=(𝑇⊗Id)𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_{−𝜆−𝐷}^⋆Λ. (148)(149) On the other hand, Taylor’s Theorem implies (Id⊗Λ)Δ_−𝜆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=(Id⊗Λ)Δ_{−𝜆−𝐷}^⋆,_{−𝜆−𝐷⋆}Δ_−𝜆⊗IdΔ_𝜇𝑑𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=^{−𝜆−𝐷⋆}_−𝐷⋆Δ_−𝜆⊗IdΔ_𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}𝑑𝜇. (150)(151) Using conformal coskew symmetry (93) and making a change of variables we have ^{−𝜆−𝐷⋆}_−𝐷⋆Δ_−𝜆⊗IdΔ_𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}𝑑𝜇=−^{−𝜆−𝐷⋆}_−𝐷⋆Δ_−𝜆⊗Id𝑇Δ_{−𝜇−𝐷}^⋆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=𝑑𝜇_𝜆0Δ_−𝜆⊗Id𝑇Δ_𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}𝑑𝜇. (152) Using (92) it is easy to check that 𝑇Δ_𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=𝑇Id⊗𝑒^{(𝐷⋆−𝜇)(𝑑/𝑑𝜆)}Δ_𝜇=𝑒^{(𝐷⋆−𝜇)(𝑑/𝑑𝜆)}⊗Id𝑇Δ_𝜇. (153) Hence, using (151) together with (152) and (153), we obtain ^−𝜆_−𝐷⋆Δ_−𝜆⊗IdΔ_𝜇𝑑𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=^𝜆₀Δ_−𝜆⊗Id𝑒^{(𝐷⋆−𝜇)(𝑑/𝑑𝜆)}⊗Id𝑇Δ_𝜇𝑑𝜇, (154) which due to Taylor’s Theorem implies ^−𝜆_−𝐷⋆Δ_−𝜆⊗IdΔ_𝜇𝑑𝜇𝑒^{𝐷⋆(𝑑/𝑑𝜆)}=∫^𝜆₀Δ_{−𝜆+𝜇−𝐷}^⋆⊗Id𝑇Δ_𝜇𝑑𝜇. (155) Replacing (148), (149), (150), and (155) in (144) and using Taylor’s Theorem we obtain (Id⊗Λ)Δ_{−𝜆−𝐷}^⋆=Δ_{−𝜆−𝐷}^⋆⊗𝑒^{𝐷⋆(𝑑/𝑑𝜆)}+()𝑒𝑇Λ𝑇⊗Id^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_{−𝜆−𝐷}^⋆Λ+^𝜆₀Δ_{−𝜆+𝜇−𝐷}^⋆⊗Id𝑇Δ_𝜇𝑑𝜇. (156) Next, using conformal coskew symmetry (93) we can rewrite (156) as follows: (Id⊗Λ)𝑇Δ_𝜆=𝑇Δ_𝜆⊗𝑒^{𝐷⋆(𝑑/𝑑𝜆)}+()𝑒𝑇Λ𝑇⊗Id^{𝐷⋆(𝑑/𝑑𝜆)}⊗𝑇Δ_𝜆Λ+^𝜆₀𝑇Δ_𝜆−𝜇⊗Id𝑇Δ_𝜇𝑑𝜇. (157) Finally, it is easy to check that if we apply (Id⊗𝑇)(𝑇⊗Id) to (157), then we obtain (139).
(2) Cocompatibility (96) implies ()()Λ=()()Λ+Λ⊗IdΛ−Id⊗ΛΛ⊗Id𝑇Λ−Id⊗𝑇)(Id⊗Λ_0−𝐷⋆(Λ⊗Id)Δ_𝜆−𝑑𝜆_{0Id⊗(−𝐷⋆)}Id⊗Δ_𝜆𝑑𝜆Λ. (158) On the other hand, coleft symmetry (97) and cocompatibility (96) imply ()()Λ()()Λ()Λ⊗Id𝑇Λ−Id⊗𝑇)(Id⊗Λ=−Id⊗𝑇)(Λ⊗IdΛ+Id⊗𝑇)(𝑇Λ⊗Id=−Id⊗𝑇⁰_{(−𝐷⋆)⊗Id}Δ_𝜆Λ()⊗Id𝑑𝜆=−𝑇⊗Id⁰_{Id⊗(−𝐷⋆)}Id⊗Δ_𝜆𝑑𝜆𝑇Λ. (159) Using cocompatibility again on the right hand side of (159) we obtain ()()Λ()Λ⊗Id𝑇Λ−Id⊗𝑇)(Id⊗Λ=−𝑇⊗Id⁰_{Id⊗(−𝐷⋆)}Id⊗Δ_𝜆Λ+()𝑑𝜆𝑇⊗Id⁰_{Id⊗(−𝐷⋆)}Id⊗Δ_𝜆×𝑑𝜆_0−𝐷⋆Δ_𝜇𝑑𝜇. (160) Now, note that by using (92) it is clear that for 𝑗≥0Id⊗−𝐷^⋆𝑗Δ_𝜇=Δ_𝜇−𝐷^⋆−𝜇^𝑗. (161)
Therefore, the double integral in (160) satisfies ⁰_{Id⊗(−𝐷⋆)}Id⊗Δ_𝜆𝑑𝜆⁰_−𝐷⋆Δ_𝜇=𝑑𝜇_{0−𝐷⋆0Id⊗(−𝐷⋆)}Id⊗Δ_𝜆𝑑𝜆Δ_𝜇=𝑑𝜇_{0−𝐷⋆0−𝐷⋆−𝜇}Id⊗Δ_𝜆Δ_𝜇=𝑑𝜆𝑑𝜇_{0−𝐷⋆𝜇−𝐷⋆}Id⊗Δ_𝜆−𝜇Δ_𝜇𝑑𝜆𝑑𝜇=⁰_−𝐷⋆⁰_𝜆Id⊗Δ_𝜆−𝜇Δ_𝜇𝑑𝜆𝑑𝜆=−⁰_−𝐷⋆⁰_𝜆Id⊗Δ_𝜆−𝜇Δ_𝜇𝑑𝜆𝑑𝜆. (162) Applying (𝑇⊗Id) to (162) and using (139), we obtain (𝑇⊗Id)⁰_{Id⊗(−𝐷⋆)}Id⊗Δ_𝜆𝑑𝜆⁰_−𝐷⋆Δ_𝜇𝑑𝜇=−⁰_−𝐷⋆(Λ⊗Id)Δ_𝜆−𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆Λ−(𝑇⊗Id)𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆Λ𝑑𝜆. (163) Now, using (163) and (160) in (158) we have ()()Λ=Λ⊗IdΛ−Id⊗Λ_0−𝐷⋆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆−Λ𝑑𝜆_{0Id⊗(−𝐷⋆)}Id⊗Δ_𝜆Λ+()𝑑𝜆𝑇⊗Id⁰_−𝐷⋆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆−()Λ𝑑𝜆𝑇⊗Id⁰_{Id⊗(−𝐷⋆)}Id⊗Δ_𝜆𝑑𝜆Λ. (164) Note that by using (90) we can rewrite ⁰_−𝐷⋆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆=∫Λ𝑑𝜆_{0−(𝐷⋆⊗Id+Id⊗𝐷⋆)}𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆𝑑𝜆Λ. (165)
Next, observe that Taylor’s Theorem implies ⁰_{−(𝐷⋆⊗Id+Id⊗𝐷⋆)}𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆=𝑑𝜆_{0−(𝐷⋆⊗Id+Id⊗𝐷⋆)}𝑒^{(𝐷⋆⊗Id)(𝑑/𝑑𝜆)}Id⊗Δ_𝜆=𝑑𝜆_{0−(𝐷⋆⊗Id+Id⊗𝐷⋆)}Id⊗Δ_𝜆+(𝐷^⋆_⊗Id)=𝑑𝜆_{0−(𝐷⋆⊗Id+Id⊗𝐷⋆)𝑘≥0}Id⊗Δ_𝑘𝜆+𝐷^⋆⊗Id^(𝑘)=𝑑𝜆_𝑘≥0Id⊗Δ_𝑘𝜆+𝐷^⋆⊗Id^(𝑘+1)_(𝐷^𝜆=0_{𝜆=−⋆⊗Id+Id⊗𝐷⋆)}=_𝑘≥0Id⊗Δ_𝑘𝐷^⋆(𝑘+1)⊗Id−_𝑘≥0Id⊗Δ_𝑘Id⊗−𝐷^⋆(𝑘+1)=^𝐷⋆⊗Id₀Id⊗Δ_𝜆𝑑𝜆+⁰_{Id⊗(−𝐷⋆)}Id⊗Δ_𝜆𝑑𝜆. (166) Hence, in light of (165) and (166) we have ⁰_−𝐷⋆𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆−Λ𝑑𝜆_{0Id⊗(−𝐷⋆)}Id⊗Δ_𝜆Λ=𝑑𝜆_{𝐷⋆⊗Id0}Id⊗Δ_𝜆𝑑𝜆Λ. (167) Finally, using (167) in (164) it is clear that (140) holds, finishing the proof of Proposition 23.

Now, we can finish the proof of Theorem 21. On the one hand, we have already proved that under the axioms of Definition 20, the 𝐷^⋆ formula (52) holds. This implies that ((𝐷^⋆)^(𝑖)⊗Id)Δ₋₁=Δ_−1−𝑖 for all 𝑖≥0, if we consider the expansion ⅄(𝑥)=∑_𝑛∈ℤΔ_𝑛𝑥^−𝑛−1 with ⅄(𝑥) defined as in (117). Therefore, in the proof of Proposition 19(9) we proved that (110) is equivalent to (133) for 𝑛=−1 and 𝐹=𝑧⁻¹. In light of Proposition 23(2), it is clear that, under the axioms of Definition 20, that case holds. To finish the proof of Theorem 21, we need to show that (133) holds for 𝑛=−1 and 𝐹=𝑒^𝜆𝑧. We will prove that Proposition 23 (1) implies this last case.

First, note that considering ⅄(𝑧) as in (117), we have (𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Id)⅄(𝑧)=𝑒^{(𝑑/𝑑𝑧)(𝑑/𝑑𝜆)}⅄(𝑧) due to the 𝐷^⋆ formula (52) already proven to hold. This implies (𝑇⊗Id)𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆Λ=Res_𝑧𝑧⁻¹(𝑇⊗Id)𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆⅄(𝑧)=Res_𝑧𝑧⁻¹𝑒^{(𝑑/𝑑𝑧)(𝑑/𝑑𝜆)}(𝑇⊗Id)Id⊗Δ_𝜆⅄(𝑧)=Res_𝑧(𝑇⊗Id)Id⊗Δ_{𝜆−(𝑑/𝑑𝑧)}𝑧⁻¹⅄(𝑧), (168) where the parenthesis indicate that 𝑑/𝑑𝑧 only acts on 𝑧⁻¹. Second, using that Id⊗Δ_𝜆−𝜇𝑒^𝜇𝑧=Id⊗Δ_{𝜆−(𝑑/𝑑𝑧)}𝑒^𝜇𝑧, (169) we have ^𝜆₀(𝑇⊗Id)Id⊗Δ_𝜆−𝜇Δ_𝜇=𝑑𝜇_𝜆0Res_𝑧(𝑇⊗Id)Id⊗Δ_𝜆−𝜇𝑒^𝜇𝑧⅄(𝑧)𝑑𝜇=Res_𝑧(𝑇⊗Id)Id⊗Δ_{𝜆−(𝑑/𝑑𝑧)}^𝜆₀𝑒^𝜇𝑧𝑑𝜇⅄(𝑧). (170)

Third, straightforward computation shows that 𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗Δ_𝜆Λ=Res_𝑧𝑒^{𝐷⋆(𝑑/𝑑𝜆)}⊗⅄(𝑧)𝑒^𝜆𝑧Λ=Res_𝑧𝑒^𝜆𝑧𝑒^𝑧𝐷⋆⊗(𝑧)Λ. (171) Replacing (168), (170), and (171) in (139) and using that 𝑧⁻¹+∫^𝜆₀𝑒^𝜇𝑧𝑑𝜇=𝑒^𝜆𝑧𝑧⁻¹, we obtain Res_𝑧(Λ⊗Id)⅄(𝑧)𝑒^𝜆𝑧=Res_𝑧𝑒^𝜆𝑧𝑒^𝑧𝐷⋆⊗⅄(𝑧)Λ+Res_𝑧()×𝑇⊗IdId⊗Δ_{𝜆−(𝑑/𝑑𝑧)}𝑧⁻¹+^𝜆₀𝑒^𝜇𝑧𝑑𝜇⅄(𝑧)=Res_𝑧𝑒^𝜆𝑧𝑒^𝑧𝐷⋆⊗⅄(𝑧)Λ+Res_𝑧()×𝑇⊗IdId⊗Δ_{𝜆−(𝑑/𝑑𝑧)}𝑒^𝜆𝑧𝑧⁻¹⅄(𝑧). (172) Now, note that using that 𝑑𝜆−𝑑𝑧^𝑛𝑒^𝜆𝑧=𝑒^𝜆𝑧−𝑑𝑑𝑧^𝑛 (173) we can rewrite the last term in (172) as follows: Res_𝑧(𝑇⊗Id)Id⊗Δ_{−(𝑑/𝑑𝑧)}𝑧⁻¹⅄(𝑧)𝑒^𝜆𝑧. (174) Next, we use (174) in (172) to obtain Res_𝑧(Λ⊗Id)⅄(𝑧)𝑒^𝜆𝑧=Res_𝑧𝑒^𝑧𝐷⋆(𝑧)()×⊗⅄Λ+𝑇⊗IdId⊗Δ_{−(𝑑/𝑑𝑧)}𝑧⁻¹⅄(𝑧)𝑒^𝜆𝑧. (175)

Finally, note that using the definition of ⅄(𝑧) in (117), it is easy to check that 𝑒^𝑧𝐷⋆⊗⅄(𝑧)Λ+(𝑇⊗Id)Id⊗Δ_{−(𝑑/𝑑𝑧)}𝑧⁻¹⅄(𝑧)=cop(−1)[⅄(𝑧)]. (176) Therefore, replacing (176) in (175) we obtain Res_𝑧(Λ⊗Id)⅄(𝑧)𝑒^𝜆𝑧=Res_𝑧cop(−1)[⅄(𝑧)]𝑒^𝜆𝑧, (177) which implies (133) for 𝑛=−1 and 𝐹=𝑒^𝜆𝑧, finishing the proof of Theorem 21.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Florencia Orosz Hunziker was supported by a scholarship granted by Conicet, Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina. José I. Liberati was supported in part by Conicet and ANPCyT.