#### 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 [24]. 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 [68]. 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 formulas. Finally, in Section 5, we define a vertex coalgebra in terms of axioms regarding the -coproducts for .

#### 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 , , and commuting formal variables and define the formal -function to be Given an integer , we define where for and . Note that is a formal power series in nonnegative powers of . The formal residue “,” of a series refers to the coefficient of the negative first power; that is, We will also use Taylor’s Theorem, which states that given a -vector space and a formal series , we have We define the linear map to be the transposition operator described by for all .

Next, we enumerate basic properties of the formal series that we will need throughout this work: We will also make use of the following result (cf. Proposition [11]).

Given a formal Laurent series with coefficients which are homomorphisms from a vector space to a vector space , if exists (i.e., when applied to any element of setting leads to only finite sums in ), we have In particular, we have

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 -vector space , together with linear maps called the coproduct and covacuum map, respectively, satisfying the following axioms for all .(i)Left counit: (ii)Cocreation: (iii)Truncation: (iv)Co-Jacobi identity:

Remarks 1. (a) The operator is linear so that, for example, acting on the coefficients of is well defined.
(b) Note that when each expression is applied to any element of , 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 be a vertex coalgebra as in Definition 1. Then, the following formulas hold.
(1) Cocommutator formula:
(2) Coassociator formula:

Proof. (1) Taking to the co-Jacobi identity (15), we obtain Using (5) twice on the right hand side of (18) we obtain (16).
(2) Taking to the Co-Jacobi identity (15) we obtain
Using (5) and (7) on the left hand side of (19) we get 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, , to the comultiplication operator. In order to study that effect we need to introduce a map. Given a vertex coalgebra , we define the linear map as The following proposition obtained in [1] describes the main properties of the operator and its relationship to the formal derivative.

Proposition 3. The map satisfies the following properties:
(1)
(2)
(3)
(4) coskew symmetry:
(5)
(6)

We consider the following expansion of the map : where is the coefficient of in the series . If we look for the coefficient of in the co-Jacobi identity (15) we obtain what we will call the co-Borcherds identity: 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 -vector space endowed with a family of linear coproducts indexed by and a linear map :
satisfying the following axioms. (i)Left counit: (ii)Cocreation: (iii)Truncation: for each there exists such that (iv)Co-Borcherds identity: for all ,

Under this coefficient approach the definition of corresponds to the linear map

Proposition 5. Let be a vertex coalgebra as in Definition 4 and defined as in (35). Then, the following properties hold. (1) For all , (2) If we denote for , then(3) The operator is locally nilpotent. Namely, for each there exists such that     for  all  .

Proof. (1) If we take the generating function in of (36) we obtain 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 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 be a vertex coalgebra as in Definition 4. Then, the following formulas hold for every .
(1) Coefficient cocommutator formula:
(2) Coefficient coassociator formula:
(3) Coefficient coskew symmetry:

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: Note that with the notation introduced above, the co-Borcherds identity (34) for the indices corresponds to We have the following formulas which are analogous to the results obtained in Section 3.2 [10].

Proposition 7. For all , for .

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 then, for , We want to prove that for , Note that (47) is equivalent to for , 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 ,  , and  imply the co-Borcherds identity for the other index.

Now, we can prove the following statement.

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

Proof. We assume the co-Borcherds identity for all and with fixed and for all and with fixed. Using Proposition 8 we have that the co-Borcherds identity also holds for for all . Using that the co-Borcherds identity holds for indices and together with Proposition 8, we obtain that the co-Borcherds identity also holds for the indices for all . Inductively we obtain that the co-Borcherds identity holds for for all for every .
Analogously, we can prove that the co-Borcherds identity also holds for for all . Inductively we obtain that the co-Borcherds identity holds for the indices for all for every . Thus, it is clear that the co-Borcherds identity holds for as long as or .
Now, we need to analyze the case in which and . First, we note that the co-Borcherds identity holds for indices and for all . Using Proposition 8 those identities imply the co-Borcherds identity for the index for all . Again, using that the co-Borcherds identity holds for indices and we obtain that co-Borcherds identity holds for the index . Inductively, we obtain that the co-Borcherds identity holds for indices for all . As this identity holds for all , we obtain that the co-Borcherds identity holds for indices for all and .
Next, the fact that the co-Borcherds identity holds for indices and for all implies that the co-Borcherds identity holds for the index for all . Inductively, we obtain that the co-Borcherds identity holds for indices for all for every . Thus, we have proved that if the co-Borcherds identity holds for indices as long as or , then it also holds for as long as or . Continuing with this procedure, we inductively obtain that the co-Borcherds identity holds for every index , 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 -vector space , together with linear maps satisfying the following axioms for all .(i)Left counit: (ii)Cocreation: (iii) formula: (iv)Coskew symmetry: (v)Coassociator formula:

Remarks 3. (a) Note that (51) trivially implies (13). That is, Definition 10 implies that for all ,
(b) Note that (51) implies that the map is the map defined in (21). In fact, if we take the coefficient of in (51), we get .
(c) Using Proposition 3(3), it is clear that (51) is also satisfied under the conditions of Definition 1.

Hence, is the map we introduced earlier and the conditions regarding the map 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 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 formula (52) imply that the map satisfies the following properties.
(1) -bracket formula:
(2) Conjugation formula:

Proof. (1) Using (53), the product rule, (52), and the fact that and reapplying (53), we have
(2) Exponentiating the -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 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 after replacing by to obtain which due to (53) and (57) implies Thus, we have that Using that and applying to (61), we obtain Using that we have that (62) is equivalent to 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 with fixed . 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 with fixed . Therefore, using Theorem 9, we have that the axioms of Definition 10 imply the co-Borcherds identity for all . 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 and .

Definition 14. A vertex coalgebra consists of a -vector space , together with linear maps satisfying the following axioms.(i)Left counit: (ii)Cocreation: (iii) formula: (iv)-bracket formula: (v)Cocommutator formula:

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 [12] for the case of vertex algebras.

Lemma 16. In the presence of (68) and (69), the coefficient coskew symmetry (41) for indices imply the coefficient coskew symmetry for every .

Proof. Note that it is enough to show that for fixed , the coefficient coskew symmetry for the index implies the coefficient coskew symmetry for the index . We will denote the coefficient coskew symmetry for fixed by coskew(); that is, We assume coskew() for and we want to prove coskew(). First, we write due to (68). Using coskew() we can rewrite the right hand side of (72) to obtain Now, we apply to (69) and use that to get Dividing by , which we have assumed to be nonzero, we obtain from (74) coskew(), finishing the proof of the lemma.

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

Proposition 17. Let be a vertex coalgebra as in Definition 14. Then, the coefficient coskew symmetry, denoted coskew() in (71), holds for .

Proof. First, let us recall that the cocommutator formula (70) is equivalent to the coefficient cocommutator formula (39) for every . Second, note that the coefficient cocommutator formula for fixed indices is exactly the co-Borcherds identity (34) for indices and . Now, due to Proposition 8, it is clear that the co-Borcherds identity for every and implies the co-Borcherds identity for every and . In particular, the cocommutator formula implies the co-Borcherds identity for ,  , and : Applying to (75) and reordering the terms, we obtain As we mentioned before, the co-Borcherds identity (34) holds for and . Therefore, evaluating , , and in (34), the right hand side of (76) can be rewritten to get Next, using that for , we apply to (77) and reorder the terms to obtain Using Remarks 2 and 3(b), together with (67) and (68), it is clear that for , Replacing (79) in (78) we have which due to (67) implies Therefore, we have proved that the cocommutator formula (70) implies coskew() for , in the presence of (67) and (68).
In order to finish the proof, we need to show that coskew() also follows from the axioms of Definition 14. To prove that, we use the coefficient cocommutator formula (39) for , : Applying to (82) and using (79) together with for , we obtain Due to (67), it follows that (83) implies which is equivalent to coskew(). Therefore, we have seen that under the axioms of Definition 14, the cocommutator formula implies coskew() for every .

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 -Lie conformal algebra and a left-symmetric -differential algebra with certain compatibilities. In light of (36), it is clear that the -coproducts for can be described using the -coproduct and the map . Therefore, the coalgebra structure is fully determined by the coproducts for . That is the motivation for the definition that we present in this section, where the axioms are reformulated to consider properties regarding the -coproducts only for .

Let be a vertex coalgebra, where In order to consider the -coproduct and the -coproducts for , we define two linear maps:

Remarks 4. Note that the map defined in (87) may contain infinitely many powers of , unlike the case of vertex algebras (cf. Section 4 in [9]), where the -bracket that encodes the -products for 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 coproduct identity: We have the following result.

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

Proof. Using that and taking the coefficient of for each in (54), the equivalence is clear.

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

Proposition 19. Let be a vertex coalgebra and let , be the maps defined in (86) and (87), respectively. Then, the following conditions are satisfied.
(1) Counit:
(2) Coderivation with respect to  :
(3) Counit holomorphic condition:
(4) -sesquilinearity:
(5) Conformal coskew symmetry:
(6) Conformal co-Jacobi formula:
(7) Co-Wick formula:
(8) Cocompatibility:
(9) Coleft symmetry:

Remarks 5. (a) The expression is interpreted as follows: with the powers of on the right, unlike the case of -Lie conformal algebras in [9]. Moreover, using that is locally nilpotent (see Proposition 5(3)), it is clear that makes sense in Hom.
(b) Note that the expression is well defined in Hom.
(c) The integral notation in (7) and (8) is interpreted as follows: unlike the case of Lie conformal algebras, where the parameter and its evaluation on are always placed on the left. Indeed, from now on expansions inside definite integrals will be interpreted with all the powers of the parameters and placed on the right.
(d) The names of properties in Proposition 19 are motivated by the Definition of vertex algebra presented in Section 6 [9]. Formulas , , and are the dual version of the axioms of a left symmetric -differential algebra with unit (cf. Section 4 [9]). Formula is related to the fact that is holomorphic. On the other hand, , , and are dual to the axioms of a -Lie conformal algebra. Finally, and 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 in (22) and (26).
(3) In light of (13), it is clear that .
(4) Using (22), we have Using (26) and part (4) proved above we have
(5) Using (25) we have
(6) Note that for the coproduct formula (88) can be rewritten as follows: Changing the variables for and for in (103) we obtain that the collection of coproduct identities (88) for is equivalent to Multiplying (104) by and taking we obtain
(7) Multiplying (104) by , using that and taking we have Using the conformal co-Jacobi formula (6) and making a change of variables we can rewrite (106) as follows:
(8) Using that and coskew symmetry (25) we have Using conformal coskew symmetry proved in (5) and making a change of variables we can rewrite (108) as follows:
(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: where the integral notation is interpreted as follows: first we expand the coproduct and put the powers of on the right inside the integral and then we take the definite integral by the usual rules inside the parenthesis. Namely, In order to prove (110) we will use the -coproduct formula proved in Proposition 18: We multiply (112) by , take , and use that for to obtain Using that for , which follows from (36), we can rewrite the right hand side of (113) to get Using (111) it is easy to check that the right hand side of (114) equals 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 . Therefore, its left hand side is invariant under as well. This clearly implies coleft symmetry (97), finishing the proof of Proposition 19.

Let us suppose that we start with a -vector space and four linear maps: such that is locally nilpotent and properties of Proposition 19 hold. Then, we define a new linear map where the expression is interpreted as evaluated in acting on . Namely, if we write , then We want to prove that with defined in (117) is a vertex coalgebra. Indeed, we will show that providing a -vector space with the structure described above is equivalent to providing with a coalgebra structure.

Remarks 6. (a) If we consider the expansion for the map defined in (117), then we are defining for .
(b) In this context, we can recover the maps and using the formulas , with defined as in (117).
(c) If we have a vertex coalgebra and define maps and as we did in (86), (87), then due to (36) we can reobtain as in (117).

We introduce the last definition.

Definition 20. A vertex coalgebra consists of a -vector space together with linear maps such that is locally nilpotent and conditions 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 and as in (86)-(87).

In order to prove the other implication let us first define as in (117). Using that is locally nilpotent, it is clear that the image of actually lies in . Second, note that using (89) and (90) it is clear that . Indeed, (89) and (90) imply Using that together with (91) and (92), we have . The fact that together with 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 satisfies the -bracket formula If we define the linear map it follows that also satisfies Next, we define Using the coefficient version of (123) (cf. proof of Proposition 5(1)) and Remarks 6 it is clear that we can reconstruct Now, conformal coskew symmetry (93) implies whereas the fact that together with conformal coskew symmetry (93) and cocompatibility (96) implies Therefore, using (126) and (127) in (125) together with the definition of in (117) it follows that , 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 coproduct identity (88) for all . If we denote then the coproduct identity is equivalent to In order to prove that it holds for every we introduce the following result.

Lemma 22. Let be a locally nilpotent map and let be a linear map satisfying Then, satisfies the coproduct identity (88) for all if and only if