Cocommutative Vertex Bialgebras

Home , Bialgebra

arXiv:2107.07290v1 [math.QA] 15 Jul 2021 contains ler sntrlyavre i ler.O h te ad o any for hand, other the On algebra. natura Lie is vertex algebra a associative algeb naturally an Lie is as vertex Just algebra where type.) [4], certain in a studied of was algebras algebra Lie algebra vertex Lie vertex of each notion to s wer Associated algebras, were [11]). conformal algebras, (see Lie Kac called Lie structures, vertex same called the and affi algebras, [17]), the Lie and of algebra Virasoro analogues the algebra including t type, associated certain be a can of algebras algebras vertex hig Especially, also hand. are other they and the hand, on one on algebras associative and algebras n a etxalgebra vertex a has one etxagba r nlgu n lsl eae ocasclalgeb classical to related closely and analogous are algebras Vertex e od n phrases. and words Key 2010 V ⊕ to once etxbagba n h aeoyo etxLealgeb Lie vertex that of condition category the the and bialgebras vertex connected V h anrsls ti rvdta if that proved is it results, main the atclr hssosta vr oomttv once vert connected cocommutative every that shows this particular, i ler.For algebra. Lie sntrlya bla eiru,weesteset the whereas semigroup, abelian an naturally is o eea etxbialgebra vertex general a For Abstract. P P g ahmtc ujc Classification. Subject Mathematics C ∈ ( ( V V V G P ) ) savre i uagba eutdet rm sta o n vert any for that is Primc to due result A subalgebra. Lie vertex a as ( ( V ⊗ V soitdt h etxLealgebra Lie vertex the to associated ) ) C V g [ n ec salse h qiaec ewe h aeoyo co of category the between equivalence the establishes hence and G where , ( nti ae,tesrcueo oomttv etxbialgebras vertex cocommutative of structure the paper, this In OOMTTV ETXBIALGEBRAS VERTEX COCOMMUTATIVE V ]a oler hr h etxagbasrcuei xlctyde explicitly is structure algebra vertex the where coalgebra a as )] g V ∈ G 1 V INH A,HIHN I UU XIAO YUKUN LI, HAISHENG HAN, JIANZHI ( G C savre ubagbawihi smrhct h etxbialgebra vertex the to isomorphic is which subbialgebra vertex a is iler,vre iler,vre i algebra. Lie vertex bialgebra, vertex Bialgebra, V ( hc scntutdo eeaie em ouefor module Verma generalized a on constructed is which , sagopadle ntecne of center the in lies and group a is ) V ,dnt by denote ), V ti rvdta h set the that proved is it , 1. Introduction 76,1B8 16T10. 17B68, 17B69, V V g saccmuaievre iler,then bialgebra, vertex cocommutative a is h once opnn containing component connected the P 1 ( V ,and ), P ( C V V n a i algebra Lie a has one , g fpiiieeeet savertex a is elements primitive of ) sa is G V xbialgebra ex 1 ( V V mdl for -module a.Frhroe under Furthermore, ras. ti rvdthat proved is it , fgoplk elements group-like of ) tde neednl by independently studied e Lie (infinite-dimensional) o l i ler,avertex a algebra, Lie a lly l ocascli nature in nonclassical hly acssessc sLie as such systems raic eLeagba.Vertex algebras. Lie ne uidb rm (see Primc by tudied a r enda Lie as defined are ras etxLealgebra Lie vertex V sinvestigated. is g sisomorphic is commutative ∈ g termined. G Among . L ( V C .In ). V V (Another . xalgebra ex = = L C and C , 2 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

V and for any vertex Lie algebra homomorphism ψ : C → V , there exists a (unique) vertex algebra homomorphism Ψ : VC → V such that Ψ|C = ψ. This illustrates the relationship between vertex algebras and Lie algebras. It has been well understood that vertex algebras are analogues and generalizations of commutative associative algebras with identity. Noncommutative associative algebra analogues of vertex algebras were studied by Baklov and Kac under the name “ﬁeld algebra” (see [2]) and independently by the second named author of this current paper under the name

“G1-axiomatic vertex algebra” in [12]. These were further studied under the name “nonlocal vertex algebras” in [13]. Theoretically, vertex algebra analogues of coalgebras, bialgebras, and Hopf algebras are also important objects to study. In [8, 9], Hubbard introduced and studied a notion of vertex operator coalgebra, which is dual to that of vertex operator algebra. Then the second named author of the current paper initiated a study on vertex algebra analogues of bialgebras in [14], where a notion of (nonlocal) vertex bialgebra was introduced. By definition, a (nonlocal) vertex bialgebra is simply a (nonlocal) vertex algebra equipped with a classical coalgebra structure such that the comultiplication map and the counit map are homomor- phisms of (nonlocal) vertex algebras. (Note that the vertex coalgebra maps of a vertex operator coalgebra in the sense of Hubbard can not be interpreted as vertex algebra morphisms.) Furthermore, a notion of (nonlocal) vertex module-algebra for a (nonlocal) vertex bialgebra was introduced and a smash product construction of (nonlocal) vertex algebras was obtained therein. Some initial applications were also given. The study of vertex bialgebras was continued in recent work [10]. For a vertex bialgebra H, a notion of right H-comodule nonlocal vertex algebra was introduced and a general construction of quantum vertex algebras was obtained by using a right comodule nonlocal vertex algebra structure and a compatible (left) module vertex algebra structure on a nonlocal vertex algebra. As an application, a family of deformations of lattice vertex algebras was obtained. Currently, this is used in a sequel to establish a natural connection between twisted quantum affine algebras and quantum vertex algebras. We mention that Hopf action on vertex operator algebras was studied by Dong and Wang in [5], and studied further by Wang (see [19, 20]), from a different direction with different perspectives. This current work is part of a program to fully investigate the structure of vertex bialgebras. Among the main results of this paper, we prove that if V is a cocommutative vertex COCOMMUTATIVE VERTEX BIALGEBRAS 3 bialgebra, then V = ⊕g∈G(V )Vg, where V1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra VP (V ) associated to the vertex Lie algebra P (V ), and Vg is a V1-module for g ∈ G(V ). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to VP (V ) and establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. Fur- thermore, under the condition that G(V ) is a group and lies in the center of V , we prove that V = VP (V ) ⊗ C[G(V )] (canonically) as a coalgebra, where the vertex algebra structure is explicitly determined. Now, we describe the main contents of this paper with more technical details. Let us start with some well known facts in the classical Hopf algebra theory. First of all, as we work on C, any cocommutative coalgebra is pointed, so that an irreducible coalgebra is the same as a connected coalgebra. Among the important examples of Hopf algebras are the group algebras (of groups) and the universal enveloping algebras of Lie algebras. The closely related notions of group-like element and primitive element play a crucial role. A basic fact is that for every bialgebra B, the set G(B) of group-like elements is a semigroup with identity, whereas the set P (B) of primitive elements is a Lie algebra. Another fact is that a cocommutative bialgebra B is a Hopf algebra if and only if G(B) is a group. A classical result is that every cocommutative (pointed) Hopf algebra H is canonically isomorphic to the smash product U(P (H))♯C[G(H)] with respect to a natural action of G(H) on U(P (H)) (see [16]). This in particular shows that any cocommutative connected bialgebra H is isomorphic to the universal enveloping algebra U(P (H)). In terms of categories, this gives the equivalence between the category of cocommutative connected bialgebras and that of Lie algebras. In this paper, to begin we show that for every vertex bialgebra V , P (V ) is a vertex Lie algebra and G(V ) is an abelian semigroup with identity. While the proof for the part on P (V ) is straightforward, the proof for the part on G(V ) is nonclassical. Note that for a classical bialgebra, the left multiplication associated to a group-like element is a coalgebra morphism, due to the fact that the multiplication map is a coalgebra morphism. However, this is not the case for a vertex bialgebra; the “vertex multiplication” is not a coalgebra morphism. As the first key step we prove (Proposition 4.1) that if g, h ∈ G(V ), then gnh =0 for all n ≥ 0, or equivalently [Y (g, x1),Y (h, x2)]=0. Then we show that G(V ) is an abelian semigroup with the operation defined by gh = g−1h for g, h ∈ G(V ). We also show that the vertex subalgebra of V generated by G(V ) is a commutative and cocommutative vertex subbialgebra. 4 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

For g ∈ G(V ), let Vg denote the connected component of V containing g. Assume that

V is a cocommutative vertex bialgebra. By a classical result we have V = ⊕g∈G(V )Vg. As the ﬁrst main result (Theorem 4.13), we prove that V1 is a vertex subbialgebra while Vg is a module for V1 as a vertex algebra for every g ∈ G(V ). Note that the proof in the classical case no longer works for the same reason that the “vertex multiplication” is not a coalgebra morphism. The main idea here is to employ the universal enveloping algebra of the Lie algebra LP (V ). We prove that the natural vertex algebra morphism Ψ : VP (V ) → V with

Ψ|P (V ) = 1 gives a vertex bialgebra isomorphism from VP (V ) to V1. This in particular implies that any cocommutative connected vertex bialgebra V is isomorphic to VP (V ) (Corollary 4.16). Furthermore, assuming that G(V ) is a group and lies in the center of V , we prove that

V = VP (V ) ⊗ C[G(V )] as a coalgebra, on which the corresponding vertex algebra structure is explicitly exhibited (Theorem 4.19). Motivated by this result, we give a general construction of vertex (bi)algebras (Theorem 4.20); we construct a vertex (bi)algebra V ⊗φ C[L] from a general vertex (bi)algebra V and an abelian semigroup L with identity with a map φ : L → P (V ) satisfying certain conditions. Note that the category of commutative vertex algebras is canonically isomorphic to that of commutative differential algebras (which are commutative associative algebras equipped with one derivation). Commutative and cocommutative vertex bialgebras are essentially commutative and cocommutative differential bialgebras. This naturally leads us to a study on commutative and cocommutative differential bialgebras. To any abelian semigroup L with identity, we associate a commutative and cocommutative differential bialgebra BL and we obtain a characterization of BL. This paper is organized as follows: In Section 2, we recall basic notions and some fundamental results in the classical Hopf algebra theory, including the basic results on group-like elements, primitive elements, irreducible coalgebras and connected coalgebras. In Section 3, we prove that for any vertex Lie algebra C, the associated vertex algebra VC is a cocommutative connected vertex bialgebra with P (VC )= C. Section 4 is the core of this paper, in which we prove that for any vertex bialgebra V , G(V ) is an abelian semigroup with identity and especially we prove that every cocommutative connected vertex bialgebra B is canonically isomorphic to the associated vertex bialgebra VP (B). In Section 5, we study commutative and cocommutative vertex bialgebras, or equivalently commutative and cocommutative differential bialgebras. In particular, we study those associated to abelian semigroups, in order to determine the structure of commutative and cocommutative vertex bialgebras. COCOMMUTATIVE VERTEX BIALGEBRAS 5

For this paper, we work on the ﬁeld C of complex numbers; all vector spaces are assumed to be over C. In addition to the standard notation Z for the set of integers, we use N for the set of nonnegative integers and Z+ for the set of positive integers.

2. Preliminaries

This section is preliminary. In this section, we recall the basic notions about classical coalgebras and bialgebras, and collect the basic results we shall need. References [1], [16], and [18] have been good resources for us, and we refer the readers to these books for details. A coalgebra is a vector space M equipped with linear maps ∆ : M → M ⊗ M, called the comultiplication, and ε : M → C, called the counit, such that

(idM ⊗ ∆)∆ = (∆ ⊗ idM )∆,

ε(a(1))a(2) = a = ε(a(2))a(1) for a ∈ M, where ∆(a) = a(1) ⊗Xa(2) in the SweedlerX notation. The comultiplication and the counit are often referredP to as the coalgebra structure maps. A coalgebra M is said to be cocommutative if

a(1) ⊗ a(2) = a(2) ⊗ a(1) for every a ∈ M. A subcoalgebra of aX coalgebra M is aX subspace N such that N with respect to the coalgebra structure of M becomes a coalgebra itself, or equivalently, ∆(N) ⊂ N ⊗ N. A nonzero coalgebra M is said to be simple if M and {0} are the only subcoalgebras.

Remark 2.1. The Fundamental Theorem of Coalgebras states that every coalgebra is the sum of its finite-dimensional subcoalgebras (cf. [1]). This implies that every simple coalgebra is finite-dimensional. Furthermore, every nonzero minimal subcoalgebra of a coalgebra is simple and finite-dimensional.

Deﬁnition 2.2. A coalgebra is said to be irreducible if it has a unique simple subcoalgebra. Furthermore, a maximal irreducible subcoalgebra is called an irreducible component.

The following results can be found in [1]:

Theorem 2.3. Let C be a coalgebra. Then (1) Every irreducible subcoalgebra of C is contained in an irreducible component of C. (2) A sum of distinct irreducible components is a direct sum. (3) If C is cocommutative, then C is the direct sum of its irreducible components. 6 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

Note that a simple coalgebra is an irreducible coalgebra from deﬁnition. It is clear that every one-dimensional subcoalgebra of a coalgebra is simple.

Deﬁnition 2.4. A coalgebra M is said to be pointed if every simple subcoalgebra of M is one-dimensional.

Remark 2.5. Note that a coalgebra version of the Nullstellensatz states that every cocommutative coalgebra over an algebraically closed ﬁeld is pointed.

Deﬁnition 2.6. For a coalgebra M, denote by M (0) the sum of all simple subcoalgebras of M, which is called the coradical of M. A coalgebra M is said to be connected if M (0) is one-dimensional, or equivalently, if M is irreducible and pointed.

Let M be a coalgebra. An element g of M is said to be group-like if ∆(g) = g ⊗ g and ε(g)=1. Let G(M) be the set of all group-like elements of M:

(2.1) G(M)= {g ∈ M | ∆(g)= g ⊗ g, ε(g)=1}.

Note that each group-like element spans a one-dimensional (simple) subcoalgebra and on the other hand, every one-dimensional subcoalgebra is spanned by a group-like element. Thus, a coalgebra M is pointed if and only if every simple subcoalgebra is spanned by a group-like element. Furthermore, every nonzero pointed coalgebra M contains at least one group-like element, and every connected coalgebra C has one and only one group-like element.

For g ∈ G(M) with M a nonzero coalgebra, let Mg denote the irreducible (connected) component containing g. It is known that ⊕g∈G(M)Mg is a subcoalgebra of M and furthermore, if M is also cocommutative, then M = ⊕g∈G(M)Mg.

Let (M, ∆M ,εM ) and (N, ∆N ,εN ) be two coalgebras. A coalgebra morphism from M to N is a linear map f : M → N such that

(2.2) ∆N f =(f ⊗ f)∆M and εM = εN f.

The following are some basic results on connected coalgebras (cf. [1]):

Theorem 2.7. (i) The image of a connected coalgebra under a coalgebra morphism is connected. (ii) If C and E are connected coalgebras, then C ⊗ E is connected.

A bialgebra is an associative algebra with an identity, equipped with a coalgebra structure such that the two coalgebra structure maps are associative algebra morphisms. A bialgebra morphism is a linear map which is both an algebra morphism and a coalgebra morphism. COCOMMUTATIVE VERTEX BIALGEBRAS 7

A Hopf algebra is a bialgebra H equipped with a linear endomorphism S : H → H, called the antipode, such that

(2.3) S(c(1))c(2) = ε(c)1 = c(1)S(c(2)) for c ∈ H. X X Example 2.8. Let g be any Lie algebra. Then the universal enveloping algebra U(g) is a cocommutative Hopf algebra, where the Hopf algebra structure is uniquely determined by

∆(x)= x ⊗ 1+1 ⊗ x, ε(x)=0, S(x)= −x for x ∈ g.

(0) (n) 1 n Let x ∈ g be a nonzero vector. Set x = 1 and x = n! x ∈ U(g) for n ≥ 1. Then n (n) (n−i) (i) (n) (2.4) ∆(x )= x ⊗ x , ε(x )= δn,0 for n ∈ N. i=0 X Example 2.9. Let G be any semigroup with identity 1. Then the semigroup algebra C[G] is a bialgebra where the coalgebra structure is given by

(2.5) ∆(g)= g ⊗ g, ε(g)=1 for g ∈ G.

Furthermore, if G is a group, then C[G] is a Hopf algebra where the antipode S is given by

(2.6) S(g)= g−1 for g ∈ G.

The following are some basic facts about group-like elements (cf. [1]):

Proposition 2.10. (1) For any coalgebra M, G(M) is a linearly independent subset. (2) For any bialgebra B, G(B) is a semigroup with an identity and C[G(B)] is a subbialgebra. (3) A cocommutative bialgebra B is a Hopf algebra if and only if G(B) is a group.

Let M be a coalgebra. For g, h ∈ G(M), set

(2.7) Pg,h(M)= {u ∈ M | ∆(u)= u ⊗ g + h ⊗ u}.

Suppose that B is a bialgebra. Note that 1 ∈ G(B). Deﬁne P (B)= P1,1(B):

(2.8) P (B)= {u ∈ B | ∆(u)= u ⊗ 1+1 ⊗ u}.

A basic result (cf. [1]) is that for any bialgebra B, P (B) is a Lie subalgebra of BLie, where

BLie denotes the Lie algebra with [a, b]= ab − ba for a, b ∈ B. The following is a well known fact (cf. [16]): 8 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

Proposition 2.11. For any Lie algebra g, the universal enveloping algebra U(g) as a coalgebra is cocommutative and connected with P (U(g)) = g.

The following is one of the fundamental results on Hopf algebras (see [18], [16]):

Theorem 2.12. Let H be a cocommutative (pointed) Hopf algebra and let H1 be the irreducible connected component containing 1. Then H1 is a Hopf subalgebra of H such that ∼ ∼ P (H1)= P (H) and H1 = U(P (H)). Furthermore, H = H1♯C[G(H)] (a smash product).

Let U be a vector space (over C) and let {vλ}λ∈Λ be a basis with Λ totally ordered. Set NΛ N (2.9) fin = {f :Λ → | f(λ) = 0 for all but finitely many λ ∈ Λ}. N NΛ As with the ordinary addition is a semigroup with 0 as its identity, fin is naturally an NΛ abelian semigroup with the zero function as its identity: For f,g ∈ fin, f + g is defined by (f + g)(λ)= f(λ)+ g(λ) for λ ∈ Λ.

NΛ For f ∈ fin, set |f| = f(λ), λ X∈Λ NΛ and for f,g ∈ fin, set f f(λ) = , g g(λ) λ Y∈Λ m m(˙m−1)···(m−k+1) N where k = k! for m, k ∈ . The following result can be found in [18]: C NΛ Theorem 2.13. Define B(U) to be the vector space (over ) with a basis {v(f) | f ∈ fin}. Then B(U) is a bialgebra with the algebra and coalgebra structure maps given by f + g (2.10) v v = v , v =1, (f) (g) f (f+g) (0) NΛ (2.11) ∆(v(f))= v(g) ⊗ v(h), ε(v(f))= δ|f|,0 for f,g ∈ fin. g h f +X= Remark 2.14. Note that the bialgebra B(U) was originally defined as the (unique) maximal cocommutative subcoalgebra, namely, the sum of all cocommutative subcoalgebras, of the “shuffle algebra” Sh(U). As here we only need to use B(U), we simply introduce this bialgebra this way. COCOMMUTATIVE VERTEX BIALGEBRAS 9 N N Note that B(U)= i∈N B(U)(i) is an -graded bialgebra, where for i ∈ , NΛ (2.12) ` B(U)(i) = span{v(f) | f ∈ fin with |f| = i}.

In particular, B(U)(0) = C and B(U)(1) = U. Then we have a canonical projection map

(2.13) π : B(U) → U.

The following result can be found in [18]:

Proposition 2.15. Let U be a vector space. Then B(U) is a commutative and cocommutative connected bialgebra with P (B(U)) = U. On the other hand, B(U) is a cofree cocommutative coalgebra in the sense that for any cocommutative coalgebra C together with a linear map ψ : C → U, there exists a unique coalgebra morphism Ψ: C → B(U) such that π ◦ Ψ= ψ.

Let M be any connected coalgebra. Then there exists a unique group-like element e and M (0) = Ce. Then P (M) is well deﬁned as

(2.14) P (M)= {u ∈ M | ∆(u)= u ⊗ e + e ⊗ u}.

The following result can be found in [18]:

Theorem 2.16. Assume that V is a vector space. Let M be any cocommutative connected coalgebra and ψ : B(V ) → M any coalgebra morphism. Then ψ is injective (resp. surjective, bijective) whenever ψ|V : V → P (M) is injective (resp. surjective, bijective).

Let g be a Lie algebra. Pick a basis {vλ | λ ∈ Λ} for g with Λ totally ordered. Recall that (n) 1 n N NΛ vλ = n! vλ ∈ U(g) for n ∈ . For f ∈ ﬁn, set

(f) (f(λ)) (2.15) v = vλ ∈ U(g). λ Y∈Λ Deﬁne a linear map

(f) NΛ (2.16) Ψg : B(g) → U(g); v(f) 7→ v for f ∈ ﬁn.

It follows from the coalgebra structure of U(g) and the P-B-W theorem that Ψg is a coalgebra isomorphism. As an immediate consequence of Theorem 2.16, we have:

Corollary 2.17. Let g be a Lie algebra and let M be any cocommutative connected coalgebra. Suppose ψ : U(g) → M is a coalgebra morphism. Then ψ is injective (resp. surjective, bijective) if and only if the restriction ψ|g : g → P (M) is injective (resp. surjective, bijective). 10 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

3. Vertex bialgebras associated to vertex Lie algebras

In this section, we recall from [14] the vertex bialgebra VC associated to a vertex Lie algebra (namely, a Lie conformal algebra) C and show that VC as a coalgebra is connected

(namely, pointed and irreducible) with P (VC )= C. We begin by recalling the deﬁnition of a vertex algebra (see [3], [7], [6]; cf. [15]).

Deﬁnition 3.1. A vertex algebra is a vector space V equipped with a linear map Y (·, x): V → (End V )[[x, x−1]]

−n−1 v 7→ Y (v, x)= vnx (where vn ∈ End V ) n Z X∈ and a distinguished vector 1 ∈ V, called the vacuum vector, satisfying all the following conditions: The truncation condition:

Y (u, x)v ∈ V ((x)) for u, v ∈ V.

The vacuum property: Y (1, x)=1. The creation property:

Y (v, x)1 ∈ V [[x]] and lim Y (v, x)1 = v for v ∈ V x→0 and for u, v ∈ V, x − x x − x (3.1) x−1δ 1 2 Y (u, x )Y (v, x ) − x−1δ 2 1 Y (v, x )Y (u, x ) 0 x 1 2 0 −x 2 1 0 0 x − x = x−1δ 1 0 Y (Y (u, x )v, x ) 2 x 0 2 2 (the Jacobi identity), where x − x n x−1δ 1 2 = x−n−1(x − x )n = (−1)ix−n−1xn−ixi . 0 x 0 1 2 i 0 1 2 0 n Z n Z i≥0 X∈ X∈ X Let V be a vertex algebra. Denote by D the linear operator on V deﬁned by Dv = v−21 for v ∈ V . Then d [D,Y (v, x)] = Y (Dv, x)= Y (v, x), dx Y (u, x)v = exDY (v, −x)u (the skew-symmetry) for u, v ∈ V . In terms of components, we have (Du)m = −mum−1 for m ∈ Z. COCOMMUTATIVE VERTEX BIALGEBRAS 11

Remark 3.2. Let u, v ∈ V (a vertex algebra). We have x − x [Y (u, x ),Y (v, x )] = Res x−1δ 1 0 Y (Y (u, x )v, x ) 1 2 x0 2 x 0 2 2 1 ∂ j x = Y (u v, x ) x−1δ 2 j 2 j! ∂x 1 x j≥0 2 1 X (the Borcherds commutator formula), or equivalently in the component form, m [u , v ]= (u v) m n j j m+n−j j N X∈ for m, n ∈ Z.

Example 3.3. Let A be a unital commutative associative algebra with a derivation ∂. A result due to Borcherds (see [3]) is that A is a vertex algebra with the identity as the vacuum vector and with the vertex operator map Y (·, x) deﬁned by 1 Y (a, x)b = ex∂a b = (∂na)bxn for a, b ∈ A. n! n≥0 X This construction of vertex algebras is known as the Borcherds construction.

Let V and K be vertex algebras. A vertex algebra (homo)morphism from V to K is a linear map ψ such that

ψ(1)= 1, ψ(Y (u, x)v)= Y (ψ(u), x)ψ(v) for u, v ∈ V.

Recall from [6] that for any vertex algebras U and V , U ⊗ V is a vertex algebra with 1 ⊗ 1 as the vacuum vector and with

(3.2) Y (u ⊗ v, x)= Y (u, x) ⊗ Y (v, x) for u ∈ U, v ∈ V.

Now, we introduce the main objects of this paper, which are a family of nonlocal vertex bialgebras introduced in [14].

Deﬁnition 3.4. A vertex bialgebra is a vertex algebra V equipped with a coalgebra structure such that the coalgebra structure maps ∆ : V → V ⊗ V and ε : V → C are vertex algebra morphisms. 12 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

A vertex bialgebra is said to be cocommutative (resp. pointed, irreducible, connected) if it is cocommutative (resp. pointed, irreducible, connected) as a coalgebra. A vertex bialgebra morphism is both a vertex algebra morphism and a coalgebra morphism. Let (A, ∂) be a diﬀerential algebra where A is a unital commutative associative algebra and ∂ is a derivation of A. Then A ⊗ A and C are naturally diﬀerential algebras with

∂A⊗A = ∂ ⊗ 1+1 ⊗ ∂ and ∂C =0. A diﬀerential algebra morphism is an algebra morphism f : A → B such that f∂A = ∂Bf. A diﬀerential bialgebra is a bialgebra B equipped with a derivation ∂ of B as an algebra such that

(3.3) ε∂ =0 (= ∂ε), ∆∂ =(∂ ⊗ 1+1 ⊗ ∂)∆, namely, ∆ and ε are differential algebra morphisms. A differential bialgebra morphism is defined to be a differential algebra morphism which is also a coalgebra morphism.

Remark 3.5. Recall that every commutative diﬀerential algebra A is naturally a vertex algebra by Borcherds’ construction (see Example 3.3). On the other hand, as it was shown in [14], every diﬀerential bialgebra is naturally a vertex bialgebra.

Next, we discuss vertex Lie algebras. A notion of vertex Lie algebra was introduced by Primc (see [17]) and an equivalent notion, called Lie conformal algebra, was independently introduced by Kac (see [11]). (Another essentially equivalent notion of vertex Lie algebra was studied in [4].) Here, we shall use Primc’s deﬁnition.

Deﬁnition 3.6. A vertex Lie algebra is a vector space U equipped with a linear operator D and a linear map

Y −(·, x): U → x−1(End U)[[x−1]]

− −n−1 u 7→ Y (u, x)= unx (where un ∈ End U), n N X∈ satisfying the following conditions for u,v,w ∈ U, m, n ∈ N:

unv =0 for n suﬃciently large,

(Du)nv = −nun−1v, unDv = D(unv) − nun−1v, 1 u v = (−1)n+j+1 Djv u, n j! n+j j≥0 X COCOMMUTATIVE VERTEX BIALGEBRAS 13

m u v w − v u w = (u v) w. m n n m j j m+n−j j≥0 X Note that every vertex algebra V is naturally a vertex Lie algebra with Y −(u, x) = −n−1 n≥0 unx , the singular part of the vertex operator Y (u, x) for u ∈ V . PLet U and V be two vertex Lie algebras. A vertex Lie subalgebra of U is a subspace W such that D(W ) ⊂ W and umv ∈ W for any u, v ∈ W, m ∈ N. A vertex Lie algebra morphism from U to V is a linear map f such that

− − (3.4) f(YU (u, x)v)= YV (f(u), x)f(v) and f(DU u)= DV (f(u)) for u, v ∈ U. Let C be a vertex Lie algebra. Set L(C) = C ⊗ C[t, t−1], a vector space, and set Dˆ = d D ⊗ 1+1 ⊗ dt , a linear operator on L(C). We have a Lie algebra LC (see [17]), where

(3.5) LC = L(C)/DˆL(C) as a vector space and the Lie bracket is given by m (3.6) [a(m), b(n)] = (a b)(m + n − j) j j j N X∈ r for a, b ∈ C, m, n ∈ Z, where u(r) denotes the image of u ⊗ t in LC for u ∈ C, r ∈ Z. + − Lie algebra LC has a polar decomposition LC = LC ⊕LC into Lie subalgebras, where

+ − (3.7) LC = span{a(n) | a ∈ C, n ≥ 0}, LC = span{a(n) | a ∈ C, n< 0}.

+ View C as a trivial LC -module and then form the induced LC -module:

(3.8) VC = U (LC) ⊗ + C. U(LC ) − Set 1 = 1 ⊗ 1 ∈ VC . By the P-B-W theorem, the natural U(LC )-module homomorphism − − − from U(LC ) to VC , sending X to X1 for X ∈ U(LC ), is an isomorphism. Thus U(LC ) ≃VC − as a U(LC )-module. We view C as a subspace of VC by identifying a with a(−1)1 for a ∈ C. The following result can be found in [4, 17]:

Proposition 3.7. On the LC -module VC , there exists a vertex algebra structure which is uniquely determined by the condition that 1 is the vacuum vector and

Y (u, x)= u(n)x−n−1 for u ∈ C. n Z X∈ Furthermore, VC is generated by C. 14 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

The following result was obtained in [14]:

Proposition 3.8. Let C be a vertex Lie algebra. Then there exist vertex algebra morphisms

∆ : VC →VC ⊗VC and ε : VC → C, which are uniquely determined by

∆(1)= 1 ⊗ 1, ε(1)=1, ∆(a)= a ⊗ 1 + 1 ⊗ a, ε(a)=0 for a ∈ C.

Furthermore, VC equipped with ∆ and ε is a cocommutative vertex bialgebra.

Let C be a vertex Lie algebra as before. It is straightforward to show that the linear map

C → LC; a 7→ a(−1) is one-to-one. Then we view C as a subspace of LC by identifying a − N with a(−1) for a ∈ C. Recall LC = span{a(−n − 1) | a ∈ C, n ∈ }. Noticing that 1 (3.9) a(−n − 1) = (Dna)(−1) for a ∈ C, n ∈ N, n! we have

− (3.10) LC = {u(−1) | u ∈ C}

− and LC = C as vector spaces.

− Proposition 3.9. Let C be any vertex Lie algebra. Then the U(LC )-module isomorphism − η : U(LC ) →VC , deﬁned by

− (3.11) η(X)= X1 for X ∈ U(LC ), is a coalgebra isomorphism. Furthermore, the associated vertex bialgebra VC is cocommutative and connected with P (VC)= C.

Proof. The furthermore assertion follows immediately from the ﬁrst assertion. Note that − − U(L ) as an algebra is generated by L = C. Since ∆ − , ε − are algebra morphisms, C C U(LC ) U(LC ) and ∆VC , εVC are vertex algebra morphisms, it follows that η is a coalgebra morphism, and hence it is a coalgebra isomorphism.

Remark 3.10. Let V be a vertex bialgebra. Note that

∆D(v)=(D ⊗ 1+1 ⊗D)∆(v) ∈D(V ) ⊗ V + V ⊗D(V ), ε(Dv)= Dε(v)=0 for v ∈ V. COCOMMUTATIVE VERTEX BIALGEBRAS 15

Then D(V ) is a coideal of V , by which we have a factor coalgebra V/D(V ). On the other hand, it is known (see [3]) that V/D(V ) is a Lie algebra with

[u + D(V ), v + D(V )] = u0v + D(V ) for u, v ∈ V.

Note that there is no canonical Lie algebra structure on V/D(V ) ⊗ V/D(V ). We know that (V ⊗ V )/(D ⊗ 1+1 ⊗D)(V ⊗ V ) is a Lie algebra with

(3.12) [a ⊗ b, u ⊗ v]= amu ⊗ b−m−1v for a, b, u, v ∈ V. m Z X∈ We have ∆D(V )=(D ⊗ 1+1 ⊗D)∆(V ) ⊂ (D ⊗ 1+1 ⊗D)(V ⊗ V ), but (D ⊗ 1+1 ⊗D)∆(V ) =(6 D ⊗ 1+1 ⊗D)(V ⊗ V ) in general. On the other hand, the Lie algebra bracket on the Lie algebra V/D(V ) is not a coalgebra morphism.

Remark 3.11. Recall that a commutative Poisson algebra is a commutative associative algebra A equipped with a Lie algebra structure {·, ·} such that

{ab, c} = a{b, c} + {a, c}b for a, b, c ∈ A.

Let V be a vertex algebra. Set

C2(V ) = span{u−2v | u, v ∈ V }.

From [21, 22], V/C2(V ) is a commutative Poisson algebra with

(u + C2(V ))(v + C2(V )) = u−1v + C2(V ), {u + C2(V ), v + C2(V )} = u0v + C2(V ) for u, v ∈ V .

Proposition 3.12. Let V be a vertex bialgebra. Then C2(V ) is a coideal of V and V/C2(V ) is a bialgebra. Furthermore, (V/C2(V )) ⊗ (V/C2(V )) is a Lie algebra with

(3.13) [a ⊗ b, u ⊗ v]= a−1u ⊗ b0v + a0u ⊗ b−1v for a, b, u, v ∈ V and the comultiplication map ∆ of V/C2(V ) is a Lie algebra morphism.

Proof. For u, v ∈ V , we have

∆(Y (u, x)v)= Y (u(1), x)v(1) ⊗ Y (u(2), x)v(2), X 16 JIANZHIHAN,HAISHENGLI,YUKUNXIAO or in terms of components,

(1) (1) (2) (2) Z (3.14) ∆(unv)= um v ⊗ un−m−1v for n ∈ . m∈Z X

Recall from [21, 22] that a−m−2b ∈ C2(V ) for a, b ∈ V, m ∈ N. Then

(1) (1) (2) (2) ∆(u−2v)= um v ⊗ u−m−3v ⊂ V ⊗ C2(V )+ C2(V ) ⊗ V. m Z X∈

We also have ε(u−2v) = 0 for u, v ∈ V , recalling that ε(umv) = δm,−1ε(u)ε(v) for m ∈ Z.

Thus C2(V ) is a coideal of V . Now, the commutative Poisson algebra V/C2(V ) is also a coalgebra. It follows from (3.14) that V/C2(V ) is a bialgebra. On the other hand, it is straightforward to show that

C2(V ⊗ V )= V ⊗ C2(V )+ C2(V ) ⊗ V.

As ∆ : V → V ⊗ V is a vertex algebra morphism, the factor comultiplication map

∆ : V/C2(V ) → (V ⊗ V )/C2(V ⊗ V )= V/C2(V ) ⊗ V/C2(V ) is a Poisson algebra morphism. Note that the Lie algebra structure on (V ⊗ V )/C2(V ⊗ V ) is given by

(3.15) [a ⊗ b + C2(V ⊗ V ),u ⊗ v + C2(V ⊗ V )] = a−1u ⊗ b0v + a0u ⊗ b−1v + C2(V ⊗ V ) for a, b, u, v ∈ V . Then V/C2(V ) ⊗ V/C2(V ) is a Lie algebra with the Lie bracket deﬁned by (3.13).

Remark 3.13. Let V be a vertex bialgebra. For n ≥ 1, set

Cn(V ) = span{u−nv | u, v ∈ V }.

This gives a descending sequence of subspaces with C1(V )= V such that

(3.16) ∆(Cn(V )) ⊂ Ci(V ) ⊗ Cn+1−i(V ) i=1 X for n ≥ 1. COCOMMUTATIVE VERTEX BIALGEBRAS 17

4. Structure of cocommutative vertex bialgebras

This section is the core of the whole paper. In this section, ﬁrst we determine the structures of G(V ) and P (V ) for a general vertex bialgebra V , and then we determine the structure of a general cocommutative vertex bialgebra V such that G(V ) is a group which lies in the center of V . The main results are Theorem 4.13, Corollary 4.16, Theorems 4.19 and 4.20. We begin with the structure of the set G(V ) of group-like elements.

Proposition 4.1. Let V be a vertex bialgebra. Then

(4.1) unv =0, u−1v ∈ G(V ) for u, v ∈ G(V ), n ≥ 0.

Furthermore, G(V ) with the operation deﬁned by gh = g−1h for g, h ∈ G(V ) is an abelian semigroup with the vacuum vector 1 as its identity.

Proof. Let u, v ∈ G(V ). We now prove unv = 0 for all n ≥ 0. Suppose that unv =6 0 for some nonnegative integer n. Then there exists a nonnegative integer k such that ukv =6 0 and umv = 0 for m>k. As ∆ is a vertex algebra homomorphism, we have

2k+1 0=∆(u2k+1v)=∆(u)2k+1∆(v) = Resxx Y (u ⊗ u, x)(v ⊗ v)

2k+1 = Resxx (Y (u, x)v ⊗ Y (u, x)v)= umv ⊗ u2k−mv = ukv ⊗ ukv, m Z X∈ which contradicts that ukv =6 0. Thus unv = 0 for all n ≥ 0. Using this property, we get

−1 ∆(u−1v)=∆(u)−1∆(v) = Resxx (Y (u, x)v ⊗ Y (u, x)v)

= umv ⊗ u−m−2v = u−1v ⊗ u−1v. m Z X∈ We also have ε(u−1v)= ε(u)−1ε(v) = 1. Therefore, u−1v ∈ G(V ).

Now, let u,v,w ∈ G(V ). We have unv =0= unw for n ≥ 0. With V a vertex algebra, from the Jacobi identity for the triple (u,v,w) we get

Y (u, x1)Y (v, x2)w = Y (v, x2)Y (u, x1)w,

Y (u, x0 + x2)Y (v, x2)w = Y (Y (u, x0)v, x2)w, which give u−1v−1w = v−1u−1w and u−1v−1w =(u−1v)−1w. It then follows that G(V ) is an abelian semigroup with 1 the identity.

Remark 4.2. Assume that V is a nonlocal vertex bialgebra. From the same argument we see that G(V ) is a (not necessarily abelian) semigroup with identity 1. 18 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

Remark 4.3. Note that for any coalgebra C, G(C) is a linearly independent subset. Then for any vertex bialgebra V , G(V ) is a linearly independent subset of V , so that C[G(V )] is naturally a subcoalgebra of V .

Remark 4.4. Recall that a cocommutative (pointed) bialgebra B (over C) is a Hopf algebra if and only if G(B) is a group. In view of this, one can deﬁne a vertex Hopf algebra as a vertex bialgebra V such that G(V ) is a group.

Lemma 4.5. Let V be a vertex bialgebra. Denote by hG(V )i the vertex subalgebra of V , generated by G(V ). Then hG(V )i is a commutative and cocommutative vertex bialgebra.

Proof. From the proof of Proposition 4.1, we have [Y (g, x1),Y (h, x2)] = 0 for g, h ∈ G(V ). Then it follows that hG(V )i is a commutative vertex subalgebra. Set

K = {v ∈ V | ∆(v) ⊂hG(V )i⊗hG(V )i}.

It is clear that G(V ) ⊂ K, in particular 1 ∈ K. Let u, v ∈ K. Then

∆(u)= u(1) ⊗ u(2), ∆(v)= v(1) ⊗ v(2) ∈hG(V )i⊗hG(V )i, X X from which we get

∆(Y (u, x)v)= Y (u(1), x)v(1) ⊗ Y (u(2), x)v(2) ∈ (hG(V )i⊗hG(V )i)((x)). X Thus K is a vertex subalgebra. Consequently, hG(V )i ⊂ K. Therefore, hG(V )i is a subcoalgebra. To show that it is cocommutative, set

E = {v ∈ V | ∆(v)= T ∆(v)}, where T denotes the ﬂip operator on V ⊗ V . It is straightforward to show that E is a vertex subalgebra of V , containing G(V ). Consequently, we have hG(V )i ⊂ E, proving that hG(V )i is cocommutative.

Remark 4.6. Note that the vertex subbialgebra hG(V )i is isomorphic to the vertex bialgebra obtained from a cocommutative (and commutative) diﬀerential bialgebra through Borcherds’ construction.

Remark 4.7. Let C be a vertex Lie algebra. Recall that the correspondence C ∋ a 7→ − a(−1) ∈LC is a linear bijection between C and the Lie subalgebra LC. This gives rise to a COCOMMUTATIVE VERTEX BIALGEBRAS 19

Lie algebra structure on C. Note that −1 1 [a(−1), b(−1)] = (a b)(−2 − i)= (−1)i Di+1a b (−1) i i (i + 1)! i i≥0 i≥0 X X − in LC for a, b ∈ C. Then 1 (4.2) [a, b]= (−1)i Di+1a b in C. (i + 1)! i i≥0 X Recall that every vertex algebra is naturally a vertex Lie algebra and that for a vertex bialgebra V , P (V )= {v ∈ V | ∆v = v ⊗ 1 + 1 ⊗ v}. Also recall that for a vertex algebra V , D is the linear operator on V , deﬁned by

−2 D(v)= v−21 = Resxx Y (v, x)1 for v ∈ V.

Note that for v ∈ V , Dv = 0 if and only if vn = 0 for all n =6 −1.

Proposition 4.8. Let V be a vertex bialgebra. Then P (V ) is a vertex Lie subalgebra of V with ε(v)=0 for v ∈ P (V ), and P (V ) is a C[D]-submodule. Furthermore, V is an

LP (V )-module with a(n) acting as an for a ∈ P (V ), n ∈ Z.

Proof. For a ∈ P (V ), with ∆(1)= 1 ⊗ 1 we have

−2 −2 ∆(Da) = ∆(a−21) = Resxx ∆(Y (a, x)1) = Resxx Y (∆(a), x)∆(1)

−2 = Resxx Y (a, x)1 ⊗ 1 + 1 ⊗ Y (a, x)1 = Da ⊗ 1 + 1 ⊗Da, which implies Da ∈ P (V ). Thus, D(P (V )) ⊂ P (V ). Now, let a, b ∈ P (V ). Then

∆(Y (a, x)b)= Y (∆(a), x)∆(b) = Y (a, x) ⊗ 1+1 ⊗ Y (a, x) (b ⊗ 1 + 1 ⊗ b) = Y (a, x)b ⊗ 1 + b ⊗ Y (a, x)1 + Y (a, x)1 ⊗ b + 1 ⊗ Y (a, x)b.

For m ∈ N, as am1 =0 we get

∆(amb)= amb ⊗ 1 + 1 ⊗ amb, so amb ∈ P (V ). Therefore, P (V ) is a vertex Lie subalgebra of V . On the other hand, for v ∈ P (V ) we have

v = (1 ⊗ ε)∆(v)=(1 ⊗ ε)(v ⊗ 1 + 1 ⊗ v)= ε(1)v + ε(v)1 = v + ε(v)1, 20 JIANZHIHAN,HAISHENGLI,YUKUNXIAO which implies ε(v) = 0. The furthermore assertion follows immediately from the construction of Lie algebra LP (V ).

Lemma 4.9. Let V be a vertex bialgebra. Deﬁne a linear map

(4.3) φ : U(LP (V )) ⊗ V → V ; X ⊗ v 7→ φ(X ⊗ v)= Xv for X ∈ U(LP (V )), v ∈ V . Then φ is a coalgebra morphism.

U V t Proof. For clarity, let ∆ , ∆ , and ∆ denote the comultiplication maps of U(LP (V )), V , and U(LP (V )) ⊗ V , respectively. Set

V t E = {X ∈ U(LP (V )) | ∆ (φ(X ⊗ v))=(φ ⊗ φ)∆ (X ⊗ v) for all v ∈ V }.

Note that ∆t = T 23(∆U ⊗ ∆V ), where T 23 denotes the ﬂip operator with respect to the indicated factors on the tensor product space (with four factors). It is clear that 1 ∈ E. Let a ∈ P (V ), n ∈ Z, X ∈ E ⊂ U(LP (V )). For any v ∈ V , we have

V V V V V ∆ (φ(anX ⊗ v))=∆ (anXv) = ∆ (a)n∆ (Xv)=(an ⊗ 1+1 ⊗ an)∆ (Xv)

23 U V =(an ⊗ 1+1 ⊗ an)(φ ⊗ φ)T (∆ (X) ⊗ ∆ (v))

23 U U V =(φ ⊗ φ)T (∆ (an)∆ (X) ⊗ ∆ (v))

23 U V =(φ ⊗ φ)T (∆ (anX) ⊗ ∆ (v))

t =(φ ⊗ φ)∆ (anX ⊗ v).

V t Thus anX ∈ E. It follows that E = U(LP (V )), so that ∆ φ = (φ ⊗ φ)∆ . On the other hand, we have εV (φ(1 ⊗ v)) = εV (v)= εt(1 ⊗ v) for v ∈ V , and

εV (φ(anX ⊗ v)) = εV (anXv)= δn,−1εV (a)εV (Xv)=0,

εt(anX ⊗ v)= εU (anX)εV (v)= εU (an)εU (X)εV (v)=0 for any a ∈ P (V ), n ∈ Z, X ∈ U(LP (V )), v ∈ V , noticing that εV (a)=0= εU (an). It then follows that εV ◦ φ = εt. Therefore, φ is a coalgebra morphism.

Similarly, we have the following straightforward result:

Lemma 4.10. Let V be a vertex bialgebra. Deﬁne a linear map

n n (4.4) φD : C[D] ⊗ V → V ; D ⊗ v 7→ D v COCOMMUTATIVE VERTEX BIALGEBRAS 21 for n ∈ N, v ∈ V . View C[D] as a coalgebra by identifying C[D] as the universal enveloping algebra of the 1-dimensional Lie algebra CD. Then φD is a coalgebra morphism.

Deﬁnition 4.11. Let V be a vertex bialgebra. For g ∈ G(V ), let Vg denote the irreducible component of V , containing Cg (an irreducible (simple) subcoalgebra).

The following is a technical result:

Lemma 4.12. Let V be a vertex bialgebra. Then

(4.5) LP (V )g ⊂ Pg,g(V ) for g ∈ G(V ). If V is cocommutative, we have

(4.6) Pg,g(V )= P (Vg) for g ∈ G(V ).

Proof. Let g ∈ G(V ), a ∈ P (V ), n ∈ Z. We have

∆(ang)=∆(a)n∆(g)=(an ⊗ 1+1 ⊗ an)(g ⊗ g)= ang ⊗ g + g ⊗ ang, which implies that ang ∈ Pg,g(V ). This proves the ﬁrst assertion. Assume that V is cocommutative. Then V = ⊕g∈G(V )Vg. Let v ∈ Pg,g(V ) with g ∈ G(V ). Write v = h∈G(V ) vh (a ﬁnite sum) with vh ∈ Vh. We have P

∆(v)= ∆(vh) ∈ Vh ⊗ Vh, h∈XG(V ) h∈MG(V ) ∆(v)= g ⊗ v + v ⊗ g = (vh ⊗ g + g ⊗ vh), h∈XG(V ) which imply v = vg ∈ Vg. Thus Pg,g(V )= Pg,g(Vg)= P (Vg).

Note that if V is an irreducible vertex bialgebra, then G(V ) = {1} and V = V1. On the other hand, if V is cocommutative (hence pointed), we have V = g∈G(V ) Vg. As the ﬁrst main result of this paper, we have: L Theorem 4.13. Let V be a cocommutative vertex bialgebra. Denote by hP (V )i the vertex subalgebra of V generated by P (V ). Then in the connected component decomposition

(4.7) V = Vg, g∈MG(V ) V1 is a vertex subbialgebra of V and V1 = hP (V )i≃VP (V ). Furthermore, for g ∈ G(V ), Vg is a submodule of V for V1 as a vertex algebra. 22 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

Proof. For convenience, set U = hP (V )i. As a subspace, U is linearly spanned by vectors

(1) (r) an1 ··· anr 1

(i) for r ∈ N, a ∈ P (V ), ni ∈ Z. That is, U = U(LP (V ))1. With P (V ) a vertex Lie subalgebra of V , by a result of Primc (see [17]), there exists a vertex algebra morphism ψ : VP (V ) → V with ψ|P (V ) = 1, where VP (V ) is the vertex algebra associated to the vertex Lie algebra

P (V ). It is clear that ψ(VP (V ))= U. By using the fact that ∆ is a vertex algebra morphism, it is straightforward to show that the subset M := {v ∈ V | ∆(v) ∈ U ⊗ U} is a vertex subalgebra containing P (V ). Consequently, we have U ⊂ M, and hence U is a subcoalgebra of V . Similarly, by showing that the subset

W := {v ∈VP (V ) | (ψ ⊗ ψ)∆(v) = ∆ψ(v), εψ(v)= ε(v)} is a vertex subalgebra containing P (V ), we conclude that W = VP (V ). Consequently, ψ is also a coalgebra morphism. Since VP (V ) is cocommutative and connected, in view of Theorem 2.7 U is irreducible (and connected) with 1 ∈ U. Then U ⊂ V1. From this we have P (V ) ⊂ U ⊂ V1. It follows that P (V )= P (V1). View P (V ) as a Lie algebra as in Remark 4.7. Recall that we have a canonical coalgebra isomorphism

η : U(P (V )) →VP (V ). Combining η with the coalgebra morphism ψ, we get a coalgebra morphism ψ ◦ η from U(P (V )) to V1. Note that V1 is cocommutative and connected with P (V1) = P (V ). Then by Corollary 2.17 ψ ◦ η is a coalgebra isomorphism and hence ψ is a coalgebra isomorphism.

Therefore, we have VP (V ) ≃hP (V )i = V1. Consequently, V1 is a vertex subalgebra of V and hence a vertex subbialgebra. Next, we prove the furthermore assertion. Recall from Proposition 4.8 that V is naturally a module for the Lie algebra LP (V ). Let g ∈ G(V ). Set

K = {w ∈ V | ∆(w) ∈ U(LP (V ))Vg ⊗ U(LP (V ))Vg}.

As Vg is a subcoalgebra, we have Vg ⊂ K. Let v ∈ P (V ), n ∈ Z, w ∈ K. Then

(4.8) ∆(vnw)=∆(v)n∆(w)=(vn ⊗ 1+1 ⊗ vn)∆(w) ∈ U(LP (V ))Vg ⊗ U(LP (V ))Vg, which implies vnw ∈ K. This proves that K is an LP (V )-submodule of V . Consequently, we have U(LP (V ))Vg ⊂ K. Thus U(LP (V ))Vg is a subcoalgebra of V . COCOMMUTATIVE VERTEX BIALGEBRAS 23

Recall from Lemma 4.9 the coalgebra morphism φ : U(LP (V )) ⊗ V → V . As both

U(LP (V )) and Vg are connected coalgebras, by Theorem 2.7 again U(LP (V ))Vg as the image of U(LP (V )) ⊗ Vg under φ is connected. Then we have U(LP (V ))Vg ⊂ Vg, proving that

Vg is a U(LP (V ))-submodule of V . Consequently, Vg is a submodule of V for V1 as a vertex algebra as V1 = hP (V )i.

Remark 4.14. Note that every coalgebra has a largest cocommutative subcoalgebra, which coincides with the sum of all cocommutative subcoalgebras. For any vertex bialgebra V , it is straightforward to show that the largest cocommutative subcoalgebra of V is a vertex subbialgebra.

Lemma 4.15. Let V be a vertex bialgebra. Deﬁne V ♯ to be the vertex subalgebra of V generated by P (V )+C[G(V )]. Then V ♯ is a vertex subbialgebra which is cocommutative with G(V ♯)= G(V ) and P (V ♯)= P (V ).

Proof. Note that the vertex subalgebra hP (V )i generated by P (V ) is canonically isomorphic to VP (V ), which is cocommutative. On the other hand, C[G(V )] is a cocommutative subcoalgebra of V . So hP (V )i + C[G(V )] is a cocommutative subcoalgebra of V . Just as in Theorem 4.13 for proving that hP (V )i is a vertex subbialgebra, we see that V ♯ is a cocommutative vertex subbialgebra. (More generally, for any cocommutative subcoalgebra C of V , hCi is a cocommutative vertex subbialgebra.) It can be readily seen that G(V ♯)= G(V ) and P (V ♯)= P (V ).

Let V be any vertex bialgebra. By Proposition 4.8, P (V ) is a vertex Lie subalgebra of V V . From [17], there exists a uniquely determined vertex algebra morphism ψ : VP (V ) → V V with ψ |P (V ) = 1. As an immediate consequence of Theorem 4.13 we have:

Corollary 4.16. Let V be any cocommutative connected vertex bialgebra. Then the vertex V algebra morphism ψ from VP (V ) to V is a vertex bialgebra isomorphism.

Remark 4.17. Let Cccva denote the category of all cocommutative connected vertex bialgebras and let Cvla denote the category of all vertex Lie algebras. Denote by P the functor from Cccva to Cvla with V 7→ P (V ) and by V the functor from Cvla to Cccva with C 7→ VC . In view of Proposition 3.9, P is a left adjoint to V. This together with Corollary 4.16 implies that functors P and V are category equivalences. 24 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

We continue to study the structure of cocommutative vertex bialgebras.

Lemma 4.18. Let V be a cocommutative vertex bialgebra. Suppose g ∈ G(V ) such that gn = 0 for n ≥ 0, or equivalently g lies in the center of V as a vertex algebra. Then g−1 is a coalgebra morphism from V to V and

(4.9) g−1Vh ⊂ Vgh for h ∈ G(V ).

Furthermore, if g is also invertible, then g−1 is a coalgebra isomorphism which gives an isomorphism from V1 to Vg as LP (V )-modules and as coalgebras, and

(4.10) Vg = U(LP (V ))g

Proof. For any v ∈ V , we have

ε(g−1v)= ε(g)ε(v)= ε(v),

∆(g−1v)=∆(Y (g, x)v)|x=0 =(Y (g, x) ⊗ Y (g, x))∆(v)|x=0 =(g−1 ⊗ g−1)∆(v).

This shows that g−1 is a coalgebra morphism from V to V . It follows that g−1Vh ⊂ Vgh for h ∈ G(V ) as g−1Vh is a connected subcoalgebra containing gh. If g is also invertible, then we obtain g−1Vh = Vgh for h ∈ G(V ). In this case, we get

Vg = g−1V1 = g−1U(LP (V ))1 = U(LP (V ))g−11 = U(LP (V ))g, as desired.

Let V be any vertex algebra. For a ∈ V , set

1 (4.11) E−(a, x) = exp a xn .  n −n  n∈Z+ X   Using Theorem 4.13 and Lemma 4.18, we have:

Theorem 4.19. Let V be a cocommutative vertex bialgebra such that G(V ) is a group which lies in the center of V as a vertex algebra. Then

(4.12) V = VP (V ) ⊗ C[G(V )] as a coalgebra, and for v ∈VP (V ), α ∈ G(V ),

(4.13) Y (v ⊗ eα, x)= E−(φ(α), x)Y (v, x) ⊗ eα, COCOMMUTATIVE VERTEX BIALGEBRAS 25

−1 where φ(α)=(α )−1D(α) ∈ V . Furthermore, φ(α) ∈ P (V ) for α ∈ G(V ) and

(4.14) φ(αβ)= φ(α)+ φ(β) for α, β ∈ G(V ).

Proof. From Theorem 4.13 and Lemma 4.18, we have V = ⊕g∈G(V )Vg, where V1 = VP (V ) as vertex bialgebras and Vg = g−1V1 as coalgebras and V1-modules for g ∈ G(V ). Thus we have

V = VP (V ) ⊗ C[G(V )] as coalgebras and as VP (V )-modules with Vg identiﬁed with V1 ⊗ g for g ∈ G(V ).

Note that for any central element u of V , as un = 0 for n ≥ 0 we have

−1 (u ⊗ u)−1 = Resxx (Y (u, x) ⊗ Y (u, x)) = u−1 ⊗ u−1.

Let g ∈ G(V ). Then

−1 −1 −1 −1 −1 ∆((g )−1D(g))=(g ⊗ g )−1∆(D(g)) = ((g )−1 ⊗ (g )−1)(D ⊗ 1+1 ⊗D)(g ⊗ g)

−1 −1 = (g )−1D(g) ⊗ 1 + 1 ⊗ (g )−1D(g)),

−1 which shows that (g )−1D(g) ∈ P (V ). Deﬁne a map

−1 −1 φ : G(V ) → P (V ); φ(g)= g D(g) := (g )−1D(g).

For g, h ∈ G(V ), we have

φ(gh)=(gh)−1D(gh)= g−1h−1(D(g)h + gD(h)) = g−1D(g)+ h−1D(h)= φ(g)+ φ(h).

As G(V ) lies in the center of V , A := hG(V )i is a commutative vertex subalgebra. Then

A is a commutative associative algebra with ab = a−1b for a, b ∈ A and with a derivation ∂ := D such that Y (a, x)b =(ex∂a)b. For g ∈ G(V ), noticing that ∂g = φ(g)g we have d ex∂ g = ex∂∂g = ex∂(φ(g)g)=(ex∂φ(g))(ex∂g), dx where 1 d 1 1 d 1 ex∂φ(g)= xn∂n(φ(g)) = xn+1 ∂n(φ(g)) = xnφ(g) . n! dx n +1 n! dx n −n n≥0 n≥0 n≥1 X X X It follows that 1 Y (g, x)= ex∂g = g exp xnφ(g) = E−(φ(g), x)g. n −n n≥1 ! X 26 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

That is,

(4.15) Y (g, x)= E−(φ(g), x) ⊗ g.

Furthermore, for any v ∈ V , as [Y (g, x1),Y (v, x2)]=0 we get

− Y (v ⊗ g, x)= Y (g−1v, x)= Y (g, x)Y (v, x)= E (φ(g), x)Y (v, x) ⊗ g, as desired.

On the other hand, we have the following result:

Theorem 4.20. Let V be a vertex algebra, L an abelian semigroup with identity 0, and φ : L → V an additive map such that φ(L) lies in the center of V . For v ∈ V, α ∈ L, deﬁne

(4.16) Y (v ⊗ eα, x)= E−(φ(α), x)Y (v, x) ⊗ eα on V ⊗ C[L]. Then V ⊗ C[L] is a vertex algebra with 1 ⊗ e0 as its vacuum vector, which contains V as a vertex subalgebra. Denote this vertex algebra by V ⊗φ C[L]. Furthermore, if V is a (cocommutative) vertex bialgebra with φ(L) ⊂ P (V ), then V ⊗φ C[L] equipped with the tensor product coalgebra structure is a (cocommutative) vertex bialgebra with

(4.17) G(V ⊗φ C[L]) = G(V ) × L and P (V ⊗φ C[L]) = P (V ).

Proof. From deﬁnition, we have Y (1 ⊗ e0, x)=1as φ(0) = 0, and

Y (v ⊗ eα, x)(1 ⊗ e0)= E−(φ(α), x)Y (v, x)1 ⊗ eα ∈ (V ⊗ C[L])[[x]],

lim Y (v ⊗ eα, x)(1 ⊗ e0)= v ⊗ eα. x→0 For u, v ∈ V, α,β ∈ L, we have

α β − − α+β Y (u ⊗ e , x1)Y (v ⊗ e , x2)= E (φ(α), x1)E (φ(β), x2)Y (u, x1)Y (v, x2) ⊗ e .

Then x − x x−1δ 1 2 Y (u ⊗ eα, x )Y (v ⊗ eβ, x ) 0 x 1 2 0 x − x −x−1δ 2 1 Y (v ⊗ eβ, x )Y (u ⊗ eα, x ) 0 −x 2 1 0 x + x = x−1δ 2 0 E−(φ(α), x )E−(φ(β), x )Y (Y (u, x )v, x ) ⊗ eα+β 1 x 1 2 0 2 1 x + x = x−1δ 2 0 E−(φ(α), x + x )E−(φ(β), x )Y (Y (u, x )v, x ) ⊗ eα+β. 1 x 2 0 2 0 2 1 COCOMMUTATIVE VERTEX BIALGEBRAS 27

On the other hand, we have x + x x−1δ 2 0 Y Y (u ⊗ eα, x )(v ⊗ eβ), x 1 x 0 2 1 x + x = x−1δ 2 0 Y E−(φ(α), x )Y (u, x )v ⊗ eα+β, x 1 x 0 0 2 1 x + x = x−1δ 2 0 E−(φ(α + β), x )Y E−(φ(α), x )Y (u, x )v, x ⊗ eα+β 1 x 2 0 0 2 1 x + x = x−1δ 2 0 E−(φ(α), x )E−(φ(β), x )Y E−(φ(α), x )Y (u, x )v, x ⊗ eα+β. 1 x 2 2 0 0 2 1 We see that the Jacobi identity follows if we can show

− − − Y E (φ(α), x0)Y (u, x0)v, x2 = E (φ(α), x2 + x0)E (−φ(α), x2)Y (Y (u, x0)v, x2).

Let a be a central element of V. Note that Dra for all r ≥ 0 are also central. Set 1 A(x)= a xn. n −n n≥1 X As am = 0 for m ≥ 0, we have d A(x)= a xn−1 = Y (a, x). dx −n n≥1 X Furthermore, for any w ∈ V we have 1 1 Y (A(x )w, x)= xnY (a w, x)= xnY ((Dn−1a) w, x) 0 n 0 −n n! 0 −1 n≥1 n≥1 X X 1 d n−1 = xn Y (a, x) Y (w, x) n! 0 dx n≥1 ! X 1 d n = xn A(x) Y (w, x) n! 0 dx n≥1 ! X d = ex0 dx A(x) − A(x) Y (w, x) = (A(x + x0) − A(x))Y (w, x).

Then we obtain

− − − Y (E (a, x0)w, x2)= E (a, x2 + x0)E (−a, x2)Y (w, x2) as desired. This proves the ﬁrst assertion. 28 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

For the second assertion, assume that V is a (cocommutative) vertex bialgebra. Then V ⊗ C[L] is a (cocommutative) coalgebra. It remains to show that the coalgebra structure maps are vertex algebra morphisms. Let u, v ∈ V, α,β ∈ L. As φ(α) ∈ P (V ), we have ε(φ(α)) = 0, so that

ε(φ(α)mw)= δm,−1ε(φ(α))ε(w)=0 for any w ∈ V, m ∈ Z. Then

ε(Y (u ⊗ eα, x)(v ⊗ eβ)) = ε E−(φ(α), x)Y (u, x)v ε(eα+β)= ε(u)ε(v) = ε(u ⊗ eα)ε(v ⊗ eβ), noticing that ε(eγ )= 1 for any γ ∈ L. On the other hand, as

∆(φ(α)mw)=(φ(α)m ⊗ 1+1 ⊗ φ(α)m)∆(w) for any w ∈ V, m ∈ Z, we have

T 23∆(Y (u ⊗ eα, x)(v ⊗ eβ)) = ∆ E−(φ(α), x)Y (u, x)v ⊗ ∆(eα+β) = (E−(φ(α), x) ⊗ E−(φ(α), x))Y (∆(u), x)∆(v) ⊗ (eα+β ⊗ eα+β) = T 23Y (∆(u ⊗ eα), x)∆(v ⊗ eβ).

It follows that ε and ∆ are vertex algebra morphisms. Therefore, V ⊗φ C[L] is a (cocommutative) vertex bialgebra.

For the last assertion, it is clear that G(V )×L ⊂ G(V ⊗φC[L]). Suppose u ∈ G(V ⊗φC[L]). α Write u = α∈S uα ⊗e , where S is a ﬁnite subset of L and uα for α ∈ S are nonzero vectors in V . We haveP 23 23 α β T ∆(u)= T (u ⊗ u)= uα ⊗ uβ ⊗ e ⊗ e , α,β S X∈ 23 α α T ∆(u)= ∆(uα) ⊗ e ⊗ e , α S X∈ which imply uα ⊗ uβ = 0 in V ⊗ V for α =6 β. Then S must be a one-element set. Therefore, α u = uα ⊗ e for some α ∈ L and ∆(uα)= uα ⊗ uα. This shows u ∈ G(V ) × L. Therefore, we have G(V ⊗φ C[L]) = G(V ) × L. Similarly, we can show P (V ⊗φ C[L]) = P (V ).

The following is a property of the vertex bialgebra V ⊗φ C[L]: COCOMMUTATIVE VERTEX BIALGEBRAS 29

Proposition 4.21. Assume that V,L,φ are given as in Theorem 4.20 (with all the assump- tions). Let U be any vertex bialgebra with a semigroup morphism ψL : L → G(U) and a vertex bialgebra morphism ψV : V → U such that

(4.18) [ψL(α)m, ψV (v)n]=0 for α ∈ L, v ∈ V, m, n ∈ Z,

(4.19) D(ψL(α)) = ψL(α)−1ψV (φ(α)) for α ∈ L.

Deﬁne a linear map Ψ: V ⊗ C[L] → U by

α Ψ(v ⊗ e )= ψL(α)−1ψV (v)= ψV (v)−1ψL(α) for v ∈ V, α ∈ L.

Then Ψ is a vertex bialgebra morphism from V ⊗φ C[L] to U.

Proof. For v ∈ V, α ∈ L, we have

α α α εΨ(v ⊗ e )= ε(ψL(α)−1ψV (v)) = ε(ψL(α))ε(ψV (v)) = ε(v)= ε(v)ε(e )= ε(v ⊗ e ) and

α ∆Ψ(v ⊗ e )=∆(ψL(α)−1ψV (v)) = (∆ψL(α))−1∆ψV (v) α =(ψL(α)−1 ⊗ ψL(α)−1)(ψV ⊗ ψV )∆(v)=(Ψ ⊗ Ψ)∆(v ⊗ e ).

Then Ψ is a coalgebra morphism. Next, we show that Ψ is also a vertex algebra morphism. Note that 0 Ψ(1 ⊗ e )=(ψL(0))−1ψV (1)= 1−11 = 1 ∈ U. Let u, v ∈ V, α,β ∈ L. We have

Ψ Y (u ⊗ eα)(v ⊗ eβ) =Ψ E−(φ(α), x)Y (u, x)v ⊗ eα+β

− =(ψL(α + β))−1ψV E (φ(α), x)Y(u, x)v − =(ψL(α + β))−1E (ψV (φ(α)), x)ψV (Y (u, x)v) and

α β Y (Ψ(u ⊗ e ), x)Ψ(v ⊗ e )= Y (ψL(α)−1ψV (v), x) ψL(β)−1ψV (v).

Since [ψL(α)m, ψV (u)n] = 0 for all m, n ∈ Z, we get

Y (ψL(α)−1ψV (u), x)= Y (ψL(α), x)Y (ψV (u), x). 30 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

With D(ψL(α)) = ψL(α)−1ψV (φ(α)) we have

− Y (ψL(α), x)= E (ψV (φ(α)), x)ψL(α)−1,

ψL(α)−1ψL(β)−1 = ψL(α + β)−1 on ψV (V ). Then

α β Y (Ψ(u ⊗ e ), x)Ψ(v ⊗ e )= Y (ψL(α), x)Y (ψV (u), x)ψL(β)−1ψV (v)

= Y (ψL(α), x)ψL(β)−1Y (ψV (u), x)ψV (v)

− = E (ψV (φ(α)), x)ψL(α)−1ψL(β)−1ψV (Y (u, x)v) − = E (ψV (φ(α)), x)ψL(α + β)−1ψV (Y (u, x)v).

It follows that Ψ is indeed a vertex algebra morphism. Therefore, Ψ is a vertex bialgebra morphism.

For any subsets A and B of a vertex algebra V , set

(4.20) CA(B)= {b ∈ B | [Y (b, x1),Y (a, x2)] = 0 for all a ∈ A}.

As an immediate consequence of Proposition 4.21, we have:

Corollary 4.22. Let V be a vertex bialgebra satisfying the conditions that G(V ) ⊂ CV (V ) and that there exists a map φ : G(V ) → CP (V )(P (V )) such that

(4.21) φ(gh)= φ(g)+ φ(h) for g, h ∈ G(V ),

(4.22) D(g)= g−1φ(g) for g ∈ G(V ).

Then there exists a vertex bialgebra morphism Ψ: VP (V )⊗φC[G(V )] → V with Ψ|P (V )+C[G(V )] = 1. Furthermore, if V as a vertex algebra is generated by P (V )+ C[G(V )], Ψ is surjective.

As a special case of Theorem 4.20 we have:

Proposition 4.23. Let C be a vertex Lie algebra, L an abelian semigroup with identity 0, and let φ : L → C be an additive map such that

(4.23) φ(α)nb =0 for α ∈ L, b ∈ C, n ∈ N.

Set

(4.24) V(C,L,φ)= VC ⊗ C[L], COCOMMUTATIVE VERTEX BIALGEBRAS 31 equipped with the tensor product coalgebra structure and the vertex algebra structure obtained in Proposition 4.20. Then V(C,L,φ) is a cocommutative vertex bialgebra with

(4.25) G(V(C,L,φ)) = L and P (V(C,L,φ)) = C.

Proof. Let α ∈ L. As φ(α)ib = 0 for b ∈ C, i ∈ N, we have [φ(α)(m), b(n)] = 0 in LC for all m, n ∈ Z. It follows that φ(α) lies in the center of the associated vertex algebra VC . Then by Proposition 4.20, V(C,L,φ) is a cocommutative vertex bialgebra. The rest is clear.

As another special case of Theorem 4.20, we immediately have:

Corollary 4.24. Let L be an abelian semigroup with identity 0 and let A be any commutative diﬀerential algebra with an additive map φ : L → A. Then A ⊗ C[L] is a vertex algebra with

(4.26) Y (a ⊗ eα)= E−(φ(α), x)ex∂a ⊗ eα for a ∈ A, α ∈ L, where 1 E−(φ(α), x) = exp (∂n−1φ(a))xn . n! n≥1 ! X As an immediate consequence of Theorem 4.19, we have:

Corollary 4.25. Let B be a cocommutative and commutative diﬀerential bialgebra such that G(B) is a group. Then

(4.27) B ≃ S(P (B)) ⊗φ C[G(B)], where φ : G(B) → P (B) with φ(g)= g−1∂(g) for g ∈ G(B).

5. Differential bialgebras associated to abelian semigroups

In this section, we study cocommutative and commutative differential bialgebras, which determine all commutative and cocommutative vertex bialgebras through Borcherds’ construction. More specifically, we associate a cocommutative and commutative differential bialgebra to any abelian semigroup L with identity and we give a universal property and a characterization of BL. We begin by recalling the differential bialgebra associated to an abelian group, which is rooted in the construction of lattice vertex algebras. Let L be any (additive) abelian group. 32 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

The group algebra C[L] is known to be a Hopf algebra. On the other hand, set h = C ⊗Z L, a vector space, and set

(5.1) α¯ =1 ⊗ α ∈ h for α ∈ L.

Furthermore, set h− = h ⊗ t−1C[t−1]. View h− as an abelian Lie algebra, so the universal enveloping algebra U(h−) coincides with the symmetric algebra S(h−). Set b b − (5.2) b BL = S(h ) ⊗ C[L], b which is naturally a Hopf algebra, in particular,b a bialgebra. As an associative algebra, BL admits a derivation ∂ which is uniquely determined by

α α (5.3) ∂e =α ¯(−1) ⊗ e , ∂a(−n)= na(−n − 1) for α ∈ L, a ∈ h, n ∈ Z+,

−n where a(−n) = a ⊗ t . This makes BL a diﬀerential algebra. Furthermore, BL is a cocommutative (and commutative) diﬀerential bialgebra (see [14]).

Remark 5.1. Note that as a cocommutative coalgebra over C, BL is pointed. It is straight- − forward (by using ∆ and ε) to show that G(BL) = L and P (BL) = h . From (5.3), it can be readily seen that BL as a diﬀerential algebra is generated by the subbialgebra C[L]. b

The following is a universal property of the diﬀerential bialgebra BL, extending the universal property of BL as a diﬀerential algebra in [14]:

Proposition 5.2. Assume that L is an abelian group. Let A be any commutative diﬀeren- tial bialgebra and let ψ : C[L] → A be a bialgebra morphism. Then there exists a unique diﬀerential bialgebra morphism ψ : BL → A with ψ|C[L] = ψ.

Proof. From [14], there exists ae unique diﬀerentiale algebra morphism ψ : BL → A with

ψ|C[L] = ψ. It remains to show the ψ is also a coalgebra morphism. Set e

e K = {x ∈ BeL | ∆Aψ(x)=(ψ ⊗ ψ)∆BL (x)}.

As ψ, ∆A and ∆BL are diﬀerential algebrae morphisms,e e it is straightforward to show that

K is a diﬀerential subalgebra of BL. Since ψ is a coalgebra morphism, we have C[L] ⊂ K. e It follows that K = BL. This proves ∆Aψ = (ψ ⊗ ψ)∆BL . Similarly, we have εAψ = εBL . Therefore, ψ is a coalgebra morphism and hence a diﬀerential bialgebra morphism. e e e e e COCOMMUTATIVE VERTEX BIALGEBRAS 33

Let B be a commutative diﬀerential bialgebra with derivation ∂. For v ∈ P (B), we have

∆(∂v)=(∂ ⊗ 1+1 ⊗ ∂)∆(v)=(∂ ⊗ 1+1 ⊗ ∂)(v ⊗ 1+1 ⊗ v)= ∂v ⊗ 1+1 ⊗ ∂v, which implies ∂v ∈ P (B). Thus ∂P (B) ⊂ P (B), so that P (B) is a C[∂]-module. Here, we consider C[∂] as the polynomial algebra with ∂ as an indeterminate. As usual, a C[∂]- module W is said to be torsion-free if f(∂)w =6 0 for any nonzero f(∂) ∈ C[∂], w ∈ W . On the other hand, assume that G(B) is a group. From Theorem 4.19, we have a linear map −1 φ : C ⊗Z G(B) → B with φ(g)= g ∂(g) for g ∈ G(B).

Proposition 5.3. Let B be a commutative diﬀerential bialgebra such that G(B) is a group which generates B as a diﬀerential algebra. Suppose that the linear map φ : C ⊗Z G(B) → B with φ(g)= g−1∂(g) for g ∈ G(B) is injective and P (B) is a torsion-free C[∂]-module. Then

B ≃ BG(B) as a diﬀerential bialgebra.

Proof. Let L be the additive copy of G(B). By Proposition 5.2, there exists a diﬀerential − α bialgebra morphism ψ : BL = S(h ) ⊗ C[L] → B such that ψ(e )= α for α ∈ L. Since B as a diﬀerential algebra is generated by G(B), ψ is surjective. Then it remains to prove that ψ b is injective.

Recall that BL is a commutative and cocommutative (pointed) bialgebra with G(BL)= L − and P (BL) = h . On the other hand, noticing that B as a homomorphism image of BL is cocommutative, by Corollary 4.25, we have B = S(P (B)) ⊗ C[L] as bialgebras. As a b − bialgebra morphism, ψ is also a C[∂]-module morphism, sending P (BL)= h to P (B). Notice that h− is a free C[∂]-module on h = h(−1). For α ∈ L = G(B), we have b b ψ(¯α(−1)) = ψ(e−α∂eα)= α−1∂α ∈ P (B), recalling thatα ¯ =1 ⊗ α ∈ h. From our assumption on φ, ψ is an injective map from h(−1) to P (B). Furthermore, as P (B) is assumed to be a torsion-free C[∂]-module, ψ is injective − − from h = P (BL) to P (B)= P (S(P (B))). Then by Corollary 2.17 ψ : S(h ) → S(P (B)) is injective. Consequently, ψ is an injective map from BL to B. Therefore, ψ is an isomorphism b b of diﬀerential bialgebras.

Next, we generalize this construction to abelian semigroups. Assume now that L is an additive (abelian) semigroup with identity 0. Let CL denote the vector space over C with a basis {(1,α) | α ∈ L}. Deﬁne C ⊗N L to be the quotient space of CL modulo the subspace 34 JIANZHIHAN,HAISHENGLI,YUKUNXIAO spanned by vectors (1,α)+(1, β) − (1,α + β) for α, β ∈ L. Set

(5.4) h = C ⊗N L and writeα ¯ for the image of (1,α) in h. Notice that in case L is a group, we have C ⊗N L =

C ⊗Z L.

Just as before, we get a commutative and cocommutative differential bialgebra BL. Iden- tify a ∈ h with a(−1) ⊗ 1 ∈ BL, to view h as a subspace of BL. It is clear that BL as a differential algebra is generated by C[L]+ h. (In general, BL as a differential algebra may not be generated by C[L] alone.) Define a map

(5.5) φ : L → h− = P (S(h−)); α 7→ φ(α)=α ¯(−1) for α ∈ L.

We have the following (universal)b propertyb for BL:

Proposition 5.4. Let B be any commutative diﬀerential bialgebra, and let ψ : C[L] → B be a bialgebra morphism, φB : h → P (B) a linear map such that

α α (5.6) ∂ψ(e )= φB(¯α)ψ(e ) for α ∈ L.

Then there exists a unique diﬀerential bialgebra morphism f : BL → B such that f|C[L] = ψ and f|h = φB.

Proof. We only need to prove the existence. Using Corollary 4.24, we can easily see that − − BL = S(h ) ⊗φ C[L] as diﬀerential bialgebras. Since h is a free C[∂]-module over h, the − linear map φB : h → P (B) extends uniquely to a C[∂]-module morphism from h to P (B) b − b and then to an algebra morphism φ˜B : S(h ) → B. It is straightforward to show that b φ˜B is also a coalgebra morphism. Then by Proposition 4.21 we have a diﬀerential bialgebra b morphism f : BL → B such that

α α − f(X ⊗ e )= φ˜B(X)ψ(e ) for X ∈ S(h ), α ∈ L, which implies f|C[L] = ψ and f|h = φB. b Using essentially the same arguments as in the proof of Proposition 5.3, we obtain the following characterization of the diﬀerential bialgebra BL: COCOMMUTATIVE VERTEX BIALGEBRAS 35

Proposition 5.5. Let B be a commutative diﬀerential bialgebra such that P (B) is a torsion- free C[∂]-module and such that P (B)+ C[G(B)] generates B as a diﬀerential algebra. In addition, assume that there is an injective linear map φ : C ⊗N G(B)+ → P (B) such that

(5.7) ∂g = φ(¯g)g for g ∈ G(B), where G(B)+ denotes an additive version of G(B) and g¯ = 1 ⊗ g ∈ C ⊗N G(B)+. Then the diﬀerential algebra morphism f : BG(B)+ → B uniquely determined by f|G(B) = 1 and f|h =1 is a diﬀerential bialgebra isomorphism.

Acknowledgment

J. Han is supported by the CSC (grant No. 202006265002). Y. Xiao is supported by NSF of China (grant No. 11971350) and the CSC (grant No. 202006260122).

References

[1] E. Abe, Hopf Algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, 1977. [2] B. Bakalov, V. Kac, Field algebras, Int. Math. Res. Not. 3 (2003), 123-159. [3] R. Borcherds, Vertex algebras, Kac-Moody algebras, and the monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071. [4] C. Dong, H.-S. Li, G. Mason, Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in: Recent Developments in Inﬁnite-Dimensional Lie Algebras and Conformal Field Theory, Proceedings of an International Conference at University of Virginia, May 2000, Contemp. Math., vol. 297, 2002, 69-96. [5] C. Dong, H. Wang, Hopf actions on vertex operator algebras, J. Algebra 514 (2018), 310-329. [6] I. Frenkel, Y.-Z. Huang, J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Mem. Am. Math. Soc. 104 (1993), 1-64. [7] I. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, 1988. [8] K. Hubbard, Constructions of vertex operator coalgebras via vertex operator algebras, J. Algebra 294 (2005), 278-293. [9] K. Hubbard, The notion of vertex operator coalgebra and a geometric interpretation, Comm. Algebra 34 (2006), 1541-1589. [10] N. Jing, F. Kong, H.-S. Li, S. Tan, Deforming vertex algebras by vertex bialgebras, arXiv:2101.01464 [math.QA]; preprint, 2021. [11] V. Kac, Vertex Algebras for Beginners, University Lecture Series, vol. 10, Amer. Math. Soc., 1997.

[12] H.-S. Li, Axiomatic G1-vertex algebras, Commun. Contemp. Math. 5 (2003), 281-327. 36 JIANZHIHAN,HAISHENGLI,YUKUNXIAO

[13] H.-S. Li, Nonlocal vertex algebras generated by formal vertex operators, Selecta Math. 11 (2005), 349-397. [14] H.-S. Li, A smash product construction of nonlocal vertex algebras, Commun. Contemp. Math. 9 (2007), 605-637. [15] J. Lepowsky, H.-S. Li, Introduction to Vertex Operator Algebras and Their Prepresentation, Progr. Math., vol. 227, Birh¨auser Boston, Inc., Boston, MA, 2004. [16] S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics, vol. 82, Amer. Math. Soc., 1993. [17] M. Primc, Vertex algebras generated by Lie algebras, J. Pure Appl. Algebra 135 (1999), 253-293. [18] M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. [19] H. Wang, Hopf actions on vertex operator algebras, II: Smash product, J. Algebra 530 (2019), 402-428. [20] H. Wang, A duality theorem for vertex operator algebras, J. Algebra 562 (2020), 627-646. [21] Y.-C. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph.D. thesis, Yale Univer- sity, 1990. [22] Y.-C. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.

School of Mathematical Sciences, Tongji University, Shanghai, 200092, China. Email address: [email protected]

Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA. Email address: [email protected]

School of Mathematical Sciences, Tongji University, Shanghai, 200092, China. Email address: [email protected]