Deforming vertex algebras by vertex bialgebras
aa r X i v : . [ m a t h . QA ] J a n DEFORMING VERTEX ALGEBRAS BY VERTEX BIALGEBRAS
NAIHUAN JING , FEI KONG , HAISHENG LI, AND SHAOBIN TAN Abstract.
This is a continuation of a previous study initiated by one of us onnonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this pa-per, we study a notion of right H -comodule nonlocal vertex algebra for a nonlocalvertex bialgebra H and give a construction of deformations of vertex algebras witha right H -comodule nonlocal vertex algebra structure and a compatible H -modulenonlocal vertex algebra structure. We also give a construction of φ -coordinatedquasi modules for smash product nonlocal vertex algebras. As an example, we givea family of quantum vertex algebras by deforming the vertex algebras associatedto non-degenerate even lattices. Introduction
Vertex algebras (see [B1], [FLM]) are analogues and generalizations of commuta-tive associative algebras, while nonlocal vertex algebras (see [BK], [Li2]; cf. [Li4])are analogues of noncommutative associative algebras. The (weak) quantum vertexalgebras [Li4], which are certain variations of quantum vertex operator algebras inthe sense of Etingof-Kazhdan [EK], are a special family of nonlocal vertex algebras.To a certain extent, the notion of nonlocal vertex algebra plays the same role inthe general vertex algebraic theory as the notion of associative algebra does in theclassical algebraic theory. On the other hand, quantum vertex (operator) algebras,which from various viewpoints are analogues of quantum groups (algebras), form aperfect category.In [Li8], certain vertex-algebra analogues of Hopf algebras were introduced andstudied. Specifically, notions of nonlocal vertex bialgebra and (left) H -module non-local vertex algebra for a nonlocal vertex bialgebra H were introduced, and thenthe smash product nonlocal vertex algebra V ♯H of an H -module nonlocal vertexalgebra V with H was constructed. As an application, a new construction of thelattice vertex algebras and their modules was given.In vertex algebra theory, among the most important notions are those of modulesand σ -twisted modules for a vertex algebra V with σ a finite order automorphism. Anotion of quasi module, generalizing that of module, was introduced in [Li3] in orderto associate vertex algebras to a certain family of infinite-dimensional Lie algebras.Indeed, with this notion vertex algebras can be associated to a wide variety of Mathematics Subject Classification.
Primary 17B69, 17B68; Secondary 17B10, 17B37.
Key words and phrases. vertex algebra, smash product, quantum vertex algebra, phi -coordinated quasi module, lattice vertex operator algebra. Partially supported by NSF of China (No.11531004) and Simons Foundation (No.198129). Partially supported by NSF of China (No.11701183). Partially supported by NSF of China (No.11531004). nfinite-dimensional Lie algebras. (From a certain point of view, the notion of quasimodule also generalizes that of twisted module (see [Li7]).) Furthermore, in orderto associate quantum vertex algebras to algebras such as quantum affine algebras,a theory of φ -coordinated quasi modules for nonlocal vertex algebras (includingquantum vertex algebras) was developed in [Li9, Li10], where weak quantum vertexalgebras were associated to quantum affine algebras conceptually .In the aforementioned notion of φ -coordinated module, the symbol φ refers to anassociate of the 1-dimensional additive formal group (law), which by definition is F ( x, y ) := x + y (an element of C [[ x, y ]]) and which satisfies F (0 , x ) = x = F ( x, , F ( x, F ( y, x )) = F ( F ( x, y ) , z ) . This very formal group law (implicitly) plays an important role in the theory ofvertex algebras and their modules. In contrast, an associate of F ( x, y ) is a formalseries φ ( x, z ) ∈ C (( x ))[[ z ]], satisfying the condition φ ( x,
0) = x, φ ( φ ( x, y ) , z ) = φ ( x, y + z ) . It was proved therein that for any p ( x ) ∈ C (( x )), the formal series φ ( x, z ) defined by φ ( x, z ) = e zp ( x ) ddx x is an associate and every associate is of this form. The essence of[Li9] is that a theory of φ -coordinated modules for a general nonlocal vertex algebrais attached to each associate φ ( x, z ), where the usual theory of modules becomes aspecial case with φ taken to be the formal group law itself. The importance lies inthe fact that a wide variety of (quantum associative) algebras can be associated to φ -coordinated (quasi) modules for some (weak) quantum vertex algebras by choosinga suitable associate φ .What is a φ -coordinated quasi module? Let φ ( x, z ) be an associate of F ( x, y ).Note that in the definition of modules for an associative algebra and for a (nonlocal)vertex algebra, the key ingredient is associativity. For a general nonlocal vertexalgebra V , the main defining property of a φ -coordinated quasi V -module W withvertex operator map Y W ( · , x ) (the representation morphism) is that for any u, v ∈ V ,there exists a nonzero polynomial q ( x , x ) such that q ( x , x ) Y W ( u, x ) Y W ( v, x ) ∈ Hom(
W, W (( x , x ))) , ( q ( x , x ) Y W ( u, x ) Y W ( v, x )) | x = φ ( x ,z ) = q ( φ ( x , z ) , x ) Y W ( Y ( u, z ) v, x ) . The usual modules including the adjoint module are simply φ -coordinated (quasi)modules with φ ( x, z ) = F ( x, z ) = x + z . In practice, for some algebras, theirmodules of highest weight type cannot be associated with vertex algebras in termsof usual modules, but can be viewed as φ -coordinated (quasi) modules for somevertex algebras or quantum vertex algebras with some φ .In this paper, we develop the theory of smash product nonlocal vertex algebrasfurther in several directions, with an ultimate goal to construct certain desired quan-tum vertex (operator) algebras. Among the main results, we first introduce a notionof right H -comodule nonlocal vertex algebra for a nonlocal vertex bialgebra H andthen using a compatible right H -comodule nonlocal vertex algebra structure on an H -module nonlocal vertex algebra V , we construct a deformed nonlocal vertex alge-bra structure on V . We show that under certain conditions, the deformed nonlocal ertex algebras are quantum vertex algebras. In another direction (representationaspect), we construct φ -coordinated quasi modules for smash product nonlocal ver-tex algebras and for the aforementioned deformed nonlocal vertex algebras. We alsoapply the general results to the vertex algebras associated to non-degenerate evenlattices, to obtain a family of quantum vertex algebras.Now, we give a more detailed description of the contents. Recall that a nonlocalvertex bialgebra is simply a nonlocal vertex algebra V equipped with a classicalcoalgebra structure on V such that the comultiplication ∆ and the counit ǫ arehomomorphisms of nonlocal vertex algebras. For a nonlocal vertex bialgebra H , an H -module nonlocal vertex algebra is a nonlocal vertex algebra V equipped with amodule structure Y HV ( · , x ) for H viewed as a nonlocal vertex algebra such that Y HV ( h, x ) v ∈ V ⊗ C (( x )) ,Y HV ( h, x ) V = ǫ ( h ) V ,Y HV ( h, x ) Y ( u, z ) v = X Y (cid:0) Y HV ( h (1) , x − z ) u, z (cid:1) Y HV ( h (2) , x ) v for h ∈ H, u, v ∈ V . (The first condition is technical, while the other two areanalogues of the classical counterparts.) Given an H -module nonlocal vertex algebra V , we have a smash product nonlocal vertex algebra V ♯H , where
V ♯H = V ⊗ H asa vector space and Y ♯ ( u ⊗ h, x )( v ⊗ k ) = X Y ( u, x ) Y HV ( h (1) , x ) v ⊗ Y ( h (2) , x ) k for u, v ∈ V, h, k ∈ H .Let H be a nonlocal vertex bialgebra. A right H -comodule nonlocal vertex algebra is defined to be a nonlocal vertex algebra V equipped with a comodue structure ρ : V → V ⊗ H for H viewed as a coalgebra such that ρ is a homomorphism ofnonlocal vertex algebras. Let V be an H -module nonlocal vertex algebra. We say aright H -comodule structure ρ : V → V ⊗ H is compatible with the (left) H -modulestructure if ρ is an H -module homomorphism with H acting only on the first factorof V ⊗ H . Assuming that ρ : V → V ⊗ H is a compatible right H -comodulestructure on an H -module nonlocal vertex algebra V , we construct a new nonlocalvertex algebra D ρY HV ( V ) with V as the underlying space, where the vertex operatormap, denoted here by Y new ( · , x ), is given by Y new ( u, x ) v = X Y ( u (1) , x ) Y HV ( u (2) , x ) v for u, v ∈ V , where ρ ( u ) = P u (1) ⊗ u (2) ∈ V ⊗ H in the Sweedler notation. (Recallthat Y HV ( · , x ) denotes the vertex operator map for H on the module V .) Fur-thermore, we show that D ρY HV ( V ) is a quantum vertex algebra, assuming that H is cocommutative, V is a vertex algebra, and Y HV ( · , x ) is invertible with respect toconvolution (plus some technical condition).In the theory of vertex algebras, an important family consists of the vertex al-gebras V L associated to non-degenerate even lattices L , which are rooted in thevertex operator realization of affine Kac-Moody algebras. These structurally sim-ple vertex algebras are often used to construct or study more complicated vertex perator algebras (see [FLM] for example). Recall that V L = S ( b h − ) ⊗ C ε [ L ] as avector space, where b h − = h ⊗ t − C [ t − ] with h = C ⊗ Z L is an abelian Lie algebraand C ε [ L ] is the ε -twisted group algebra of L with ε a certain particular 2-cocycleof L . Set B L = S ( b h − ) ⊗ C [ L ] equipped with the tensor product bialgebra structure,where C [ L ] is the ordinary group algebra. Use a natural derivation on B L and theBorcherds construction to make B L a commutative vertex algebra. In fact, B L is avertex bialgebra, which was exploited in [Li8].In this paper, we explore the vertex bialgebra B L furthermore. We prove thatthere exists a natural right B L -comodule vertex algebra structure ρ : V L → V L ⊗ B L on the vertex algebra V L . In fact, with V L identified with B L as a vector space, ρ issimply the comultiplication ∆ : B L ⊗ B L → B L . (It was proved in [Li8] that withcanonical identifications, ∆ is a vertex algebra embedding of V L into B L,ε ♯B L .) Onthe other hand, for any linear map η ( · , x ) : h → h ⊗ x C [[ x ]], we give a compatible B L -module vertex algebra structure Y ηM ( · , x ) on V L . Consequently, we obtain a familyof quantum vertex algebras V ηL (with V L as the underlying vector space).In a sequel, we shall use ~ -adic versions of the results obtained in this paperto construct certain quantum vertex operator algebras (over C [[ ~ ]]) in the senseof Etingof-Kazhdan by deforming vertex algebras V L , where φ -coordinated quasimodules for these quantum vertex operator algebras are associated to highest weightmodules for twisted quantum affine algebras.This paper is organized as follows: In Section 2, we first recall the basic notionsand results about smash product nonlocal vertex algebras, and then study rightcomodule nonlocal vertex algebras, and compatible module nonlocal vertex algebrastructure. Furthermore, we give a deformation construction of nonlocal vertex alge-bras. In Section 3, we continue to show that under certain conditions, the deformednonlocal vertex algebras are quantum vertex algebras. In Section 4, we study φ -coordinated quasi modules for smash product nonlocal vertex algebras. In Section5, for a general non-degenerate even lattice L , we give a right H -comodule vertexalgebra structure on V L and give a family of compatible H -module vertex algebrastructures. Then we give a family of deformations of the vertex algebra V L .In this paper, we work on the field C of complex numbers, and we use Z + forthe set of positive integers, while we use N for the set of nonnegative integers. Fora ring R , e.g., Z , C , C [ x, y ], we use R × for the set of nonzero elements of R . Wecontinue using the formal variable notations and conventions (including formal deltafunctions) as established in [FLM] and [FHL]. . Smash product nonlocal vertex algebras
In this section, we first recall the basic notions and results on smash productnonlocal vertex algebras from [Li8], and we then study a notion of right H -comodulenonlocal vertex algebra for a nonlocal vertex bialgebra H . As the main result, for an H -module nonlocal vertex algebra V with a compatible right H -comodule nonlocalvertex algebra structure, we establish a deformed nonlocal vertex algebra structureon V .2.1. Nonlocal vertex algebras and quantum vertex algebras.
Here, we recallsome basic notions and results on nonlocal vertex algebras and (weak) quantumvertex algebras from [Li4].We start with the notion of nonlocal vertex algebra (see Remark 2.3).
Definition 2.1. A nonlocal vertex algebra is a vector space V equipped with a linearmap Y ( · , x ) : V → (End V )[[ x, x − ]] v Y ( v, x ) = X n ∈ Z v n x − n − (where v n ∈ End( V ))and a distinguished vector ∈ V , satisfying the conditions that Y ( u, x ) v ∈ V (( x )) for u, v ∈ V, (2.1) Y ( , x ) v = v, Y ( v, x ) ∈ V [[ x ]] and lim x → Y ( v, x ) = v for v ∈ V, (2.2)and that for any u, v ∈ V , there exists a nonnegative integer k such that( x − x ) k Y ( u, x ) Y ( v, x ) ∈ Hom(
V, V (( x , x )))(2.3)and (cid:0) ( x − x ) k Y ( u, x ) Y ( v, x ) (cid:1) | x = x + x = x k Y ( Y ( u, x ) v, x ) . (2.4)The following was obtained in [JKLT] (cf. [DLMi], [LTW]): Lemma 2.2.
Let V be a nonlocal vertex algebra. Then the following weak associa-tivity holds: For any u, v, w ∈ V , there exists l ∈ N such that ( x + x ) l Y ( u, x + x ) Y ( v, x ) w = ( x + x ) l Y ( Y ( u, x ) v, x ) w. (2.5) Remark 2.3.
Note that a notion of nonlocal vertex algebra was defined in [Li4]in terms of weak associativity as in Lemma 2.2, which is the same as the notionof axiomatic G -vertex algebra introduced in [Li2] and also the same as the notionof field algebra introduced in [BK]. In view of Lemma 2.2, the notion of nonlocalvertex algebra in the sense of Definition 2.1 is stronger than the other notions.Let V be a nonlocal vertex algebra. Define a linear operator D by D ( v ) = v − for v ∈ V . Then [ D , Y ( v, x )] = Y ( D ( v ) , x ) = ddx Y ( v, x ) for v ∈ V. (2.6)The following notion singles out a family of nonlocal vertex algebras (see [Li4]): efinition 2.4. A weak quantum vertex algebra is a nonlocal vertex algebra V satisfying the following S -locality: For any u, v ∈ V , there exist u ( i ) , v ( i ) ∈ V, f i ( x ) ∈ C (( x )) ( i = 1 , . . . , r )and a nonnegative integer k such that( x − z ) k Y ( u, x ) Y ( v, z ) = ( x − z ) k r X i =1 f i ( z − x ) Y ( v ( i ) , z ) Y ( u ( i ) , x ) . (2.7)The following was proved in [Li4]: Proposition 2.5.
A weak quantum vertex algebra can be defined equivalently byreplacing the conditions (2.3) and (2.4) in Definition 2.1 (for a nonlocal vertexalgebra) with the condition that for any u, v ∈ V , there exist u ( i ) , v ( i ) ∈ V, f i ( x ) ∈ C (( x )) ( i = 1 , . . . , r ) such that x − δ (cid:18) x − x x (cid:19) Y ( u, x ) Y ( v, x ) − x − δ (cid:18) x − x − x (cid:19) r X i =1 f i ( x − x ) Y ( v ( i ) , x ) Y ( u ( i ) , x )= x − δ (cid:18) x − x x (cid:19) Y ( Y ( u, x ) v, x )(2.8) (the S -Jacobi identity ). The following is a convenient technical result (cf. [LL, Proposition 5.6.7], [Li1]):
Lemma 2.6.
Let V be a nonlocal vertex algebra and let u, v, u ( i ) , v ( i ) , w ( j ) ∈ V, f i ( x ) ∈ C (( x )) (1 ≤ i ≤ r, ≤ j ≤ s ) such that ( x − x ) k Y ( u, x ) Y ( v, x ) = ( x − x ) k r X i =1 f i ( x − x ) Y ( v ( i ) , x ) Y ( u ( i ) , x ) for some positive integer k . Then ( x − x ) n Y ( u, x ) Y ( v, x ) − ( − x + x ) n r X i =1 f i ( x − x ) Y ( v ( i ) , x ) Y ( u ( i ) , x )= s X j =0 Y ( w ( j ) , x ) 1 j ! (cid:18) ∂∂x (cid:19) j x − δ (cid:18) x x (cid:19) . (2.9) if and only if u n + j v = w ( j ) for ≤ j ≤ s and u j + n v = 0 for j > s . roof. From [Li4] the S -Jacobi identity (2.8) holds, we have that( x − x ) n Y ( u, x ) Y ( v, x ) − ( − x + x ) n r X i =1 f i ( x − x ) Y ( v ( i ) , x ) Y ( u ( i ) , x )= X j ≥ Y ( u n + j v, x ) 1 j ! (cid:18) ∂∂x (cid:19) j x − δ (cid:18) x x (cid:19) . By using this, we complete the proof. (cid:3)
Recall that a rational quantum Yang-Baxter operator on a vector space U is alinear map S ( x ) : U ⊗ U → U ⊗ U ⊗ C (( x ))such that S ( x ) S ( x + z ) S ( z ) = S ( z ) S ( x + z ) S ( x )(2.10)(the quantum Yang-Baxter equation ). Furthermore, a rational quantum Yang-Baxteroperator S ( x ) on U is said to be unitary if S ( x ) S ( − x ) = 1 , (2.11)where S ( x ) = σ S ( x ) σ with σ denoting the flip operator on U ⊗ U .For any nonlocal vertex algebra V , follow [EK] to denote by Y ( x ) the linear map Y ( x ) : V ⊗ V → V (( x )) , associated to the vertex operator map Y ( · , x ) : V → Hom(
V, V (( x ))) of V . Thefollowing is a variation of Etingof-Kazhdan’s notion of quantum vertex operatoralgebra (see [EK], [Li4]): Definition 2.7. A quantum vertex algebra is a weak quantum vertex algebra V equipped with a unitary rational quantum Yang-Baxter operator S ( x ) on V , satis-fying the conditions that for any u, v ∈ V , there exists k ∈ N such that( x − z ) k Y ( x ) (1 ⊗ Y ( z )) ( S ( x − z )( u ⊗ v ) ⊗ w ) = ( x − z ) k Y ( z ) (1 ⊗ Y ( x )) ( v ⊗ u ⊗ w )for all w ∈ V and that[ D ⊗ , S ( x )] = − ddx S ( x ) , (2.12) S ( x ) ( Y ( z ) ⊗
1) = ( Y ( z ) ⊗ S ( x ) S ( x + z )(2.13)(the hexagon identity ).A nonlocal vertex algebra V is said to be nondegenerate (see [EK]) if for everypositive integer n , the linear map Z n : C (( x )) · · · (( x n )) ⊗ V ⊗ n → V (( x )) · · · (( x n ))) , defined by Z n ( f ⊗ v ⊗ · · · ⊗ v n ) = f Y ( v , x ) · · · Y ( v n , x n ) , is injective.We have (see [EK], [Li4]): roposition 2.8. Let V be a nondegenerate weak quantum vertex algebra. Thenthere exists a linear map S ( x ) : V ⊗ V → V ⊗ V ⊗ C (( x )) such that V together with S ( x ) is a quantum vertex algebra. Furthermore, such a linear map S ( x ) is uniquelydetermined by the S -locality. The following notion (under a different name) was introduced in [Li3]:
Definition 2.9.
Let G be a group, χ : G → C × a linear character. A ( G, χ ) -modulenonlocal vertex algebra is a nonlocal vertex algebra V equipped with a representation R : G → GL( V ) of G on V such that R ( g ) = and R ( g ) Y ( v, x ) R ( g ) − = Y ( R ( g ) v, χ ( g ) x ) for g ∈ G, v ∈ V. (2.14)We sometimes denote a ( G, χ )-module nonlocal vertex algebra by a pair (
V, R ). Remark 2.10.
Note that in Definition 2.9, for g ∈ G , if χ ( g ) = 1, then g acts on V as an automorphism. Thus a ( G, χ )-module nonlocal vertex algebra with χ = 1(the trivial character) is simply a nonlocal vertex algebra on which G acts as anautomorphism group. In this case, V is a C [ G ]-module nonlocal vertex algebra inthe sense of [Li8] (see Section 3).Let ( U, R U ) and ( V, R V ) be ( G, χ )-module nonlocal vertex algebras. A (
G, χ ) -module nonlocal vertex algebra homomorphism is a nonlocal vertex algebra homo-morphism f : U → V which is also a G -module homomorphism.The following is a technical lemma formulated in [JKLT, Lemma 3.3]: Lemma 2.11.
Let G be a group equipped with a linear character χ : G → C × .Suppose that V is a nonlocal vertex algebra, ρ : G → Aut( V ) and L : G → GL( V ) are group homomorphisms such that L ( g ) = , ρ ( g ) L ( h ) = L ( h ) ρ ( g ) for g, h ∈ G,L ( g ) Y ( v, x ) L ( g ) − = Y ( L ( g ) v, χ ( g ) x ) for g ∈ G, v ∈ S, where S is a generating subset of V . Then ( V, R ) is a ( G, χ ) -module nonlocal vertexalgebra with R defined by R ( g ) = ρ ( g ) L ( g ) for g ∈ G . Smash product nonlocal vertex algebras.
We first recall from [Li8] thebasic notions and results on smash product nonlocal vertex algebras, and then weintroduce a notion of right H -comodule nonlocal vertex algebra with H a nonlocalvertex bialgebra and we establish a deformed nonlocal vertex algebra structure ona right H -comodule nonlocal vertex algebra V with a compatible (left) H -modulenonlocal vertex algebra structure.We begin with the notion of nonlocal vertex bialgebra. Definition 2.12. A nonlocal vertex bialgebra is a nonlocal vertex algebra V equippedwith a classical coalgebra structure (∆ , ε ) such that (the co-multiplication) ∆ : V → V ⊗ V and (the co-unit) ε : V → C are homomorphisms of nonlocal vertex algebras.The notion of homomorphism of nonlocal vertex bialgebras is defined in the obvi-ous way: For nonlocal vertex bialgebras ( V, ∆ , ε ) and ( V ′ , ∆ ′ , ε ′ ), a nonlocal vertex ialgebra homomorphism from V to V ′ is a homomorphism f of nonlocal vertexalgebras such that ∆ ′ ◦ f = ( f ⊗ f ) ◦ ∆ , ε ′ ◦ f = ε. (2.15)In other words, a nonlocal vertex bialgebra homomorphism is both a nonlocal vertexalgebra homomorphism and a coalgebra homomorphism. Definition 2.13.
Let ( H, ∆ , ε ) be a nonlocal vertex bialgebra. A (left) H -modulenonlocal vertex algebra is a nonlocal vertex algebra V equipped with a module struc-ture Y HV ( · , x ) on V for H viewed as a nonlocal vertex algebra such that Y HV ( h, x ) v ∈ V ⊗ C (( x )) , (2.16) Y HV ( h, x ) V = ε ( h ) V , (2.17) Y HV ( h, x ) Y ( u, x ) v = X Y ( Y HV ( h (1) , x − x ) u, x ) Y HV ( h (2) , x ) v (2.18)for h ∈ H , u, v ∈ V , where V denotes the vacuum vector of V and ∆( h ) = P h (1) ⊗ h (2) is the coproduct in the Sweedler notation.The following two results were obtained in [Li8]: Theorem 2.14.
Let H be a nonlocal vertex bialgebra and let V be an H -modulenonlocal vertex algebra. Set V ♯H = V ⊗ H as a vector space. For u, v ∈ V , h, h ′ ∈ H , define Y ♯ ( u ⊗ h, x )( v ⊗ h ′ ) = X Y ( u, x ) Y ( h (1) , x ) v ⊗ Y ( h (2) , x ) h ′ . (2.19) Then ( V ♯H, Y ♯ , ⊗ ) carries the structure of a nonlocal vertex algebra, which con-tains V and H canonically as subalgebras such that for h ∈ H , u ∈ V , Y ♯ ( h, x ) Y ♯ ( u, x ) = X Y ♯ ( Y ( h (1) , x − x ) u, x ) Y ♯ ( h (2) , x ) . (2.20) Proposition 2.15.
Let H be a nonlocal vertex bialgebra and let V be an H -modulenonlocal vertex algebra. Let W be a vector space and assume that ( W, Y VW ) is a V -module and ( W, Y HW ) is an H -module such that for any h ∈ H , v ∈ V , w ∈ W , Y HW ( h, x ) w ∈ W ⊗ C (( x )) , (2.21) Y HW ( h, x ) Y VW ( v, x ) w = X Y VW ( Y HV ( h (1) , x − x ) v, x ) Y HW ( h (2) , x ) w. (2.22) Then W is a V ♯H -module with the vertex operator map Y ♯W ( · , x ) given by Y ♯W ( v ⊗ h, x ) w = Y VW ( v, x ) Y HW ( h, x ) w for h ∈ H, v ∈ V, w ∈ W. (2.23)As an immediate consequence of Proposition 2.15, we have: Proposition 2.16.
Let H be a nonlocal vertex bialgebra and let V be an H -modulenonlocal vertex algebra. Then V is a V ♯H -module with the vertex operator map Y ♯V ( · , x ) given by Y ♯V ( u ⊗ h, x ) v = Y ( u, x ) Y HV ( h, x ) v for u, v ∈ V, h ∈ H. Furthermore, ⊗ ε : V ♯H → V is a V ♯H -module epimorphism. e shall need the following simple result: Lemma 2.17.
Let H be a nonlocal vertex bialgebra and let ( V, Y HV ) be an H -modulenonlocal vertex algebra. Then Y HV ( Y ( h, z ) h ′ , x ) = Y HV ( h, x + z ) Y HV ( h ′ , x ) for h, h ′ ∈ H. (2.24) Proof.
Assume h, h ′ ∈ H . Let v ∈ V . There exists a nonnegative integer l such that( z + x ) l Y HV ( h, z + x ) Y HV ( h ′ , x ) v = ( z + x ) l Y HV ( Y ( h, z ) h ′ , x ) v. From the assumption (2.16), we have Y HV ( h ′ , x ) v, Y HV ( h, x ) u ∈ V ⊗ C (( x ))for any u ∈ V . Consequently, we have Y HV ( h, x ) Y HV ( h ′ , x ) v ∈ V ⊗ C (( x, x )) . This implies that Y HV ( h, x + z ) Y HV ( h ′ , x ) exists in (Hom( V, V ⊗ C (( x ))))[[ z ]]. It alsoimplies that we can choose l so large that we also have x l Y HV ( h, x ) Y HV ( h ′ , x ) v ∈ V ⊗ C (( x ))[[ x ]] . Then ( z + x ) l Y HV ( h, z + x ) Y HV ( h ′ , x ) v = ( x + z ) l Y HV ( h, x + z ) Y HV ( h ′ , x ) v. Thus ( x + z ) l Y HV ( h, x + z ) Y HV ( h ′ , x ) v = ( z + x ) l Y HV ( Y ( h, z ) h ′ , x ) v. Consequently, by cancellation we get Y HV ( h, x + z ) Y HV ( h ′ , x ) v = Y HV ( Y ( h, z ) h ′ , x ) v, as desired. (cid:3) As in [Li8], by a differential bialgebra we mean a bialgebra ( B, ∆ , ε ) equippedwith a derivation ∂ such that ε ◦ ∂ = 0 and ∆ ∂ = ( ∂ ⊗ ⊗ ∂ )∆. (That is, ε and∆ are homomorphisms of differential algebras.) Remark 2.18.
Let ( B, ∆ , ε, ∂ ) be a differential bialgebra. In particular, the asso-ciative algebra B with derivation ∂ is a differential algebra. Then we have a nonlocalvertex algebra structure on B with Y ( a, x ) b = ( e x∂ a ) b for a, b ∈ B. Denote this nonlocal vertex algebra by (
B, ∂ ). Then (
B, ∂ ) equipped with ∆ and ε is naturally a nonlocal vertex bialgebra (see [Li8, Example 4.2]).Let V be a nonlocal vertex algebra. A subset U of Hom( V, V ⊗ C (( x ))) is saidto be ∆ -closed (see [Li8]) if for any a ( x ) ∈ U , there exist a (1 i ) ( x ) , a (2 i ) ( x ) ∈ U for i = 1 , , . . . , r such that a ( x ) Y ( v, x ) = r X i =1 Y ( a (1 i ) ( x − x ) v, x ) a (2 i ) ( x ) for all v ∈ V. (2.25) et B ( V ) be the sum of all ∆-closed subspaces U of Hom( V, V ⊗ C (( x ))) such that a ( x ) ∈ C for a ( x ) ∈ U. (2.26)Note that Hom( V, V ⊗ C (( x ))) ∼ = End C (( x )) ( V ⊗ C (( x ))) is an associative algebra.The following is a summary of some results of [Li8]: Proposition 2.19.
Let V be a nonlocal vertex algebra. Then B ( V ) is a ∆ -closedsubalgebra of Hom(
V, V ⊗ C (( x ))) and is closed under the derivation ∂ = ddx . Fur-thermore, if V is nondegenerate, then (a) B ( V ) is a differential bialgebra with thecoproduct ∆ and the counit ε , which are uniquely determined by a ( x ) = ε ( a ( x )) , ∆( a ( x )) = X a (1) ( x ) ⊗ a (2) ( x )(2.27) for a ( x ) ∈ B ( V ) , where a ( x ) Y ( v, z ) = P Y ( a (1) ( x − z ) v, z ) a (2) ( x ) for all v ∈ V . (b) V is a B ( V ) -module nonlocal vertex algebra with Y V ( a ( x ) , x ) = a ( x ) for a ( x ) ∈ B ( V ) . (c) For any nonlocal vertex bialgebra H , an H -module nonlocal vertex algebrastructure Y HV ( · , x ) on V amounts to a nonlocal vertex bialgebra homomorphism from H to B ( V ) . The following notions are due to [EK] and [Li6] (cf. [Li8]):
Definition 2.20.
Let V be a nonlocal vertex algebra. A pseudo-derivation (resp. pseudo-endomorphism ) of V is an element a ( x ) ∈ Hom(
V, V ⊗ C (( x ))) such that[ a ( x ) , Y ( u, z )] = Y ( a ( x − z ) u, z )(2.28)(resp. a ( x ) Y ( u, z ) = Y ( a ( x − z ) u, z ) a ( x )) for all u ∈ V . Denote by PDer( V ) (resp.PEnd( V )) the set of all pseudo-derivations (resp. pseudo-endomorphisms).From definition, we havePDer( V ) ⊂ B ( V ) , PEnd( V ) ⊂ B ( V ) . (2.29)The following construction of pseudo-derivations is due to [EK]: Proposition 2.21.
Let V be a vertex algebra. For v ∈ V , f ( x ) ∈ C (( x )) , set Φ( v, f )( x ) = Res z f ( x − z ) Y ( v, z ) = X n ≥ ( − n n ! f ( n ) ( x ) v n , (2.30) where f ( n ) ( x ) = (cid:0) ddx (cid:1) n f ( x ) . Then Φ( v, f )( x ) ∈ PDer( V ) . The following is an analogue of the notion of right comodule algebra:
Definition 2.22.
Let H be a nonlocal vertex bialgebra. A right H -comodule non-local vertex algebra is a nonlocal vertex algebra V equipped with a homomorphism ρ : V → V ⊗ H of nonlocal vertex algebras such that( ρ ⊗ ρ = (1 ⊗ ∆) ρ, (1 ⊗ ǫ ) ρ = Id V , (2.31)i.e., ρ is also a right comodule structure on V for H viewed as a coalgebra. emark 2.23. Let B be a coalgebra and let U with the map ρ : U → U ⊗ B be aright B -comodule. For u ∈ U , we have ( ρ ⊗ ρ ( u ) = (1 ⊗ ∆) ρ ( u ), i.e., X u (1) , (1) ⊗ u (1) , (2) ⊗ u (2) = X u (1) ⊗ u (2) , (1) ⊗ u (2) , (2) . Note that by applying permutation operators on both sides we get five more rela-tions. If B is cocommutative, we have P (1 ⊗ ∆) ρ = (1 ⊗ ∆) ρ , so P ( ρ ⊗ ρ = (1 ⊗ ∆) ρ, P ( ρ ⊗ ρ = ( ρ ⊗ ρ, (2.32)where P denotes the indicated flip operator on B ⊗ B ⊗ B . Definition 2.24.
Let H be a nonlocal vertex bialgebra. Assume that V is an H -module nonlocal vertex algebra with the module vertex operator map Y HV ( · , x ) : H → Hom(
V, V ⊗ C (( x )))and V is also a right H -comodule nonlocal vertex algebra with the comodule map ρ : V → V ⊗ H . We say Y HV and ρ are compatible if ρ is an H -module homomorphismwith V ⊗ H viewed as an H -module on which H acts on the first factor only, i.e., ρ ( Y HV ( h, x ) v ) = ( Y HV ( h, x ) ⊗ ρ ( v ) for h ∈ H, v ∈ V. (2.33)The following is the main result of this section: Theorem 2.25.
Let H be a cocommutative nonlocal vertex bialgebra and let V bea nonlocal vertex algebra. Suppose that ( V, Y HV ) is an H -module nonlocal vertexalgebra and ( V, ρ ) is a right H -comodule nonlocal vertex algebra such that Y HV and ρ are compatible. For a ∈ V , set D ρY HV ( Y )( a, x ) = X Y ( a (1) , x ) Y HV ( a (2) , x )(2.34) on V , where ρ ( a ) = P a (1) ⊗ a (2) ∈ V ⊗ H . Then ( V, D ρY HV ( Y ) , ) carries the struc-ture of a nonlocal vertex algebra. Denote this nonlocal vertex algebra by D ρY HV ( V ) .Furthermore, ρ is a nonlocal vertex algebra homomorphism from D ρY HV ( V ) to V ♯H .Proof.
Recall from Proposition 2.16 that Y ♯V ( · , x ) denotes the V ♯H -module structureon V . Notice that for a ∈ V , we have D ρY HV ( Y )( v, x ) = Y ♯V ( ρ ( v ) , x ) . (2.35)It follows that D ρY HV ( Y )( a, x ) ∈ Hom(
V, V (( x ))). Let u, v ∈ V . As H is cocommuta-tive, by (2.32) we have X u (1) , (1) ⊗ u (2) ⊗ u (1) , (2) = X u (1) ⊗ u (2) , (1) ⊗ u (2) , (2) . Using this and (2.33), we get ρ (cid:16) D ρY HV ( Y )( u, x ) v (cid:17) = X ρ (cid:0) Y ( u (1) , x ) Y HV ( u (2) , x ) v (cid:1) = X Y ⊗ ( ρ ( u (1) ) , x ) ρ (cid:0) Y HV ( u (2) , x ) v (cid:1) = X Y ⊗ ( ρ ( u (1) ) , x )( Y HV ( u (2) , x ) ⊗ ρ ( v ) X Y ⊗ ( u (1) , (1) ⊗ u (1) , (2) , x )( Y HV ( u (2) , x ) ⊗ v (1) ⊗ v (2) )= X Y ( u (1) , (1) , x ) Y HV ( u (2) , x ) v (1) ⊗ Y ( u (1) , (2) , x ) v (2) = X Y ( u (1) , x ) Y HV ( u (2) , (1) , x ) v (1) ⊗ Y ( u (2) , (2) , x ) v (2) = X Y ♯ ( u (1) ⊗ u (2) , x )( v (1) ⊗ v (2) )= Y ♯ ( ρ ( u ) , x ) ρ ( v ) . (2.36)Now, let u, v ∈ V . Then there exists a nonnegative integer k such that( x − x ) k Y ♯V ( ρ ( u ) , x ) Y ♯V ( ρ ( v ) , x ) ∈ Hom(
V, V (( x , x ))) , (cid:16) ( x − x ) k Y ♯V ( ρ ( u ) , x ) Y ♯V ( ρ ( v ) , x ) (cid:17) | x = x + x = x k Y ♯V ( Y ♯ ( ρ ( u ) , x ) ρ ( v ) , x ) . Using this we obtain( x − x ) k D ρY HV ( Y )( u, x ) D ρY HV ( Y )( v, x ) ∈ Hom(
V, V (( x , x )))(recalling (2.35)) and (cid:16) ( x − x ) k D ρY HV ( Y )( u, x ) D ρY HV ( Y )( v, x ) (cid:17) | x = x + x = (cid:16) ( x − x ) k Y ♯V ( ρ ( u ) , x ) Y ♯V ( ρ ( v ) , x ) (cid:17) | x = x + x = x k Y ♯V ( Y ♯ ( ρ ( u ) , x ) ρ ( v ) , x )= x k Y ♯V ( ρ ( D ρY HV ( Y )( u, x ) v ) , x )= x k D ρY HV ( Y )( D ρY HV ( Y )( u, x ) v, x ) . The vacuum and creation properties follow immediately from the counit propertyand the vacuum property (2.17). Therefore, ( V, D ρY HV ( Y ) , ) carries the structure of anonlocal vertex algebra and by (2.36) ρ is a nonlocal vertex algebra homomorphismfrom D ρY HV ( V ) to V ♯H . (cid:3) Furthermore, we have:
Proposition 2.26.
The nonlocal vertex algebra D ρY HV ( V ) obtained in Theorem 2.25with the same map ρ is also a right H -comodule nonlocal vertex algebra.Proof. We only need to prove that ρ : V → V ⊗ H is a homomorphism of nonlocalvertex algebras from D ρY HV ( V ) to D ρY HV ( V ) ⊗ H . For u, v ∈ V , we have ρ (cid:16) D ρY HV ( Y )( u, x ) v (cid:17) = ρ (cid:16)X Y ( u (1) , x ) Y HV ( u (2) , x ) v (cid:17) = X Y ⊗ ( ρ ( u (1) ) , x ) ρ (cid:0) Y HV ( u (2) , x ) v (cid:1) = X Y ⊗ ( ρ ( u (1) ) , x )( Y HV ( u (2) , x ) ⊗ v (1) ⊗ v (2) )= X Y ( u (1) , (1) , x ) Y HV ( u (2) , x ) v (1) ⊗ Y ( u (1) , (2) , x ) v (2) . n the other hand, denoting by Y ′⊗ ( · , x ) the vertex operator map of D ρY HV ( V ) ⊗ H ,we have Y ′⊗ ( ρ ( u ) , x ) ρ ( v ) = X Y ′⊗ ( u (1) ⊗ u (2) , x )( v (1) ⊗ v (2) )= X D ρY HV ( Y )( u (1) , x ) v (1) ⊗ Y ( u (2) , x ) v (2) = X Y ( u (1) , (1) , x ) Y HV ( u (1) , (2) , x ) v (1) ⊗ Y ( u (2) , x ) v (2) . Meanwhile, since H is cocommutative, (2.32) yields X u (1) , (1) ⊗ u (2) ⊗ u (1) , (2) = X u (1) , (1) ⊗ u (1) , (2) ⊗ u (2) . Consequently, we get ρ (cid:16) D ρY HV ( Y )( u, x ) v (cid:17) = Y ′⊗ ( ρ ( u ) , x ) ρ ( v ) . (2.37)This proves that ρ is also a homomorphism of nonlocal vertex algebras from D ρY HV ( V )to D ρY HV ( V ) ⊗ H , concluding the proof. (cid:3) At the end of this section, we present some technical results. First, by a straight-forward argument we have:
Lemma 2.27.
Let H be a nonlocal vertex bialgebra and let ( V, Y HV ) be an H -modulenonlocal vertex algebra. Suppose that ( W, Y VW ) is a V -module and ( W, Y HW ) is an H -module such that Y HW ( h, x ) w ∈ W ⊗ C (( x )) , (2.38) Y HW ( h, x ) Y VW ( v, x ) w = X Y VW ( Y HV ( h (1) , x − x ) v, x ) Y HW ( h (2) , x ) w (2.39) for h ∈ S , v ∈ V , w ∈ W , where S is a generating subset of H as a nonlocal vertexalgebra. Then the two relations above hold for all h ∈ H , v ∈ V , w ∈ W . The following is the second technical result:
Lemma 2.28.
Let H be a nonlocal vertex bialgebra and let V be a nonlocal vertexalgebra. Suppose that Y HV ( · , x ) : H → Hom(
V, V ⊗ C (( x ))) is an H -module nonlocalvertex algebra structure and ρ : V → V ⊗ H is an H -comodule nonlocal vertex algebrastructure, satisfying Y HV ( h, x ) v ∈ T ⊗ C (( x )) , (2.40) ρ ( Y HV ( h, x ) v ) = ( Y HV ( h, x ) ⊗ ρ ( v )(2.41) for h ∈ S, v ∈ T , where S and T are generating subspaces of H and V as nonlocalvertex algebras, respectively. Then Y HV and ρ are compatible.Proof. First, we prove that (2.40) holds for all h ∈ H, v ∈ T . Let K consist of h ∈ H such that Y HV ( h, x ) v ∈ T ⊗ C (( x )) for v ∈ T . Recall from Lemma 2.17 thatfor any a, b ∈ H, w ∈ V , we have Y HV ( Y ( a, x ) b, x ) w = Y HV ( a, x + x ) Y HV ( b, x ) w. sing this we get that a m b ∈ K for a, b ∈ K, m ∈ Z . It follows that K is a nonlocalvertex subalgebra of H , containing S . Consequently, we have K = H , confirmingour assertion.Second, we show that (2.41) holds for all h ∈ H, v ∈ T . Let H ′ consist of a ∈ H ′ such that (2.41) holds for all v ∈ T . Using Lemma 2.17 and the first assertion, we canstraightforwardly show that H ′ is a nonlocal vertex subalgebra of H . Consequently,we have H ′ = H , confirming the second assertion.Third, we prove that (2.41) holds for all h ∈ H, v ∈ V . Similarly, let V ′ consistof v ∈ V such that (2.41) holds for all h ∈ H and we then prove V ′ = V by showingthat V ′ is a nonlocal vertex subalgebra containing T . The closure of V ′ can beestablished straightforwardly by using (2.18) and the fact that ρ : V → V ⊗ H isa homomorphism of nonlocal vertex algebras. Now that (2.41) holds for all h ∈ H, v ∈ V , Y HV and ρ are compatible. (cid:3) More on nonlocal vertex algebras D ρY HV ( V )In this section, we continue studying the deformed nonlocal vertex algebra D ρY HV ( V ).As the main result, we prove that D ρY HV ( V ) is a quantum vertex algebra under thecondition that H is cocommutative, V is a vertex algebra, and Y HV is invertible in acertain sense.Throughout this section, we assume that H is a cocommutative nonlocal vertexbialgebra. By coassociativity we have(∆ ⊗ ∆)∆ = (∆ ⊗ ⊗ ⊗ ∆)∆ = (∆ ⊗ ⊗ ⊗ ⊗ ∆ ⊗ ⊗ , As P (1 ⊗ ∆ ⊗
1) = (1 ⊗ ∆ ⊗
1) by cocommutativity, consequently we have P (∆ ⊗ ∆)∆ = (∆ ⊗ ∆)∆ . (3.1) Definition 3.1.
Let (
V, ρ ) be a right H -comodule nonlocal vertex algebra. Denoteby L ρH ( V ) the set of all H -module nonlocal vertex algebra structures on V whichare compatible with ρ . Example 3.2.
Let (
V, ρ ) be an H -comodule nonlocal vertex algebra. Define a linearmap Y εM ( · , x ) : H → End C ( V ) ⊂ (End V )[[ x, x − ]] by Y εM ( h, x ) v = ε ( h ) v for h ∈ H, v ∈ V. (3.2)Then it is straightforward to show that Y εM ∈ L ρH ( V ) and D ρY εM ( V ) = V .From definition, L ρH ( V ) is a subset of the space Hom ( H, Hom(
V, V ⊗ C (( x )))).Note that Hom( V, V ⊗ C (( x ))) is naturally an associative algebra. Furthermore, with H a coalgebra Hom ( H, Hom(
V, V ⊗ C (( x )))) is an associative algebra with respectto the convolution. More generally, it is a classical fact that for any associativealgebra A with identity, Hom( H, A ) is an associative algebra with the operation ∗ defined by ( f ∗ g )( h ) = X f ( h (1) ) g ( h (2) )(3.3)for h ∈ H, f, g ∈ Hom(
H, A ), where ∆( h ) = P h (1) ⊗ h (2) , and with ε as identity. or Y M , Y ′ M ∈ Hom( H, Hom(
V, V ⊗ C (( x )))) , we say that Y M and Y ′ M commute if Y M ( h, x ) Y ′ M ( k, z ) = Y ′ M ( k, z ) Y M ( h, x ) for all h, k ∈ H. (3.4)Furthermore, a subset U of L ρH ( V ) is said to be commutative if any two elements of U commute. Proposition 3.3.
Let ( V, ρ ) be a right H -comodule nonlocal vertex algebra. For Y M ( · , x ) , Y ′ M ( · , x ) ∈ Hom ( H, Hom(
V, V ⊗ C (( x )))) , define a linear map ( Y M ∗ Y ′ M )( · , x ) : H → Hom(
V, V ⊗ C (( x ))) by ( Y M ∗ Y ′ M )( h, x ) X Y M ( h (1) , x ) Y ′ M ( h (2) , x ) for h ∈ H. Then
Hom ( H, Hom(
V, V ⊗ C (( x )))) equipped with the operation ∗ is an associativealgebra with Y εM as identity. Furthermore, if Y M , Y ′ M ∈ L ρH ( V ) and if Y M and Y ′ M commute, then Y M ∗ Y ′ M ∈ L ρH ( V ) and Y M ∗ Y ′ M = Y ′ M ∗ Y M .Proof. The first assertion immediately follows from the aforementioned general fact,so it remains to prove the second assertion. Let ( Y M , Y ′ M ) be a commuting pair in L ρH ( V ). It follows immediately from the cocommutativity of H and the commutativ-ity of Y M and Y ′ M that Y M ∗ Y ′ M = Y ′ M ∗ Y M . For h, k ∈ H , using the homomorphismproperty of ∆, Lemma 2.17, and the commutativity of Y M with Y ′ M , we have( Y M ∗ Y ′ M )( Y ( h, z ) k, x )= X Y M (cid:16) ( Y ( h, z ) k ) (1) , x (cid:17) Y ′ M (cid:16) ( Y ( h, z ) k ) (2) , x (cid:17) = X Y M (cid:16) Y ( h (1) , z ) k (1) , x (cid:17) Y ′ M (cid:16) Y ( h (2) , z ) k (2) , x (cid:17) = X Y M ( h (1) , x + z ) Y M ( k (1) , x ) Y ′ M ( h (2) , x + z ) Y ′ M ( k (2) , x )= X Y M ( h (1) , x + z ) Y ′ M ( h (2) , x + z ) Y M ( k (1) , x ) Y ′ M ( k (2) , x )= ( Y M ∗ Y ′ M )( h, x + z )( Y M ∗ Y ′ M )( k, x ) . Then it follows that Y M ∗ Y ′ M is an H -module structure on V . On the other hand,for h ∈ H, u, v ∈ V , using the relation P (∆ ⊗ ∆)∆( h ) = (∆ ⊗ ∆)∆( h ) from (3.1),we get( Y M ∗ Y ′ M )( h, x ) Y ( u, z ) v = X Y M ( h (1) , x ) Y ′ M ( h (2) , x ) Y ( u, z ) v = X Y M ( h (1) , x ) Y ( Y ′ M ( h (2) , (1) , x − z ) u, z ) Y ′ M ( h (2) , (2) , x ) v = X Y (cid:16) Y M ( h (1) , (1) , x − z ) Y ′ M ( h (2) , (1) , x − z ) u, z (cid:17) Y M ( h (1) , (2) , x ) Y ′ M ( h (2) , (2) , x ) v = X Y (cid:16) Y M ( h (1) , (1) , x − z ) Y ′ M ( h (1) , (2) , x − z ) u, z (cid:17) Y M ( h (2) , (1) , x ) Y ′ M ( h (2) , (2) , x ) v X Y (cid:16) ( Y M ∗ Y ′ M )( h (1) , x − z ) u, z (cid:17) ( Y M ∗ Y ′ M )( h (2) , x ) v. This proves that (
V, Y M ∗ Y ′ M ) is an H -module nonlocal vertex algebra. It is straight-forward to show that Y M ∗ Y ′ M is compatible with ρ . Therefore, Y M ∗ Y ′ M ∈ L ρH ( V ). (cid:3) Furthermore, we have:
Proposition 3.4.
Let ( V, ρ ) be a right H -comodule nonlocal vertex algebra and let ( Y M , Y ′ M ) be a commuting pair in L ρH ( V ) . Then Y M ∈ L ρH (cid:16) D ρY ′ M ( V ) (cid:17) and D ρY M (cid:16) D ρY ′ M ( V ) (cid:17) = D ρY M ∗ Y ′ M ( V ) . (3.5) Proof.
Let h ∈ H, u ∈ V . Recall that ρ : V → V ⊗ H is an H -module homomor-phism, where ρ ( v ) = P v (1) ⊗ v (2) ∈ V ⊗ H for v ∈ V . Then we have Y M ( h, x ) D ρY ′ M ( Y )( u, z ) = X Y M ( h, x ) Y ( u (1) , z ) Y ′ M ( u (2) , z )= X Y ( Y M ( h (1) , x − z ) u (1) , z ) Y M ( h (2) , x ) Y ′ M ( u (2) , z )= X Y ( Y M ( h (1) , x − z ) u (1) , z ) Y ′ M ( u (2) , z ) Y M ( h (2) , x )= X Y (cid:16)(cid:0) Y M ( h (1) , x − z ) u (cid:1) (1) , z (cid:17) Y ′ M (cid:16)(cid:0) Y M ( h (1) , x − z ) u (cid:1) (2) , z (cid:17) Y M ( h (2) , x )= X D ρY ′ M ( Y ) (cid:0) Y M ( h (1) , x − z ) u, z (cid:1) Y M ( h (2) , x ) . Thus, ( D ρY ′ M ( V ) , Y M ) is an H -module nonlocal vertex algebra.From Proposition 2.26, D ρY ′ M ( V ) with map ρ is a right H -comodule nonlocal vertexalgebra. As Y M is compatible with ρ , we have Y M ∈ L ρH (cid:16) D ρY ′ M ( V ) (cid:17) .Let u ∈ V . Note that from the cocommutativity we have X u (1) , (1) ⊗ u (1) , (2) ⊗ u (2) = X u (1) ⊗ u (2) , (2) ⊗ u (2) , (1) . Then using the commutativity of Y M and Y ′ M , we get D ρY M (cid:16) D ρY ′ M ( Y ) (cid:17) ( u, x )= X D ρY ′ M ( Y )( u (1) , x ) Y M ( u (2) , x )= X Y ( u (1) , (1) , x ) Y ′ M ( u (1) , (2) , x ) Y M ( u (2) , x )= X Y ( u (1) , x ) Y ′ M ( u (2) , (2) , x ) Y M ( u (2) , (1) , x )= X Y ( u (1) , x ) Y M ( u (2) , (1) , x ) Y ′ M ( u (2) , (2) , x )= D ρY M ∗ Y ′ M ( Y )( u, x ) . Therefore, we have D ρY M (cid:16) D ρY ′ M ( V ) (cid:17) = D ρY M ∗ Y ′ M ( V ). (cid:3) We also have the following result which is straightforward to prove: emma 3.5. Let Y M be an invertible element of L ρH ( V ) with inverse Y − M withrespect to the operation ∗ . Then Y ( u, x ) v = X D ρY M ( Y ) (cid:0) u (1) , x (cid:1) Y − M (cid:0) u (2) , x (cid:1) v for u, v ∈ V, (3.6) where ρ ( u ) = P u (1) ⊗ u (2) ∈ V ⊗ H . As the main result of this section, we have:
Theorem 3.6.
Assume that H is a nonlocal vertex cocommutative bialgebra. Let V be a vertex algebra with a right H -comodule vertex algebra structure ρ . Suppose that { Y ± M } is a commutative subset of L ρH ( V ) with Y M and Y − M inverses each other withrespect to the operation ∗ . Define a linear map S ( x ) : V ⊗ V → V ⊗ V ⊗ C (( x )) by S ( x )( v ⊗ u ) = X Y M ( u (2) , − x ) v (1) ⊗ Y − M ( v (2) , x ) u (1) (3.7) for u, v ∈ V . Then S ( x ) is a unitary rational quantum Yang-Baxter operator andthe nonlocal vertex algebra D ρY M ( V ) which was obtained in Theorem 2.25 with S ( x ) is a quantum vertex algebra.Proof. Let u, v ∈ V . Note that as V is a vertex algebra, the usual skew symmetryholds. As ρ : V → V ⊗ H is an H -module homomorphism, we have X (cid:0) Y M ( u (2) , x ) v (cid:1) (1) ⊗ (cid:0) Y M ( u (2) , x ) v (cid:1) (2) = X Y M ( u (2) , x ) v (1) ⊗ v (2) . By using these properties and Lemma 3.5, it is straightforward to show that D ρY M ( Y )( u, x ) v = e x D D ρY M ( Y )( − x ) S ( − x )( v ⊗ u ) . It then follows from [Li4] that S -locality holds.Recall that S ( x ) = R S ( x ) R . For any a, b ∈ V , we have S ( − x )( b ⊗ a ) = X Y − M ( a (2) , − x ) b (1) ⊗ Y M ( b (2) , x ) a (1) . On the other hand, with ( ρ ⊗ ρ = (1 ⊗ ∆) ρ , it follows that X u (1) , (2) ⊗ u (2) ⊗ u (1) , (1) = X u (2)(1) ⊗ u (2) , (2) ⊗ u (1) , X v (1) , (1) ⊗ v (1) , (2) ⊗ v (2) = X v (1) ⊗ v (2) , (1) ⊗ v (2) , (2) . The compatibilities of ρ and Y ± M yield X (cid:0) Y M ( u (2) , − x ) v (1) (cid:1) (1) ⊗ (cid:0) Y M ( u (2) , − x ) v (1) (cid:1) (2) = X Y M ( u (2) , − x ) v (1)(1) ⊗ v (1)(2) , X (cid:0) Y − M ( v (2) , x ) u (1) (cid:1) (2) ⊗ (cid:0) Y − M ( v (2) , x ) u (1) (cid:1) (1) = X u (1)(2) ⊗ Y − M ( v (2) , x ) u (1)(1) . Then using all of these relations we get S ( − x ) S ( x )( v ⊗ u )= X S ( − x ) (cid:0) Y M ( u (2) , − x ) v (1) ⊗ Y − M ( v (2) , x ) u (1) (cid:1) = X Y − M (cid:16)(cid:0) Y − M ( v (2) , x ) u (1) (cid:1) (2) , − x (cid:17) (cid:0) Y M ( u (2) , − x ) v (1) (cid:1) (1) ⊗ Y M (cid:16)(cid:0) Y M ( u (2) , − x ) v (1) (cid:1) (2) , x (cid:17) (cid:0) Y − M ( v (2) , x ) u (1) (cid:1) (1) X Y − M (cid:0) u (1) , (2) , − x (cid:1) Y M ( u (2) , − x ) v (1) , (1) ⊗ Y M (cid:0) v (1) , (2) , x (cid:1) Y − M ( v (2) , x ) u (1) , (1) = X Y − M (cid:0) u (2) , (1) , − x (cid:1) Y M ( u (2) , (2) , − x ) v (1) ⊗ Y M (cid:0) v (2) , (1) , x (cid:1) Y − M ( v (2) , (2) , x ) u (1) = X ε ( u (2) ) v (1) ⊗ ε ( v (2) ) u (1) = X ε ( v (2) ) v (1) ⊗ ε ( u (2) ) u (1) = v ⊗ u. This proves that S ( x ) is unitary.Furthermore, for u, v, w ∈ V , as ρ is a vertex algebra homomorphism we have S ( x ) (cid:0) D ρY M ( Y )( u, z ) v ⊗ w (cid:1) = X S ( x ) (cid:0) Y ( u (1) , z ) Y M ( u (2) , z ) v ⊗ w (cid:1) = X Y M (cid:0) w (2) , − x (cid:1) (cid:0) Y ( u (1) , z ) Y M ( u (2) , z ) v (cid:1) (1) ⊗ Y − M (cid:16)(cid:0) Y ( u (1) , z ) Y M ( u (2) , z ) v (cid:1) (2) , x (cid:17) w (1) = X Y M (cid:0) w (2) , − x (cid:1) Y ( u (1) , (1) , z )( Y M ( u (2) , z ) v ) (1) ⊗ Y − M (cid:0) Y ( u (1) , (2) , z )( Y M ( u (2) , z ) v ) (2) , x (cid:1) w (1) = X Y M (cid:0) w (2) , − x (cid:1) Y ( u (1) , (1) , z ) Y M ( u (2) , z ) v (1) ⊗ Y − M (cid:0) Y ( u (1) , (2) , z ) v (2) , x (cid:1) w (1) = X Y ( Y M (cid:0) w (2) , (1) , − x − z (cid:1) u (1) , (1) , z ) Y M (cid:0) w (2) , (2) , − x (cid:1) Y M ( u (2) , z ) v (1) ⊗ Y − M (cid:0) u (1) , (2) , x + z (cid:1) Y − M (cid:0) v (2) , x (cid:1) w (1) . On the other hand, we have (cid:0) D ρY M ( Y )( z ) ⊗ (cid:1) S ( x ) S ( x + z )( u ⊗ v ⊗ w )= (cid:0) D ρY M ( Y )( z ) ⊗ (cid:1) S ( x ) X Y M ( w (2) , − x − z ) u (1) ⊗ v ⊗ Y − M ( u (2) , x + z ) w (1) = (cid:0) D ρY M ( Y )( z ) ⊗ (cid:1) X Y M ( w (2) , − x − z ) u (1) ⊗ Y M (cid:16)(cid:0) Y − M ( u (2) , x + z ) w (1) (cid:1) (2) , − x (cid:17) v (1) ⊗ Y − M (cid:0) v (2) , x (cid:1) (cid:0) Y − M ( u (2) , x + z ) w (1) (cid:1) (1) = (cid:0) D ρY M ( Y )( z ) ⊗ (cid:1) X Y M ( w (2) , − x − z ) u (1) ⊗ Y M (cid:0) w (1) , (2) , − x (cid:1) v (1) ⊗ Y − M (cid:0) v (2) , x (cid:1) Y − M ( u (2) , x + z ) w (1) , (1) = X Y (cid:16)(cid:0) Y M ( w (2) , − x − z ) u (1) (cid:1) (1) , z (cid:17) Y M (cid:16)(cid:0) Y M ( w (2) , − x − z ) u (1) (cid:1) (2) , z (cid:17) Y M (cid:0) w (1) , (2) , − x (cid:1) v (1) ⊗ Y − M (cid:0) v (2) , x (cid:1) Y − M ( u (2) , x + z ) w (1) , (1) = X Y (cid:0) Y M ( w (2) , − x − z ) u (1) , (1) , z (cid:1) Y M (cid:0) u (1) , (2) , z (cid:1) Y M (cid:0) w (1) , (2) , − x (cid:1) v (1) ⊗ Y − M (cid:0) v (2) , x (cid:1) Y − M ( u (2) , x + z ) w (1) , (1) = X Y (cid:0) Y M ( w (2) , − x − z ) u (1) , (1) , z (cid:1) Y M (cid:0) u (1) , (2) , z (cid:1) Y M (cid:0) w (1) , (2) , − x (cid:1) v (1) ⊗ Y − M (cid:0) v (2) , x (cid:1) Y − M ( u (2) , x + z ) w (1) , (1) = X Y (cid:0) Y M ( w (2) , − x − z ) u (1) , (1) , z (cid:1) Y M (cid:0) w (1) , (2) , − x (cid:1) Y M (cid:0) u (1) , (2) , z (cid:1) v (1) ⊗ Y − M ( u (2) , x + z ) Y − M (cid:0) v (2) , x (cid:1) w (1) , (1) = X Y (cid:0) Y M ( w (2) , (1) , − x − z ) u (1) , (1) , z (cid:1) Y M (cid:0) w (2) , (2) , − x (cid:1) Y M (cid:0) u (2) , z (cid:1) v (1) ⊗ Y − M ( u (1) , (2) , x + z ) Y − M (cid:0) v (2) , x (cid:1) w (1) . or the last equality we are also using the propertites X w (2) ⊗ w (1) , (2) ⊗ w (1) , (1) = X w (2) , (1) ⊗ w (2) , (2) ⊗ w (1) , X u (1) , (1) ⊗ u (1) , (2) ⊗ u (2) = X u (1) , (1) ⊗ u (2) ⊗ u (1) , (2) , i.e., P ( ρ ⊗ ρ ( w ) = P P (1 ⊗ ∆) ρ ( w ), P ( ρ ⊗ ρ ( u ) = ( ρ ⊗ ρ ( u ), which followfrom the cocommutativity as before. Thus we have S ( x ) (cid:0) D ρY M ( Y )( u, z ) v ⊗ w (cid:1) = (cid:0) D ρY M ( Y )( z ) ⊗ (cid:1) S ( x ) S ( x + z )( u ⊗ v ⊗ w ) . This proves that the hexagon identity holds.Next, we show that the quantum Yang-Baxter equation holds. Let u, v, w ∈ V .Recall that the compatibilities of ρ with Y M and Y − M state that X ( Y ± M ( h, z ) a ) (1) ⊗ ( Y ± M ( h, z ) a ) (2) = X Y ± M ( h, z ) a (1) ⊗ a (2) for h ∈ H, a ∈ V . Using these relations we get S ( x ) S ( x + z ) S ( z )( u ⊗ v ⊗ w )= X S ( x ) S ( x + z ) (cid:0) u ⊗ Y M ( w (2) , − z ) v (1) ⊗ Y − M ( v (2) , z ) w (1) (cid:1) = X S ( x ) Y M (cid:16)(cid:0) Y − M ( v (2) , z ) w (1) (cid:1) (2) , − x − z (cid:17) u (1) ⊗ Y M ( w (2) , − z ) v (1) ⊗ Y − M ( u (2) , x + z ) (cid:0) Y − M ( v (2) , z ) w (1) (cid:1) (1) = X S ( x ) Y M (cid:0) w (1) , (2) , − x − z (cid:1) u (1) ⊗ Y M ( w (2) , − z ) v (1) ⊗ Y − M ( u (2) , x + z ) Y − M ( v (2) , z ) w (1) , (1) = X Y M (cid:16)(cid:0) Y M ( w (2) , − z ) v (1) (cid:1) (2) , − x (cid:17) (cid:0) Y M (cid:0) w (1) , (2) , − x − z (cid:1) u (1) (cid:1) (1) ⊗ Y − M (cid:16)(cid:0) Y M (cid:0) w (1) , (2) , − x − z (cid:1) u (1) (cid:1) (2) , x (cid:17) (cid:0) Y M ( w (2) , − z ) v (1) (cid:1) (1) ⊗ Y − M ( u (2) , x + z ) Y − M ( v (2) , z ) w (1) , (1) = X Y M (cid:0) v (1) , (2) , − x (cid:1) Y M (cid:0) w (1) , (2) , − x − z (cid:1) u (1) , (1) ⊗ Y − M (cid:0) u (1) , (2) , x (cid:1) Y M ( w (2) , − z ) v (1) , (1) ⊗ Y − M ( u (2) , x + z ) Y − M ( v (2) , z ) w (1) , (1) . Similarly, we have S ( z ) S ( x + z ) S ( x )( u ⊗ v ⊗ w )= X Y M (cid:0) w (2) , − x − z (cid:1) Y M ( v (2) , − x ) u (1) , (1) ⊗ Y M (cid:0) w (1) , (2) , − z (cid:1) Y − M ( u (2) , x ) v (1) , (1) ⊗ Y − M (cid:0) v (1) , (2) , z (cid:1) Y − M (cid:0) u (1) , (2) , x + z (cid:1) w (1) , (1) = X Y M ( v (2) , − x ) Y M (cid:0) w (2) , − x − z (cid:1) u (1) , (1) ⊗ Y − M ( u (2) , x ) Y M (cid:0) w (1) , (2) , − z (cid:1) v (1) , (1) ⊗ Y − M (cid:0) u (1) , (2) , x + z (cid:1) Y − M (cid:0) v (1) , (2) , z (cid:1) w (1) , (1) , here for the last equality we are also using the commutativity for ( Y ± M , Y ± M ) and( Y ± M , Y ∓ M ). Note that with H cocommutative we have P ( ρ ⊗ ρ ( v ) = P P ( ρ ⊗ ρ ( v ) , which states X v (1) , (2) ⊗ v (1) , (1) ⊗ v (2) = X v (2) ⊗ v (1) , (1) ⊗ v (1) , (2) . Similarly, we have X u (1) , (1) ⊗ u (1) , (2) ⊗ u (2) = X u (1) , (1) ⊗ u (2) ⊗ u (1) , (2) , X w (1) , (2) ⊗ w (2) ⊗ w (1) , (1) = X w (2) ⊗ w (1) , (2) ⊗ w (1) , (1) . Then we conclude S ( x ) S ( x + z ) S ( z )( u ⊗ v ⊗ w ) = S ( z ) S ( x + z ) S ( x )( u ⊗ v ⊗ w ) . Last, the shift condition can be proved straightforwardly. Therefore, D ρY M ( V )with S ( x ) is a quantum vertex algebra. (cid:3) The following is a result about the generating subsets of D ρY M ( V ): Lemma 3.7.
Let Y M be an invertible element of L ρH ( V ) . Suppose that S and T aregenerating subspaces of V and H as nonlocal vertex algebras, respectively, such that ρ ( S ) ⊂ S ⊗ T, ∆( T ) ⊂ T ⊗ T, Y − M ( T, x ) S ⊂ S ⊗ C (( x )) . (3.8) Then S is also a generating subset of D ρY M ( V ) .Proof. Let X consist of all subspaces U of V such that Y − M ( T, x ) U ⊂ U ⊗ C (( x )) . Denote by K the sum of all such subspaces. It is clear that K is the (unique) largestin X . We see that S and C are in X , so that C + S ⊂ K . Let K denote thelinear span of u n v in D ρY M ( V ) for u, v ∈ K, n ∈ Z . For u, v ∈ K and t ∈ T , fromthe first part of the proof of Proposition 3.4, we see that Y − M ( t, x ) D ρY M ( Y )( u, z ) v = X D ρY M ( Y )( Y − M ( t (1) , x − z ) u, z ) Y − M ( t (2) , x ) v. It follows that K is in X . As K is the largest, we have K ⊂ K . Thus K is anonlocal vertex subalgebra of D ρY M ( V ) with S ⊂ K .Now, let H ′ be the maximal subspace of H such that Y − M ( H ′ , x ) K ⊂ K ⊗ C (( x )) . We have T + C ⊂ H ′ . Note that for h, k ∈ H ′ , we have Y − M ( Y ( h, z ) k, x ) V ′ = Y − M ( h, x + z ) Y − M ( k, x ) K ⊂ K ⊗ C (( x ))[[ z ]] . It follows that H ′ is a nonlocal vertex subalgebra of H and hence H ′ = H . Thus Y − M ( H, x ) K ⊂ K ⊗ C (( x )).Let V ′′ be any subspace of V such that Y M ( H, x ) V ′′ ⊂ V ′′ ⊗ C (( x )) and ρ ( V ′′ ) ⊂ V ′′ ⊗ H. (3.9) hat is, V ′′ is an H -submodule and a sub-comodule. For u, v ∈ V ′′ , we have ρ (cid:0) D ρY M ( V )( u, z ) v (cid:1) = Y ♯ ( ρ ( u ) , z ) ρ ( v )= X Y ( u (1) , z ) Y M ( u (2) , (1) , z ) v (1) ⊗ Y ( u (2) , (2) , z ) v (2) = X Y ( u (1) , (1) , z ) Y M ( u (1) , (2) , z ) v (1) ⊗ Y ( u (2) , z ) v (2) = X D ρY M ( Y )( u (1) , z ) v (1) ⊗ Y ( u (2) , z ) v (2) . It follows that (3.9) holds for the nonlocal vertex subalgebra of D ρY M ( V ) generatedby V ′′ . Since (3.9) holds for V ′′ = S , it holds for V ′′ = K .From Lemma 3.5, for u ∈ S, v ∈ K we have Y ( u, x ) v = X D ρY M ( Y )( u (1) , x ) Y − M ( u (2) , x ) v ∈ K [[ x, x − ]] . As C + S ⊂ K , it follows that K contains the nonlocal vertex subalgebra of V generated by S . Thus K = V . Therefore, S is also a generating subspace of thequantum vertex algebra D ρY M ( V ). (cid:3) φ -coordinated quasi modules for smash product nonlocal vertexalgebras In this section, we first recall from [Li9] the basic notions and results on φ -coordinated quasi modules for nonlocal vertex algebras and then study (equivariant) φ -coordinated modules for smash product nonlocal vertex algebras.We begin with formal group. A one-dimensional formal group (law) over C (see[H]) is a formal power series F ( x, y ) ∈ C [[ x, y ]] such that F (0 , y ) = y, F ( x,
0) = x, F ( F ( x, y ) , z ) = F ( x, F ( y, z )) . A particular example is the additive formal group F a ( x, y ) := x + y .Let F ( x, y ) be a one-dimensional formal group over C . An associate of F ( x, y )(see [Li9]) is a formal series φ ( x, z ) ∈ C (( x ))[[ z ]], satisfying the condition that φ ( x,
0) = x, φ ( φ ( x, y ) , z ) = φ ( x, F ( y, z )) . The following result was obtained therein:
Proposition 4.1.
Let p ( x ) ∈ C (( x )) . Set φ ( x, z ) = e zp ( x ) ddx x = X n ≥ z n n ! (cid:18) p ( x ) ddx (cid:19) n x ∈ C (( x ))[[ z ]] . Then φ ( x, z ) is an associate of F a ( x, y ) . On the other hand, every associate of F a ( x, y ) is of this form with p ( x ) uniquely determined. Example 4.2.
Taking p ( x ) = 1 in Proposition 4.1, we get φ ( x, z ) = x + z = F a ( x, z )(the formal group itself), and taking p ( x ) = x we get φ ( x, z ) = xe z . More generally,for r ∈ Z , from [FHL] we have φ r ( x, z ) := e zx r +1 ddx x = ( x (1 − rzx r ) − r if r = 0 xe z if r = 0 . (4.1) rom now on, we shall always assume that φ ( x, z ) is an associate of F a ( x, y ) with φ ( x, z ) = x , or equivalently, φ ( x, z ) = e zp ( x ) ddx x with p ( x ) = 0.From [Li9], for f ( x , x ) ∈ C (( x , x )), f ( φ ( x, z ) , x ) exists in C (( x ))[[ z ]] and thecorrespondence f ( x , x ) f ( φ ( x, z ) , x ) gives a ring embedding ι x = φ ( x,z ) : C (( x , x )) → C (( x ))[[ z ]] ⊂ C (( x ))(( z )) . Denote by C ∗ (( x , x )) the fraction field of C (( x , x )). Then ι x = φ ( x,z ) naturallyextends to a field embedding ι ∗ x = φ ( x,z ) : C ∗ (( x , x )) → C (( x ))(( z )) . (4.2)As a convention, for F ( x , x ) ∈ C ∗ (( x , x )), we write F ( φ ( x, z ) , x ) = ι ∗ x = φ ( x,z ) ( F ( x , x )) ∈ C (( x ))(( z )) . (4.3)For f ( x , x ) ∈ C (( x , x )), by definition f ( φ ( x, z ) , φ ( x, z )) = (cid:0) e z p ( x ) ∂ x e z p ( x ) ∂ x f ( x , x ) (cid:1) | x = x = x , which exists in C (( x ))[[ z , z ]], and the correspondence f ( x , x ) f ( φ ( x, z ) , φ ( x, z ))is a ring embedding of C (( x , x )) into C (( x ))[[ z , z ]] (see [JKLT]). Then this ringembedding gives rise to a field embedding C ∗ (( x , x )) → C (( x )) ∗ (( z , z )) , (4.4)where C (( x )) ∗ (( z , z )) denotes the fraction field of C (( x ))(( z , z )). As a conven-tion, for F ( x , x ) ∈ C ∗ (( x , x )), we view F ( φ ( x, z ) , φ ( x, z )) as an element of C (( x )) ∗ (( z , z )). Definition 4.3.
Let V be a nonlocal vertex algebra. A φ -coordinated quasi V -module is a vector space W equipped with a linear map Y W ( · , x ) : V → (End W )[[ x, x − ]] v Y W ( v, x ) , satisfying the conditions that Y W ( u, x ) w ∈ W (( x )) for u ∈ V, w ∈ W,Y W ( , x ) = 1 W (the identity operator on W ) , and that for any u, v ∈ V , there exists f ( x , x ) ∈ C (( x , x )) × such that f ( x , x ) Y W ( u, x ) Y W ( v, x ) ∈ Hom(
W, W (( x , x ))) , (4.5) ( f ( x , x ) Y W ( u, x ) Y W ( v, x )) | x = φ ( x ,z ) = f ( φ ( x , z ) , x ) Y W ( Y ( u, z ) v, x ) . (4.6)The following notion was introduced in [JKLT] (cf. [Li10]): Definition 4.4.
Let V be a ( G, χ )-module nonlocal vertex algebra and let χ φ be alinear character of G such that φ ( x, χ ( g ) z ) = χ φ ( g ) φ ( χ φ ( g ) − x, z ) for g ∈ G. (4.7) ( G, χ φ ) -equivariant φ -coordinated quasi V -module is a φ -coordinated quasi V -module ( W, Y W ) satisfying the conditions that Y W ( R ( g ) v, x ) = Y W ( v, χ φ ( g ) − x ) for g ∈ G, v ∈ V (4.8)and that for u, v ∈ V , there exists q ( x ) ∈ C χ φ ( G ) [ x ] such that q ( x /x ) Y W ( u, x ) Y W ( v, x ) ∈ Hom(
W, W (( x , x ))) , (4.9)where C χ φ ( G ) [ x ] denotes the multiplicative monoid generated in C [ x ] by x − χ φ ( g )for g ∈ G . Remark 4.5.
It was proved (see loc. cit.) that the condition (4.7) is equivalent to p ( χ φ ( g ) x ) = χ ( g ) − χ φ ( g ) p ( x ) for g ∈ G, (4.10)where φ ( x, z ) = e zp ( x ) ddx x . In case p ( x ) = x r +1 with r ∈ Z , the compatibilitycondition (4.7) is equivalent to χ = χ − rφ .The following is a straightforward analogue of Lemma 2.11 for ( G, χ φ )-equivariant φ -coordinated quasi V -modules: Lemma 4.6.
Under the setting of Definition 4.4, assume that all the conditionshold except (4.8), and instead assume Y φW ( R ( g ) v, x ) = Y φW ( v, χ φ ( g ) − x ) f or g ∈ G, v ∈ S, (4.11) where S is a generating subset of V . Then W is a ( G, χ φ ) -equivariant φ -coordinatedquasi V -module. Next, we study (
G, χ )-module nonlocal vertex algebra structures on smash prod-uct nonlocal vertex algebra
V ♯H and equivariant φ -coordinated quasi modules for V ♯H . For the rest of this section, we fix a group G with a linear character χ . Wefirst formulate the following notion: Definition 4.7.
A (
G, χ ) -module nonlocal vertex bialgebra H is a nonlocal vertexbialgebra and a ( G, χ )-module nonlocal vertex algebra on which G acts as an auto-morphism group with H viewed as a coalgebra.The following two propositions can be proved straightforwardly: Proposition 4.8.
Let H be a ( G, χ ) -module nonlocal vertex bialgebra. Supposethat V is a ( G, χ ) -module nonlocal vertex algebra and an H -module nonlocal vertexalgebra with the H -module structure denoted by Y HV ( · , x ) such that R V ( g ) Y HV ( h, x ) v = Y HV ( R H ( g ) h, χ ( g ) x ) R V ( g ) v f or g ∈ G, h ∈ H, v ∈ V, (4.12) where R H and R V denote the representations of G on H and V , respectively. Then V ♯H is a ( G, χ ) -module nonlocal vertex algebra with R = R V ⊗ R H . Proposition 4.9.
Let
H, V be given as in Proposition 4.8. In addition, assume that V is a right H -comodule nonlocal vertex algebra whose right H -comodule structure ρ : V → V ⊗ H is compatible with Y HV and is a G -module homomorphism. Then D ρY HV ( V ) is a ( G, χ ) -module nonlocal vertex algebra with the same representation R of on V . Furthermore, the map ρ : V → V ⊗ H is a homomorphism of ( G, χ ) -modulenonlocal vertex algebras from D ρY HV ( V ) to V ♯H . Remark 4.10.
Let H be a ( G, χ )-module nonlocal vertex bialgebra and let V bea ( G, χ )-module nonlocal vertex algebra. As a convention, we shall always assumethat the condition (4.12) holds for an H -module nonlocal vertex algebra structure Y HV on V and that ρ is also a G -module homomorphism for an H -comodule nonlocalvertex algebra structure ρ .Recall that C ∗ (( x , x )) denotes the fraction field of C (( x , x )) and that for F ( x , x ) ∈ C ∗ (( x , x )), F ( φ ( x, z ) , x ) was defined as an element of C (( x ))(( z )).Here, we interpret this definition in a different way. For f ( x , x ) ∈ C (( x , x )), wealternatively define f ( φ ( x, z ) , x ) in two steps via substitutions C (( x , x )) s x φ ( x,z ) −→ C (( x, x ))[[ z ]] s x x −→ C (( x ))[[ z ]] ⊂ C (( x ))(( z )) , where s x = φ ( x,z ) and s x = x denote the indicated substitution maps. The compositionof the two substitutions was proved to be one-to-one. But, note that the substitution C (( x, x ))[[ z ]] s x x −→ C (( x ))[[ z ]] is not one-to-one. Set C (( x, x ))[[ z ]] o = { f ( x, x , z ) ∈ C (( x, x ))[[ z ]] | f ( x, x, z ) = 0 } . (4.13)Note that f ( φ ( x, z ) , x ) ∈ C (( x, x ))[[ z ]] o for f ( x , x ) ∈ C (( x , x )) × . Then setΛ( x, x , z ) = (cid:26) fg | f ∈ C (( x, x ))[[ z ]] , g ∈ C (( x, x ))[[ z ]] o (cid:27) . (4.14)The substitution map s x = x extends uniquely to a ring embedding˜ s x = x : Λ( x, x , z ) −→ C (( x ))(( z )) . (4.15)Now, for any F ( x , x ) ∈ C ∗ (( x , x )), we have F ( φ ( x, z ) , x ) ∈ Λ( x, x , z ) and F ( φ ( x, z ) , x ) | x = x := ˜ s x = x ( F ( φ ( x, z ) , x )) = F ( φ ( x, z ) , x ) . (4.16)We have the following result: Lemma 4.11.
Let φ ( x, z ) = e zp ( x ) ddx x with p ( x ) ∈ C (( x )) × and let F ( x , x ) ∈ C ∗ (( x , x )) . Then F ( φ ( x , z ) , x ) = F ( x , φ ( x , − z ))(4.17) holds in C ∗ (( x , x ))[[ z ]] if and only if p ( x ) ∂∂x F ( x , x ) = − p ( x ) ∂∂x F ( x , x ) . (4.18) Furthermore, assuming either one of the two equivalent conditions, we have F ( φ ( x, z ) , x ) = f ( z ) , (4.19) ι x,z ,z F ( φ ( x, z ) , φ ( x, z )) = f ( z − z )(4.20) for some f ( z ) ∈ C (( z )) . roof. From definition we have F ( φ ( x , z ) , x ) = e zp ( x ) ∂ x F ( x , x ) and F ( x , φ ( x , − z )) = e − zp ( x ) ∂ x F ( x , x ) . This immediately confirms the first assertion. For the second assertion, assuming F ( φ ( x , z ) , x ) = F ( x , φ ( x , − z )), we have p ( x ) ∂∂x F ( φ ( x, z ) , x ) = (cid:18) p ( x ) ∂∂x F ( φ ( x, z ) , x ) + p ( x ) ∂∂x F ( φ ( x, z ) , x ) (cid:19) | x = x = (cid:18) ∂∂z F ( φ ( x, z ) , x ) + p ( x ) ∂∂x F ( x, φ ( x , − z )) (cid:19) | x = x = (cid:18) ∂∂z F ( φ ( x, z ) , x ) − ∂∂z F ( x, φ ( x , − z )) (cid:19) | x = x = ∂∂z ( F ( φ ( x, z ) , x ) − F ( x, φ ( x , − z ))) | x = x = 0 . As p ( x ) = 0, we get ∂∂x F ( φ ( x, z ) , x ) = 0. Thus F ( φ ( x, z ) , x ) ∈ C (( z )). Setting f ( z ) = F ( φ ( x, z ) , x ), using (4.17) we have F ( φ ( x, z ) , φ ( x, z )) = F ( φ ( φ ( x, z ) , − z ) , x ) = F ( φ ( x, z − z ) , x ) = f ( z − z ) , where we are using obvious expansion conventions. This completes the proof. (cid:3) Definition 4.12.
Let φ ( x, z ) = e zp ( x ) ddx x with p ( x ) ∈ C (( x )) × . Denote by C φ (( x , x ))the set of all F ( x , x ) ∈ C ∗ (( x , x )) such that (4.18) holds.Note that with φ ( x, z ) = e zp ( x ) ddx x , we have ∂ z φ ( x, z ) = ∂ z e zp ( x ) ddx x = e zp ( x ) ddx p ( x ) = p ( φ ( x, z )) . (4.21)For F ( x , x ) ∈ C φ (( x , x )), we get ∂ z F ( φ ( x , z ) , x ) = ( ∂ x F ( x , x )) | x = φ ( x ,z ) ∂ z φ ( x , z )= ( p ( x ) ∂ x F ( x , x )) | x = φ ( x ,z ) . Then by a straightforward argument we have:
Lemma 4.13.
The set C φ (( x , x )) is a subalgebra of C ∗ (( x , x )) with p ( x ) ∂ x asa derivation and the map π φ : C φ (( x , x )) → C (( z )) defined by π φ ( F ( x , x )) = F ( φ ( x, z ) , x )(4.22) for F ( x , x ) ∈ C φ (( x , x )) is an embedding of differential algebras, where C (( z )) isviewed as a differential algebra with derivation ddz . With Lemma 4.13 we define a differential subalgebra of C (( z )) C φ (( z )) = π φ ( C φ (( x , x ))) ⊂ C (( z )) . (4.23)We also have the following result, whose proof is straightforward: emma 4.14. Let F ( x , x ) ∈ C φ (( x , x )) , λ ∈ C × such that p ( λx ) = µp ( x ) forsome µ ∈ C × . Then F ( λx , λx ) ∈ C φ (( x , x )) with π φ ( F ( λx , λx ))( z ) = π φ ( F ( x , x ))( λµ − z ) . (4.24) Remark 4.15.
Let G be a group with two linear characters χ and χ φ such that p ( χ φ ( g ) x ) = χ ( g ) − χ φ ( g ) p ( x ) for g ∈ G (see (4.10)). Assume F ( x , x ) ∈ C φ (( x , x )). By Lemma 4.14 we have π φ ( F ( χ φ ( g ) x , χ φ ( g ) x ))( z ) = π φ ( F ( x , x ))( χ ( g ) z ) . (4.25)Let G act on C [[ x, x − ]] by( σf )( x ) = f ( χ ( σ ) − x ) for σ ∈ G, f ( x ) ∈ C [[ x, x − ]] . (4.26)On the other hand, let G act on C [[ x ± , x ± ]] by( σF )( x , x ) = F ( χ φ ( σ ) − x , χ φ ( σ ) − x )(4.27)for σ ∈ G, F ∈ C [[ x ± , x ± ]]. It is straightforward to see that C φ (( x )) and C φ (( x , x )) are G -submodules of C [[ x ± ]] and C [[ x ± , x ± ]], respectively, and π φ is a G -module isomorphism from C φ (( x , x )) onto C φ (( x )). For f ( x ) ∈ C φ (( x )), set b f = π − φ ( f ) ∈ C φ (( x , x )) . (4.28)Then c σf = σ b f for f ∈ C φ (( x )) , σ ∈ G . Remark 4.16.
Let r ∈ Z . Recall (4.1): φ r ( x, z ) = e zx r +1 ddx x = ( x (1 − rzx r ) − r if r = 0 xe z if r = 0 . Set F r ( z , z ) = ( − r ( z − r − z − r ) if r = 0 z /z if r = 0 , (4.29)which are elements of C [ z ± , z ± ], and set f r ( z ) = ( z if r = 0 e z if r = 0 . (4.30)It can be readily seen that F r ( x , x ) ∈ C φ r (( x , x )) with π φ r ( F r ) = f r . Remark 4.17.
Consider the special case φ ( x, z ) = e zx ddx x = xe z . It is straightfor-ward to show that for F ( x , x ) ∈ C [[ x ± , x ± ]], x ∂∂x F ( x , x ) = − x ∂∂x F ( x , x )if and only if F ( x , x ) = f ( x /x ) for some f ( z ) ∈ C [[ z, z − ]]. From this we get C (( x , x )) ∩ C φ (( x , x )) = C [ x ± ](4.31) the Laurent polynomial ring in x ), where x = x /x . It is clear that C ( x /x ) ⊂ C φ (( x , x )) , (4.32)where C ( x /x ) denotes the field of rational functions in x /x . Notice that C ( x /x ) = C ( x /x ) inside C ( x , x ) ( ⊂ C ∗ (( x , x ))).Note that in case φ ( x, z ) = e z ddx x = x + z , we have C φ (( x , x )) ⊃ C (( x − x )) , C (( x − x )) , C ( x − x ) . (4.33)For any subgroup Γ of C × , set C Γ ∗ (( x , x )) = (cid:26) F ( x , x ) q ( x /x ) | F ( x , x ) ∈ C (( x , x )) , q ( x ) ∈ C Γ [ x ] (cid:27) , (4.34)recalling that C Γ [ x ] denotes the monoid generated multiplicatively by polynomials x − α for α ∈ Γ. Furthermore, set C Γ φ (( x , x )) = C φ (( x , x )) ∩ C Γ ∗ (( x , x )) . (4.35)It is straightforward to see that C Γ φ (( x , x )) is a differential subalgebra of C φ (( x , x )).Consequently, π φ (cid:0) C Γ φ (( x , x )) (cid:1) is a differential subalgebra of C φ (( x )). Definition 4.18.
Let H be a nonlocal vertex bialgebra. A φ -compatible H -modulenonlocal vertex algebra is an H -module nonlocal vertex algebra ( V, Y HV ) such that Y HV ( h, z ) v ∈ V ⊗ C φ (( z )) for h ∈ H, v ∈ V. (4.36)If H is a ( G, χ )-module nonlocal vertex bialgebra, we in addition assume that for h ∈ H, v ∈ V , there exists q ( x ) ∈ C χ φ ( G ) [ x ] such that q ( x /x ) b Y HV ( h, x , x ) v ∈ V ⊗ C (( x , x )) , (4.37)where b Y HV ( h, x , x ) v = (1 ⊗ π − φ ) Y HV ( h, x ) v .Note that we can alternatively combine the conditions (4.36) and (4.37) to write Y HV ( h, z ) v ∈ V ⊗ π φ ( C χ φ ( G ) φ (( x , x ))) for h ∈ H, v ∈ V. (4.38)For convenience, we formulate the following technical result: Lemma 4.19.
Let ( V, Y HV ) be an H -module nonlocal vertex algebra. Assume that(4.36) holds for h ∈ S, v ∈ U , where S is a generating subspace of H as a nonlocalvertex algebra such that ∆( S ) ⊂ S ⊗ S and U is a generating subset of V . Then ( V, Y HV ) is φ -compatible.Proof. Set V ′ = { v ∈ V | Y HV ( h, x ) v ∈ V ⊗ C φ (( x )) for h ∈ S } . Then U ∪ { } ⊂ V ′ .Let h ∈ S, u, v ∈ V ′ . With V an H -module nonlocal vertex algebra, we have Y HV ( h, x ) Y ( u, z ) v = X Y ( Y HV ( h (1) , x − z ) u, z ) Y HV ( h (2) , x ) v. Writing Y HV ( h (1) , x ) u = r X i =1 a ( i ) ⊗ f i ( x ) , Y HV ( h (2) , x ) v = s X j =1 b ( j ) ⊗ g i ( x ) , here a ( i ) , b ( j ) ∈ V, f i ( x ) , g j ( x ) ∈ C φ (( x )), we have Y HV ( h, x ) Y ( u, z ) v = X i,j g j ( x ) e − z ∂∂x f i ( x ) Y ( a ( i ) , z ) b ( j ) . From this we get Y HV ( h, x ) u m v ∈ V ⊗ C φ (( x )) for all m ∈ Z , recalling that C φ (( x ))is a subalgebra of C (( x )), which is closed under the derivation ddx . As U generates V , it follows that V ′ = V . Thus Y HV ( h, x ) v ∈ V ⊗ C φ (( x )) for all h ∈ S, v ∈ V .Furthermore, for h, h ′ ∈ H, v ∈ V , by Lemma 2.17 we have Y HV ( Y ( h, z ) h ′ , x ) v = Y HV ( h, x + z ) Y HV ( h ′ , x ) v. Using a similar argument we get Y HV ( h m h ′ , x ) v ∈ V ⊗ C φ (( x )) for all m ∈ Z . It followsthat (4.36) holds for all h ∈ H, v ∈ V . Therefore, ( V, Y HV ) is φ -compatible. (cid:3) Let (
V, Y HV ) be a φ -compatible H -module nonlocal vertex algebra. From defini-tion, we have b Y HV ( h, x , x ) v ∈ V ⊗ C φ (( x , x )) for h ∈ H, v ∈ V. (4.39)The following are immediate consequences: Lemma 4.20.
Let ( V, Y HV ) be a φ -compatible H -module nonlocal vertex algebra. For h, h ′ ∈ H, v ∈ V , we have b Y HV ( h, φ ( x , z ) , x ) = b Y HV ( h, x , φ ( x , − z )) , (4.40) b Y HV ( Y ( h, z ) h ′ , x , x ) = b Y HV ( h, φ ( x , z ) , x ) b Y HV ( h ′ , x , x ) , (4.41) b Y HV ( h, x , x ) Y ( v, x ) = X Y ( b Y HV ( h (1) , x , φ ( x , x )) v, x ) b Y HV ( h (2) , x , x ) . (4.42) Furthermore, we have g ( φ ( x, z ) , x ) b Y HV ( h, φ ( x, z ) , x ) v = g ( φ ( x, z ) , x ) Y HV ( h, z ) v (4.43) for any g ( x , x ) ∈ C (( x , x )) such that g ( x , x ) b Y HV ( h, x , x ) v ∈ V ⊗ C (( x , x )) . Definition 4.21.
Let H be a ( G, χ )-module nonlocal vertex bialgebra, let (
V, Y HV )be a φ -compatible H -module nonlocal vertex algebra, and let χ φ be a linear charac-ter of G satisfying (4.7). A ( G, χ φ ) -equivariant φ -coordinated quasi ( H, V ) -module is a ( G, χ φ )-equivariant φ -coordinated quasi H -module ( W, Y HW ), equipped with a( G, χ φ )-equivariant φ -coordinated quasi V -module structure Y VW ( · , x ), such that Y HW ( h, x ) w ∈ W ⊗ C (( x )) , (4.44) Y HW ( h, x ) Y VW ( v, x ) = X Y VW (cid:16) ι x ,x b Y HV ( h (1) , x , x ) v, x (cid:17) Y HW ( h (2) , x )(4.45)for h ∈ H , v ∈ V, w ∈ W .The following lemma follows immediately from the proof of Lemma 2.17: emma 4.22. Let ( W, Y HW ) be a φ -coordinated quasi H -module such that (4.44)holds for any h ∈ H, w ∈ W . Then Y HW ( h, x ) Y HW ( k, x ) | x = φ ( x ,z ) = Y HW ( Y ( h, z ) k, x )(4.46) for h, k ∈ H . As the main result of this section, we have (cf. Proposition 2.15):
Theorem 4.23.
Let H be a ( G, χ ) -module nonlocal vertex bialgebra, let ( V, Y HV ) bea φ -compatible H -module nonlocal vertex algebra, and let ( W, Y HW , Y VW ) be a ( G, χ φ ) -equivariant φ -coordinated quasi ( H, V ) -module. In addition, assume R V ( g ) Y HV ( h, x ) v = Y HV ( R H ( g ) h, χ ( g ) x ) R V ( g ) v for g ∈ G, h ∈ H, v ∈ V (see Proposition 4.8). For h ∈ H, v ∈ V , set Y ♯W ( v ⊗ h, x ) = Y VW ( v, x ) Y HW ( h, x ) ∈ (End W )[[ x, x − ]] . (4.47) Then ( W, Y ♯W ) is a ( G, χ φ ) -equivariant φ -coordinated quasi V ♯H -module.Proof.
First of all, for v ∈ V, h ∈ H , with the property (4.44) we have Y ♯W ( v ⊗ h, x ) = Y VW ( v, x ) Y HW ( h, x ) ∈ Hom(
W, W (( x ))) . We also have Y ♯W ( ⊗ , x ) = Y VW ( , x ) Y HW ( , x ) = 1 . Let u, v ∈ V , h, k ∈ H . From definition, there exists q ( x ) ∈ C χ ( G ) [ x ] such that q ( x /x ) b Y HV (cid:0) h (1) , x , x (cid:1) v ∈ V ⊗ C (( x , x )) ,q ( x /x ) Y VW ( u, x ) Y VW (cid:16) b Y HV ( h (1) , x , x ) v, x (cid:17) ∈ Hom(
W, W (( x , x ))) . On the other hand, from (4.44) we have Y HW ( h (2) , x ) Y HW ( k, x ) ∈ Hom(
W, W ⊗ C (( x , x ))) . Then q ( x /x ) Y ♯W ( u ⊗ h, x ) Y ♯W ( v ⊗ k, x )= q ( x /x ) X Y VW ( u, x ) Y VW (cid:16) ι x ,x b Y HV ( h (1) , x , x ) v, x (cid:17) Y HW ( h (2) , x ) Y HW ( k, x ) , which implies that the common quantity on both sides lies in Hom( W, W (( x , x ))).Using all of these, Lemma 4.22, and (4.43), we get (cid:16) q ( x /x ) Y ♯W ( u ⊗ h, x ) Y ♯W ( v ⊗ k, x ) (cid:17) | x = φ ( x ,z ) = (cid:16) q ( x /x ) X Y VW ( u, x ) Y VW (cid:16) b Y HV ( h (1) , x , x ) v, x (cid:17) Y HW ( h (2) , x ) Y HW ( k, x ) (cid:17) | x = φ ( x ,z ) = q ( φ ( x , z ) /x ) Y VW (cid:16) Y ( u, z ) b Y HV ( h (1) , φ ( x , z ) , x ) v, x (cid:17) Y HW ( Y ( h (2) , z ) k, x )= q ( φ ( x , z ) /x ) Y VW ( Y ( u, z ) Y HV ( h (1) , z ) v, x ) Y HW ( Y ( h (2) , z ) k, x )= q ( φ ( x , z ) /x ) Y ♯W ( Y ♯ ( u ⊗ h, z )( v ⊗ k ) , x ) . This proves that (
W, Y ♯W ) carries the structure of a φ -coordinated quasi V ♯H -module. urthermore, for g ∈ G, v ∈ V, h ∈ H , we have Y ♯W (( R V ( g ) ⊗ R H ( g ))( v ⊗ h ) , x ) = Y VW ( R V ( g ) v, x ) Y HW ( R H ( g ) h, x )= Y VW ( v, χ φ ( g ) − x ) Y HW ( h, χ φ ( g ) − x ) = Y ♯W ( u ⊗ h, χ φ ( g ) − x ) . This proves that (
W, Y ♯W ) is ( G, χ φ )-equivariant. (cid:3) Recall from Theorem 2.25 that the right H -comodule map ρ : V → V ⊗ H isalso a nonlocal vertex algebra homomorphism from D ρY HV ( V ) to V ♯H . Note that D ρY HV ( V ) is a ( G, χ )-module nonlocal vertex algebra. As an immediate consequenceof Theorem 4.23 we have:
Corollary 4.24.
Under the setting of Theorem 4.23, for v ∈ V , set Y ♯ρW ( v, x ) = X Y VW ( v (1) , x ) Y HW ( v (2) , x ) (cid:16) = Y ♯W ( ρ ( v ) , x ) (cid:17) . (4.48) In addition we assume that H is cocommutative. Then ( W, Y ♯ρW ) is a ( G, χ φ ) -equivariant φ -coordinated quasi D ρY HV ( V ) -module. For any subgroup Γ of C × , set C Γ [ x , x ] = h x − αx | α ∈ Γ i , (4.49)the multiplicative moniod generated by x − αx for α ∈ Γ in C [ x , x ]. Lemma 4.25.
Let V be a ( G, χ ) -module nonlocal vertex algebra and let ( W, Y φW ) bea ( G, χ φ ) -equivariant φ -coordinated quasi V -module. Suppose that u, v ∈ V , and u ( i ) , v ( i ) ∈ V, f i ( x ) ∈ C φ (( x )) ( i = 1 , . . . , r ) such that for some nonnegative integer k , ( x − x ) k Y ( u, x ) Y ( v, x )=( x − x ) k r X i =1 f i ( − x + x ) Y ( v ( i ) , x ) Y ( u ( i ) , x ) . (4.50) Then there exists q ( x , x ) ∈ C χ φ ( G ) [ x , x ] such that q ( x , x ) Y φW ( u, x ) Y φW ( v, x )= q ( x , x ) r X i =1 ι x ,x (cid:16) b f i ( x , x ) (cid:17) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x ) . (4.51) Furthermore, (4.51) holds for any q ( x , x ) ∈ C (( x , x )) such that q ( x , x ) Y φW ( u, x ) Y φW ( v, x ) ∈ Hom(
W, W (( x , x ))) . Proof.
With (4.50), from [Li4, Corollary 5.3] we have Y ( u, x ) v = r X i =1 f i ( x ) e x D Y ( v ( i ) , − x ) u ( i ) . rom definition, there exists h ( x , x ) ∈ C χ φ ( G ) [ x , x ] such that h ( x , x ) Y φW ( u, x ) Y φW ( v, x ) ∈ Hom(
W, W (( x , x ))) ,h ( x , x ) b f i ( x , x ) ∈ C (( x , x )), and( h ( x , x ) b f i ( x , x )) Y φW ( v ( i ) , x ) Y W ( u ( i ) , x ) ∈ Hom(
W, W (( x , x )))for i = 1 , . . . , r . Then using [Li9, Lemma 3.7] we get( h ( x , x ) Y φW ( u, x ) Y φW ( v, x )) | x = φ ( x ,z ) = h ( φ ( x , z ) , x ) Y φW ( Y ( u, z ) v, x )= r X i =1 ( h ( φ ( x , z ) , x ) f i ( z ) Y φW ( e z D ( v ( i ) , − z ) u ( i ) , x )= r X i =1 ( h ( φ ( x , z ) , x ) f i ( z ) Y φW ( Y ( v ( i ) , − z ) u ( i ) , φ ( x , z )) . On the other hand, we have h ( x , x ) r X i =1 ι x ,x ( b f i ( x , x )) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x ) ! | x = φ ( x , − z ) = r X i =1 h ( x , φ ( x , − z )) b f i ( x , φ ( x , − z )) Y φW ( Y ( v ( i ) , − z ) u ( i ) , x )= r X i =1 h ( x , φ ( x , − z )) f i ( z ) Y φW ( Y ( v ( i ) , − z ) u ( i ) , x ) . Then using [Li9, Remark 2.8] we have( h ( x , x ) Y φW ( u, x ) Y φW ( v, x )) | x = φ ( x ,z ) = h ( x , x ) r X i =1 b f i ( x , x ) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x ) ! | x = φ ( x , − z ) ! | x = φ ( x ,z ) = h ( x , x ) r X i =1 ι x ,x ( b f i ( x , x )) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x ) ! | x = φ ( x ,z ) , noticing that φ ( φ ( x , z ) , − z ) = φ ( x ,
0) = x . By [Li9, Remark 2.8] again we get h ( x , x ) Y φW ( u, x ) Y φW ( v, x ) = h ( x , x ) r X i =1 ι x ,x ( b f i ( x , x )) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x ) . This proves the first assertion. As for the second assertion we have h ( x , x ) h q ( x , x ) Y φW ( u, x ) Y φW ( v, x ) i = h ( x , x ) q ( x , x ) r X i =1 b f i ( x , x ) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x ) . ultiplying both sides by the inverse of h ( x , x ) in C (( x ))(( x )) we obtain thedesired relation. (cid:3) Remark 4.26.
Let p ( x ) ∈ C (( x )) × . It follows from induction that for every positiveinteger n , there exist A n, ( x ) , . . . , A n,n ( x ) ∈ C (( x )) with A n,n ( x ) = p ( x ) − n such that (cid:18) ddx (cid:19) n = A n, ( x ) + A n, ( x ) p ( x ) ddx + · · · + A n,n ( x ) (cid:18) p ( x ) ddx (cid:19) n (4.52)on C (( x )). For B n, ( x ) , . . . , B n,n ( x ) ∈ C (( x )), it can be readily seen that B n, ( x ) + B n, ( x ) ddx + · · · + B n,n ( x ) (cid:18) ddx (cid:19) n = 0if and only if B n,i ( x ) = 0 for i = 0 , . . . , n . Then it follows that those coefficients A n,i ( x ) for 0 ≤ i ≤ n in (4.52) are uniquely determined.The following result generalizes the corresponding results in [Li4] and [Li10]: Proposition 4.27.
Let V be a ( G, χ ) -module nonlocal vertex algebra, ( W, Y φW ) a ( G, χ φ ) -equivariant φ -coordinated quasi V -module, and let G be a complete set ofcoset representatives of ker χ φ in G . Assume u, v ∈ V , and u ( i ) , v ( i ) ∈ V, f i ( x ) ∈ C φ (( x )) ( i = 1 , . . . , r ) such that ( x − x ) k Y ( u, x ) Y ( v, x ) = ( x − x ) k r X i =1 f i ( − x + x ) Y ( v ( i ) , x ) Y ( u ( i ) , x ) for some nonnegative integer k . Then Y φW ( u, x ) Y φW ( v, x ) − r X i =1 ι x ,x ( b f i ( x , x )) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x )= X g ∈ G X j ≥ Y φW (( R ( g ) u ) j v, x ) 1 j ! ( p ( x ) ∂ x ) j p ( x ) x − δ (cid:18) χ φ ( g ) − x x (cid:19) . (4.53) Proof.
From Lemma 4.25, there exist distinct (nonzero complex numbers) c , . . . , c s ∈ χ φ ( G ) and positive integers k , . . . , k s such that (4.51) holds with q ( x , x ) = ( x − c x ) k · · · ( x − c s x ) k s . By [Li3, Lemma 2.5], we have p ( x ) − Y φW ( u, x ) Y φW ( v, x ) − p ( x ) − r X i =1 ι x ,x ( b f i ( x , x )) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x )= s X j =1 k j − X t =0 C j,t ( x ) (cid:18) ∂∂x (cid:19) t x − δ (cid:18) c j x x (cid:19) , here C j,t ( x ) ∈ E ( W ). Furthermore, using Remark 4.26, we get Y φW ( u, x ) Y φW ( v, x ) − r X i =1 ι x ,x ( b f i ( x , x )) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x )= s X j =1 k j − X t =0 A j,t ( x ) 1 t ! (cid:18) p ( x ) ∂∂x (cid:19) t p ( x ) x − δ (cid:18) c j x x (cid:19) , (4.54)where A j,t ( x ) ∈ E ( W ). By [JKLT, Theorems 2.19, 2.21] and Lemma 2.6 we have Y φW ( u, c j x ) φt Y φW ( v, x ) = ( A j,t ( x ) for 1 ≤ j ≤ s, ≤ t ≤ k j − , ≤ j ≤ s, t ≥ k j . (4.55)Let g , . . . , g r ∈ G such that c j = χ φ ( g j ) − for 1 ≤ j ≤ s . Then using [Li9,Proposition 4.11] we get Y φW ( u, c j x ) φt Y φW ( v, x ) = Y φW ( R ( g j ) u, x ) φt Y φW ( v, x ) = Y φW (( R ( g j ) u ) t v, x ) . Consequently, we have Y φW ( u, x ) Y φW ( v, x ) − r X i =1 ι x ,x ( b f i ( x , x )) Y φW ( v ( i ) , x ) Y φW ( u ( i ) , x )= s X j =1 k j − X t =0 Y φW (( R ( g j ) u ) t v, x ) 1 t ! ( p ( x ) ∂ x ) t p ( x ) x − δ (cid:18) χ φ ( g j ) − x x (cid:19) . Suppose g ∈ G such that χ φ ( g ) − / ∈ { c , . . . , c s } . By [JKLT, Theorems 2.19, 2.21]also (by adding a term with zero coefficients corresponding to c on the right-handside of (4.54)) we have Y φW (( R ( g ) u ) t v, x ) = 0 for t ≥
0. Therefore (4.53) holds. Thiscompletes the proof. (cid:3) Deformations of lattice vertex algebras V L In this section, we use the vertex bialgebra B L introduced in [Li8] to study thelattice vertex algebra V L . More specifically, we give a right B L -comodule nonlocalvertex algebra structure and a family of compatible B L -module nonlocal vertexalgebra structures on V L . As an application, we obtain a family of deformations of V L .5.1. Vertex algebra V L and associative algebra A ( L ) . We first recall the latticevertex algebra V L (see [B1], [FLM]). Let L be a finite rank even lattice, i.e., a freeabelian group of finite rank equipped with a symmetric Z -valued bilinear form h· , ·i such that h α, α i ∈ Z for α ∈ L . We assume that L is nondegenerate in the obvioussense. Set h = C ⊗ Z L (5.1) nd extend h· , ·i to a symmetric C -valued bilinear form on h . View h as an abelianLie algebra with h· , ·i as a nondegenerate symmetric invariant bilinear form. Thenwe have an affine Lie algebra b h , where b h = h ⊗ C [ t, t − ] + C k as a vector space, and where k is central and[ α ( m ) , β ( n )] = mδ m + n, h α, β i k (5.2)for α, β ∈ h , m, n ∈ Z with α ( m ) denoting α ⊗ t m . Set b h ± = h ⊗ t ± C [ t ± ] , (5.3)which are abelian Lie subalgebras. Identify h with h ⊗ t . Furthermore, set b h ′ = b h + + b h − + C k , (5.4)which is a Heisenberg algebra. Then b h = b h ′ ⊕ h , which is a direct sum of Lie algebras.Let ε : L × L → C × be a 2-cocycle such that ε ( α, β ) ε ( β, α ) − = ( − h α,β i , ε ( α,
0) = 1 = ε (0 , α ) for α, β ∈ L. Denote by C ε [ L ] the ε -twisted group algebra of L , which by definition has a desig-nated basis { e α | α ∈ L } with relations e α · e β = ε ( α, β ) e α + β for α, β ∈ L. (5.5)Make C ε [ L ] an b h -module by letting b h ′ act trivially and letting h (= h (0)) act by h (0) e β = h h, β i e β for h ∈ h , β ∈ L. (5.6)Note that S ( b h − ) is naturally an b h -module of level 1. Set V L = S ( b h − ) ⊗ C ε [ L ] , (5.7)the tensor product of b h -modules, which is an b h -module of level 1. Set = 1 ⊗ e ∈ V L . Identify h and C ε [ L ] as subspaces of V L via the correspondence a a ( − ⊗ a ∈ h ) and e α ⊗ e α ( α ∈ L ) . For h ∈ h , set h ( z ) = X n ∈ Z h ( n ) z − n − . (5.8)On the other hand, for α ∈ L set E ± ( α, z ) = exp X n ∈ Z + α ( ± n ) ± n z ∓ n (5.9)on V L . For α ∈ L , define a linear operator z α : C ε [ L ] → C ε [ L ][ z, z − ] by z α · e β = z h α,β i e β for β ∈ L. (5.10) hen there exists a vertex algebra structure on V L , which is uniquely determined bythe conditions that is the vacuum vector and that ( h ∈ h , α ∈ L ): Y ( h, z ) = h ( z ) , Y ( e α , z ) = E − ( − α, z ) E + ( − α, z ) e α z α . (5.11)Next, we recall from [LL, Section 6.5] the associative algebra A ( L ). By definition, A ( L ) is the associative algebra over C with unit 1, generated by { h [ n ] , e α [ n ] | h ∈ h , α ∈ L, n ∈ Z } , where h [ n ] is linear in h , subject to a set of relations written in terms of generatingfunctions h [ z ] = X n ∈ Z h [ n ] z − n − , e α [ z ] = X n ∈ Z e α [ n ] z − n − . The relations are(AL1) e [ z ] = 1 , (AL2) [ h [ z ] , h ′ [ z ]] = h h, h ′ i ∂∂z z − δ (cid:18) z z (cid:19) , (AL3) [ h [ z ] , e α [ z ]] = h α, h i e α [ z ] z − δ (cid:18) z z (cid:19) , (AL4) [ e α [ z ] , e β [ z ]] = 0 if h α, β i ≥ , (AL5) ( z − z ) −h α,β i [ e α [ z ] , e β [ z ]] = 0 if h α, β i < , for h, h ′ ∈ h , α, β ∈ L . An A ( L )-module W is said to be restricted if for every w ∈ W , we have h [ z ] w, e α [ z ] w ∈ W (( z )) for all h ∈ h , α ∈ L .The following result was obtained in [LL]: Proposition 5.1.
For any V L -module ( W, Y W ) , W is a restricted A ( L ) -module with h [ z ] = Y W ( h, z ) , e α [ z ] = Y W ( e α , z ) for h ∈ h , α ∈ L , such that the following relations hold on W for α, β ∈ L : (AL6) ∂ z e α [ z ] = α [ z ] + e α [ z ] + e α [ z ] α [ z ] − , (AL7) Res x (cid:0) ( x − z ) −h α,β i− e α [ x ] e β [ z ] − ( − z + x ) −h α,β i− e β [ z ] e α [ x ] (cid:1) = ε ( α, β ) e α + β [ z ] , where α [ z ] + = P n< α [ n ] z − n − and α [ z ] − = P n ≥ α [ n ] z − n − . On the other hand,if W is a restricted A ( L ) -module satisfying (AL6) and (AL7), then W admits a V L -module structure Y W ( · , z ) which is uniquely determined by Y W ( h, z ) = h [ z ] , Y W ( e α , z ) = e α [ z ] for h ∈ h , α ∈ L. Let (
W, Y W ) be any V L -module. From [LL, Proposition 6.5.2], the following rela-tions hold on W for α, β ∈ L :( x − z ) −h α,β i− Y W ( e α , x ) Y W ( e β , z ) − ( − z + x ) −h α,β i− Y W ( e β , z ) Y W ( e α , x )= ε ( α, β ) Y W ( e α + β , z ) x − δ (cid:16) zx (cid:17) . (5.12) t can be readily seen that this implies the relations (AL4-5) and (AL7) with e α [ z ] = Y W ( e α , z ) for α ∈ L . In fact, (5.12) is equivalent to (AL7) together with( x − z ) −h α,β i Y W ( e α , x ) Y W ( e β , z ) = ( − z + x ) −h α,β i Y W ( e β , z ) Y W ( e α , x ) . (5.13)The following is a universal property of the vertex algebra V L (see [JKLT]): Proposition 5.2.
Let V be a nonlocal vertex algebra and let ψ : h ⊕ C ε [ L ] → V bea linear map such that ψ ( e ) = , the relations (AL1-3) and (AL6) hold with h [ z ] = Y ( ψ ( h ) , z ) , e α [ z ] = Y ( ψ ( e α ) , z ) for h ∈ h , α ∈ L, and such that the following relation holds for α, β ∈ L : ( x − z ) −h α,β i− e α [ x ] e β [ z ] − ( − z + x ) −h α,β i− e β [ z ] e α [ x ]= ε ( α, β ) e α + β [ z ] x − δ (cid:16) zx (cid:17) . (5.14) Then ψ can be extended uniquely to a nonlocal vertex algebra homomorphism from V L to V . Vertex bialgebra B L . View L as an abelian group and let C [ L ] be its groupalgebra with basis { e α | α ∈ L } . Recall that h = C ⊗ Z L , a vector space, and b h − = h ⊗ t − C [ t − ], an abelian Lie algebra. Note that both C [ L ] and S ( b h − ) (= U ( b h − ))are Hopf algebras. Set B L = S ( b h − ) ⊗ C [ L ] , (5.15)which is a Hopf algebra. In particular, B L is a bialgebra, where the comultiplication∆ and counit ε are uniquely determined by∆( h ( − n )) = h ( − n ) ⊗ ⊗ h ( − n ) , ∆( e α ) = e α ⊗ e α , (5.16) ε ( h ( − n )) = 0 , ε ( e α ) = 1(5.17)for h ∈ h , n ∈ Z + , α ∈ L . On the other hand, B L admits a derivation ∂ which isuniquely determined by ∂ ( h ( − n )) = nh ( − n − , ∂ ( u ⊗ e α ) = α ( − u ⊗ e α + ∂u ⊗ e α (5.18)for h ∈ h , n ∈ Z + , u ∈ S ( b h − ) , α ∈ L . Then B L becomes a commutative vertexalgebra where the vertex operator map, denoted by Y B L ( · , x ), is given by Y B L ( a, x ) b = ( e x∂ a ) b for a, b ∈ B L . We sometimes denote this vertex algebra by ( B L , ∂ ).The following result was obtained in [Li8]: Proposition 5.3.
For h ∈ h , α ∈ L , we have Y B L ( h ( − , x ) = X n ≥ h ( − n ) x n − , (5.19) Y B L ( e α , x ) = e α exp X n ≥ α ( − n ) n x n ! = e α E − ( − α, x ) . (5.20) n the other hand, ( B L , ∂ ) is a vertex bialgebra with the comultiplication ∆ andcounit ε of B L , which is both commutative and cocommutative. Recall that the vertex algebra V L = S ( b h − ) ⊗ C ǫ [ L ] contains h and C ǫ [ L ] as sub-spaces. As the main result of this section, we have: Theorem 5.4.
There exists a right B L -comodule vertex algebra structure ρ on vertexalgebra V L , which is uniquely determined by ρ ( h ) = h ⊗ ⊗ h ( − , ρ ( e α ) = e α ⊗ e α for h ∈ h , α ∈ L. (5.21) Proof.
The uniqueness is clear as V L as a vertex algebra is generated by the subspace h + C ǫ [ L ]. Denote by Y ⊗ the vertex operator map of the tensor product vertexalgebra V L ⊗ B L . Let ρ be the linear map from h ⊕ C ǫ [ L ] to V L ⊗ B L , defined by ρ ( h ) = h ⊗ ⊗ h ( − , ρ ( e α ) = e α ⊗ e α for h ∈ h , α ∈ L. We are going to apply Proposition 5.2 with V = V L ⊗ B L and ψ = ρ . As B L is acommutative vertex algebra, for h, h ′ ∈ h , α ∈ L we have[ Y ⊗ ( ρ ( h ) , x ) , Y ⊗ ( ρ ( h ′ ) , z )] = h h, h ′ i ∂∂z x − δ (cid:16) zx (cid:17) , [ Y ⊗ ( ρ ( h ) , x ) , Y ⊗ ( ρ ( e α ) , z )] = h h, α i x − δ (cid:16) zx (cid:17) Y ⊗ ( ρ ( e α ) , z ) . Let α, β ∈ L . Using Proposition 5.3 we get Y ⊗ ( ρ ( e α ) , x ) Y ⊗ ( ρ ( e β ) , x )= Y ( e α , x ) Y ( e β , x ) ⊗ e α E − ( − α, x ) e β E − ( − β, x )= Y ( e α , x ) Y ( e β , x ) ⊗ e α + β E − ( − α, x ) E − ( − β, x )and Y ⊗ ( ρ ( e β ) , x ) Y ⊗ ( ρ ( e α ) , x ) = Y ( e β , x ) Y ( e α , x ) ⊗ e α + β E − ( − β, x ) E − ( − α, x ) . Note that E − ( − α, x ) E − ( − β, x ) = E − ( − β, x ) E − ( − α, x ). Furthermore, we have( x − x ) −h α,β i− Y ⊗ ( ρ ( e α ) , x ) Y ⊗ ( ρ ( e β ) , x ) − ( − x + x ) −h α,β i− Y ⊗ ( ρ ( e β ) , x ) Y ⊗ ( ρ ( e α ) , x )= ǫ ( α, β ) Y ( e α + β , x ) x − δ (cid:18) x x (cid:19) ⊗ e α + β E − ( − α, x ) E − ( − β, x )= ǫ ( α, β ) Y ( e α + β , x ) ⊗ e α + β E − ( − α, x ) E − ( − β, x ) x − δ (cid:18) x x (cid:19) = ǫ ( α, β ) Y ( e α + β , x ) ⊗ e α + β E − ( − α − β, x ) x − δ (cid:18) x x (cid:19) = ǫ ( α, β ) Y ⊗ ( e α + β , x ) x − δ (cid:18) x x (cid:19) and ddx Y ⊗ ( e α , x ) = ddx Y ( e α , x ) ⊗ e α E − ( − α, x ) + Y ( e α , x ) ⊗ e α α ( x ) + E − ( − α, x ) (cid:0) α ( x ) + Y ( e α , x ) + Y ( e α , x ) α ( x ) − (cid:1) ⊗ Y ( e α , x ) + Y ( e α , x ) ⊗ α ( x ) + Y ( e α , x )= Y ⊗ ( ρ ( α ) , x ) + Y ⊗ ( ρ ( e α ) , x ) + Y ⊗ ( ρ ( e α ) , x ) Y ⊗ ( ρ ( α ) , x ) − , where Y ⊗ ( ρ ( α ) , x ) + = P n< ρ ( α ) n x − n − and Y ⊗ ( ρ ( α ) , x ) − = P n ≥ ρ ( α ) n x − n − .(Notice that Y B L ( h, x ) − = 0 for h ∈ h .) Then by Proposition 5.2 there exists avertex algebra homomorphism ρ from V L to V L ⊗ B L , which extends ρ uniquely.Recall that ∆ : B L → B L ⊗ B L is a homomorphism of vertex algebras. Noticethat both ( ρ ⊗ ρ and (1 ⊗ ∆) ρ are vertex algebra homomorphisms from V L to V L ⊗ B L ⊗ B L . It can be readily seen that ( ρ ⊗ ρ and (1 ⊗ ∆) ρ agree on thesubspace h + C ǫ [ L ] of V L . Since V L is generated by h + C ǫ [ L ], we conclude( ρ ⊗ ρ = (1 ⊗ ∆) ρ (on V L ). Similarly, we can prove(1 ⊗ ε ) ρ ( u ) = u for u ∈ V L . Therefore, ρ is a right B L -comodule vertex algebra structure on V L . (cid:3) Recall the ǫ -twisted group algebra C ǫ [ L ]. Set B L,ǫ = S ( b h − ) ⊗ C ǫ [ L ] , (5.22)viewed as an associative algebra. From [Li8], B L,ǫ admits a derivation ∂ such that ∂e α = α ( − ⊗ e α , ∂h ( − n ) = nh ( − n − α ∈ L, h ∈ h , n ≥
1. Then we have a nonlocal vertex algebra ( B L,ǫ , ∂ ), where Y ( e α , x ) = E − ( − α, x ) e α for α ∈ L. Furthermore, it was proved in [Li8] that B L,ǫ is a B L -module nonlocal vertex algebrawhere the B L -module structure Y B L B L,ǫ ( · , x ) on B L,ǫ is uniquely determined by Y B L B L,ǫ ( e α , x ) = E + ( − α, x ) x α (0) for α ∈ L. (5.23)It was proved therein that the comultiplication ∆ : B L → B L ⊗ B L gives rise toa vertex algebra homomorphism from V L into B L,ǫ ♯B L , with V L and B L,ǫ beingcanonically identified with B L .The following is an analogue of Theorem 5.4: Proposition 5.5.
There exists a nonlocal vertex algebra homomorphism ρ : B L,ǫ → B L,ǫ ⊗ B L , which is uniquely determined by ρ ( h ( − n )) = h ( − n ) ⊗ ⊗ h ( − n ) , ρ ( e α ) = e α ⊗ e α (5.24) for h ∈ h , n ∈ Z + , α ∈ L . Furthermore, B L,ǫ with ρ becomes a right B L -comodulenonlocal vertex algebra and ρ is compatible with Y B L B L,ε ( · , x ) .Proof. Let ∆ be the comultiplication map of S ( b h − ), which is an algebra homomor-phism. Define a linear map ρ : B L,ǫ → B L,ǫ ⊗ B L by ρ ( u ⊗ e α ) = ∆( u )( e α ⊗ e α ) for u ∈ S ( b h − ) , α ∈ L. s B L is commutative, it can be readily seen that the linear map from C ε [ L ] to C ε [ L ] ⊗ C [ L ], sending e α to e α ⊗ e α for α ∈ L , is an algebra homomorphism. Con-sequently, ρ is an algebra homomorphism. It is straightforward to see that ρ ( ∂h ( − n )) = ( ∂ ⊗ ⊗ ∂ ) ρ ( h ( − n )) , ρ ( ∂e α ) = ( ∂ ⊗ ⊗ ∂ ) ρ ( e α )for h ∈ h , n ∈ Z + , α ∈ L . Since B L,ε as an algebra is generated by b h − + C ε [ L ], itfollows that ρ is a homomorphism of differential algebras. Thus ρ is a nonlocal vertexalgebra homomorphism. The same argument at the end of the proof of Theorem5.4 shows that B L,ǫ with ρ is a right B L -comodule. Therefore, B L,ǫ with ρ becomesa right B L -comodule nonlocal vertex algebra.Let α, β ∈ L . As Y B L B L,ε ( e α , x ) e β = E + ( − α, x ) x α (0) e β = x h α,β i e β , we have ρ ( Y B L B L,ε ( e α , x ) e β ) = ρ (cid:0) x h α,β i e β (cid:1) = x h α,β i ( e β ⊗ e β ) = ( Y B L B L,ε ( e α , x ) ⊗ ρ ( e β ) . On the other hand, we have Y B L B L,ε ( e α , x ) e β ∈ C ǫ [ L ] ⊗ C (( x )) . Since B L as a vertex algebra is generated by C [ L ] and B L,ε as a nonlocal vertexalgebra is generated by C ǫ [ L ], from Lemma 2.28 Y B L B L,ε ( · , x ) is compatible with ρ . (cid:3) Remark 5.6.
With Proposition 5.5, by Theorem 2.25 we get a new nonlocal vertexalgebra structure on B L,ǫ . It was essentially proved in [Li8] that vertex algebra V L coincides with (is canonically isomorphic to) this deformation of B L,ǫ .Recall that for a ∈ h , f ( x ) ∈ C (( x )),Φ( a, f )( z ) = X n ≥ ( − n n ! f ( n ) ( z ) a n on V L . We have Φ( a, f )( z ) e β = h β, a i f ( z ) e β for β ∈ L. (5.25)Now we define Φ( G ( x ))( z ) for G ( x ) ∈ h ⊗ C (( x )) by linearity. On the other hand,we extend the bilinear form h· , ·i on h to a C (( x ))-valued bilinear form on h ⊗ C (( x )).Then we have Φ( G ( x ))( z ) e β = h β, G ( z ) i e β for G ( x ) ∈ h ⊗ C (( x )) , β ∈ L .Note that for any g ( x ) ∈ x C [[ x ]], e g ( x ) exists in C [[ x ]]. Recall from Theorem 5.4the right B L -comodule vertex algebra structure ρ : V L → V L ⊗ B L on V L . Proposition 5.7.
Let f : h → h ⊗ x C [[ x ]]; α f ( α, x ) be any linear map. Thenthere exists a B L -module structure Y fM ( · , x ) on V L , which is uniquely determined by Y fM ( e α , z ) = exp(Φ( f ( α, x ))( z )) for α ∈ L, (5.26) and V L with this B L -module structure becomes a B L -module vertex algebra. Further-more, Y fM ( · , x ) is compatible with ρ and Y fM is an invertible element of L ρB L ( V L ) . roof. Since V L as a V L -module is irreducible, by [Li5, Corollary 3.10] V L is non-degenerate. By Proposition 2.19, B ( V L ) is a vertex bialgebra and V L is naturally a B ( V L )-module vertex algebra.For α ∈ h , as f ( α, x ) ∈ h ⊗ x C [[ x ]], exp(Φ( f ( α, x ))( z )) exists and from [Li8,Proposition 2.10], we haveexp(Φ( f ( α, x ))( z )) ∈ PEnd( V L ) ⊂ B ( V L ) , a group-like element, i.e., ∆(exp(Φ( f ( α, x )))) = exp(Φ( f ( α, x ))) ⊗ exp(Φ( f ( α, x ))).Since Φ is linear and f is linear in α ∈ h , we haveexp(Φ( f ( α, x ))) exp (Φ ( f ( β, x ))) = exp(Φ( f ( α + β, x ))) for α, β ∈ L. Then the linear map ψ : C [ L ] → B ( V L ) defined by ψ ( e α ) = exp(Φ( f ( α, x ))) for α ∈ L is an algebra homomorphism. Note that though B ( V L ) is not necessarilycommutative, the subalgebra generated byΦ( f ( r ) ( α, x )) , exp(Φ( f ( α, x ))) for r ≥ , α ∈ h is a commutative differential subalgebra of B ( V L ). Then by [Li8, Lemma 5.4], ψ can be extended uniquely to a homomorphism ¯ ψ of differential algebras from B L to B ( V L ). Furthermore, ¯ ψ is a homomorphism of vertex algebras. We have( ¯ ψ ⊗ ¯ ψ )∆ B L ( e α ) = exp(Φ( f ( α, x ))) ⊗ exp(Φ( f ( α, x ))) = ∆ B ¯ ψ ( e α ) ,ε B ( ¯ ψ ( e α )) = ε B (exp(Φ( f ( α, x )))) = 1 = ε B L ( e α )for α ∈ L , where (∆ B L , ε B L ) and (∆ B , ε B ) denote the coalgebra structures of B L and B ( V L ), respectively. Since C [ L ] generates B L as a differential algebra, it follows that¯ ψ is a homomorphism of differential bialgebras. Consequently, ¯ ψ is a homomorphismof vertex bialgebras. Then, by [Li8, Proposition 4.8], ¯ ψ gives rise to a B L -modulestructure Y fM ( · , x ) on V L , which makes V L a B L -module vertex algebra.Next, we prove that Y fM ( · , x ) is compatible with ρ , i.e., ρ ( Y fM ( b, x ) v ) = (cid:16) Y fM ( b, x ) ⊗ (cid:17) ρ ( v ) for b ∈ B L , v ∈ V L . (5.27)Let α, β ∈ L . By (5.25) we have Φ( f ( α, x )) e β = h β, f ( α, x ) i e β , so that Y fM ( e α , x ) e β = exp(Φ( f ( α, x ))) e β = e h β,f ( α,x ) i e β . Then ρ ( Y fM ( e α , x ) e β ) = ρ (cid:0) e h β,f ( α,x ) i e β (cid:1) = e h β,f ( α,x ) i ( e β ⊗ e β ) = ( Y fM ( e α , x ) ⊗ ρ ( e β ) . This shows that (5.27) holds for b = e α , v = e β ∈ L . On the other hand, we have Y fM ( e α , x ) e β ∈ C ǫ [ L ] ⊗ C (( x )) . Since B L as a vertex algebra is generated by C [ L ], and V L as a vertex algebra isgenerated by C ǫ [ L ], it follows from Lemma 2.28 that (5.27) holds for all u ∈ B L , v ∈ V L . Thus Y fM ( · , x ) is compatible with ρ . Namely, Y fM ∈ L ρB L ( V L ).Note that with − f in place of f we get another B L -module vertex algebra structure Y − fM on V L such that Y − fM ∈ L ρB L ( V L ), where Y − fM ( e α , x ) = exp(Φ( − f ( α, x )) = exp( − Φ( f ( α, x ))(5.28) or α ∈ L . It follows that ( Y fM ∗ Y − fM )( e α ) = 1 = ε ( e α ). Again, since B L as a vertexalgebra is generated by C [ L ], we have Y fM ∗ Y − fM = Y εM . That is, Y − fM is an inverseof Y fM in L ρB L ( V L ). (cid:3) Remark 5.8.
Let φ ( x, z ) be an associate given as before. Note that if the linearmap f : h → h ⊗ x C [[ x ]] in Proposition 5.7 is assumed to be a linear map f : h → h ⊗ ( x C [[ x ]] ∩ C φ (( x ))), then using Lemma 4.19 (a technical result) we can show thatthe B L -module structure Y fM ( · , x ) on V L is also φ -compatible.Combining Theorems 2.25 and 3.6, we have: Theorem 5.9.
Let f : h → h ⊗ x C [[ x ]] be any linear map. Then there existsa quantum vertex algebra structure on V L , whose vertex operator map, denoted by Y f ( · , x ) , is uniquely determined by Y f ( e α , x ) = Y ( e α , x ) exp(Φ( f ( α, x ))) for α ∈ L, (5.29) Y f ( h, x ) = Y ( h, x ) + Φ( f ′ ( h, x )) for h ∈ h . (5.30) Denote this quantum vertex algebra by V fL . Then V fL is an irreducible V fL -moduleand in particular, V fL is non-degenerate. Furthermore, the following relations hold: [ Y f ( a, x ) , Y f ( b, z )]= h a, b i ∂∂z x − δ (cid:16) zx (cid:17) − h f ′′ ( a, x − z ) , b i + h f ′′ ( b, z − x ) , a i , (5.31) [ Y f ( a, x ) , Y f ( e β , z )] = h a, β i Y f ( e β , z ) x − δ (cid:16) zx (cid:17) − Y f ( e β , z )( h f ′ ( β, z − x ) , a i + h f ′ ( a, x − z ) , β i ) , (5.32) Res x x − Y f ( α, x ) e α = D e α + h f ′ ( α, , α i e α , (5.33) ( x − z ) −h α,β i− Y f ( e α , x ) Y f ( e β , z ) − ( − z + x ) −h α,β i− e h β,f ( α,x − z ) i−h α,f ( β,z − x ) i Y f ( e β , z ) Y f ( e α , x )= ε ( α, β ) Y f ( e α + β , z ) x − δ (cid:16) zx (cid:17) (5.34) for a, b ∈ h , α, β ∈ L .Proof. For the first assertion, the uniqueness is clear, so we only need to show theexistence. With Propositions 5.4 and 5.7, by Theorems 2.25 and 3.6 we have aquantum vertex algebra D ρY fM ( V L ) with V L as its underlying space. From Theorem2.25, for h ∈ h , α ∈ L , we have Y f ( h, x ) = Y ♯ ( ρ ( h ) , x ) = Y ♯ ( h ⊗ + ⊗ h, x ) = Y ( h, x ) + Y fM ( h, x )= Y ( h, x ) + Φ( f ′ ( h, x )) ,Y f ( e α , x ) = Y ♯ ( ρ ( e α ) , x ) = Y ♯ ( e α ⊗ e α , x ) = Y ( e α , x ) Y fM ( e α , x )= Y ( e α , x ) exp(Φ( f ( α, x ))) . Thus we have a quantum vertex algebra structure on V L as desired. or the second assertion, since V L is an irreducible V L -module, it suffices to provethat any V fL -submodule U of V fL is also a V L -submodule of V L . For h ∈ h , since f ′ ( h, x ) ∈ h ⊗ x C [[ x ]], we have Y f ( h, x ) − = Y ( h, x ) − = X n ≥ h ( n ) x − n − , (5.35) Y f ( h, x ) + = Y ( h, x ) + + Φ( f ′ ( h, x )) . (5.36)From (5.35) we conclude h ( n ) U ⊂ U for h ∈ h , n ≥
0. Furthermore, we getΦ( f ′ ( h, x )) U ⊂ U [[ x ]] for h ∈ h . Then from (5.36) we deduce Y ( h, x ) + U ⊂ U [[ x ]].This proves that U is an b h -submodule of V L . Furthermore, for α ∈ L , as Y ( e α , x ) = Y f ( e α , x ) exp( − Φ( f ( α, x ))) we obtain Y ( e α , x ) U ⊂ U [[ x, x − ]]. Then it follows that U is also a V L -submodule of V L . Therefore, V fL is an irreducible V fL -module, whichimplies that V fL is a simple quantum vertex algebra and it is non-degenerate by [Li5].For the furthermore assertion, note that for u, v ∈ h , g ( x ) ∈ C (( x )), we haveΦ( u ⊗ g ( x )) v = X n ≥ ( − n n ! g ( n ) ( x ) u n v = −h u ⊗ g ′ ( x ) , v i . (5.37)Combining this with (5.25) and Theorem 3.6, we get that S ( x )( b ⊗ a ) = b ⊗ a + ⊗ ⊗ ( h f ′′ ( b, x ) , a i − h f ′′ ( a, − x ) , b i ) , (5.38) S ( x )( b ⊗ e α ) = b ⊗ e α − ⊗ e α ⊗ ( h f ′ ( α, − x ) , b i + h f ′ ( b, x ) , α i ) , (5.39) S ( x )( e α ⊗ b ) = e α ⊗ b + e α ⊗ ⊗ ( h f ′ ( α, x ) , b i + h f ′ ( b, − x ) , α i ) , (5.40) S ( x )( e β ⊗ e α ) = ( e β ⊗ e α ) ⊗ e h β,f ( α, − x ) i−h α,f ( β,x ) i , (5.41)where a, b ∈ h and α, β ∈ L and S ( x ) is the quantum Yang-Baxter operator of V fL .Since f ( h , x ) ⊂ h ⊗ x C [[ x ]], we have thatΦ( f ( a, x )) e β ∈ C e β ⊗ x C [[ x ]] and Φ( f ( a, x )) b ∈ C ⊗ x C [[ x ]] . It follows thatSing x Φ( f ′ ( a, x )) b = 0 = Sing x Φ( f ′ ( a, x )) β = Sing x exp (Φ( f ( a, x ))) b (5.42)and exp (Φ( f ( a, x ))) e β ∈ e β + C e β ⊗ x C [[ x ]] . Then we get thatSing x Y f ( a, x ) b = Sing x Y ( a, x ) b, Sing x Y f ( a, x ) e β = Sing x Y ( a, x ) e β , Res x x − Y f ( α, x ) e α = ∂e α + h f ′ ( α, , α i e α , Sing x x h α,β i Y f ( e α , x ) e β = ε ( α, β ) e α + β . Combining these with Lemma 2.6, Definition 2.7 and relations (5.38), (5.39), (5.41),we complete the proof. (cid:3)
Combining Proposition 5.2 and Theorem 5.9, we immediately get that orollary 5.10. Let f : h → h ⊗ x C [[ x ]] be a linear map such that h f ( a, x ) , b i = h f ( b, − x ) , a i , for a, b ∈ h . Then the identity map on h ⊕ C ε [ L ] determines a nonlocal vertex algebra isomorphismfrom V L to V fL . Recall that h = C ⊗ Z L ⊂ V L . Let G be an automorphism group of V L with alinear character χ such that G preserves h . It follows that G preserves the bilinearform on h and the standard conformal vector ω of V L . Thus gL (0) = L (0) g on V L for g ∈ G . Then V L is a ( G, χ )-module vertex algebra with R ( g ) = χ ( g ) L (0) g for g ∈ G . Furthermore, we have: Proposition 5.11.
Let G be an automorphism group of V L such that G ( h ) = h andlet χ be a linear character of G . Assume η ( · , x ) : h → h ⊗ x C [[ x ]] is a linear mapsatisfying the condition ( µ ⊗ η ( h, x ) = η ( µ ( h ) , χ ( µ ) x ) for µ ∈ G, h ∈ h . (5.43) Then V ηL is a ( G, χ ) -module quantum vertex algebra with R defined by R ( g ) = χ ( g ) L (0) g for g ∈ G .Proof. Recall that for v ∈ V L , f ( t ) ∈ C (( t )),Φ( v ⊗ f ( t ))( z ) = Res x Y ( v, x ) f ( z − x ) . For µ ∈ G , set R µ = χ ( µ ) L (0) µ . For h ∈ h , we have R µ Φ( η ( h, t ))( z ) = Res x R µ Y ( η ( h, z − x ) , x )= Res x χ ( µ ) Y (( µ ⊗ η ( h, z − x ) , χ ( µ ) x ) R µ µ = Res x χ ( µ ) Y ( η ( µ ( h ) , χ ( µ )( z − x )) , χ ( µ ) x ) R µ = Res z Y ( η ( µ ( h ) , χ ( µ ) z − z ) , z ) R µ = Φ( η ( µ ( h ) , t ))( χ ( µ ) z ) R µ . (5.44)Then for α ∈ L we get R µ Y η ( e α , z ) R − µ = R µ Y ( e α , z ) exp(Φ( η ( α, t ))( z )) R − µ = Y ( R µ e α , χ ( µ ) z ) exp (cid:0) R µ Φ( η ( α, t ))( z ) R − µ (cid:1) = χ ( µ ) h α,α i Y ( µe α , χ ( µ ) z ) exp (Φ( η ( µ ( α ) , t )( χ ( µ ) z ))= λ µ,α χ ( µ ) h µ ( α ) ,µ ( α ) i Y ( e µ ( α ) , χ ( µ ) z ) exp (Φ( η ( µ ( α ) , t ))( χ ( µ ) z ))= λ µ,α χ ( µ ) h µ ( α ) ,µ ( α ) i Y η ( e µ ( α ) , χ ( µ ) z )= Y η ( R µ e α , χ ( µ ) z ) , where µ ( e α ) = λ µ,α e µ ( α ) with λ µ,α ∈ C . Note that (5.43) implies( µ ⊗ η ′ ( h, x ) = χ ( µ ) η ′ ( µ ( h ) , χ ( µ ) x ) . Following the argument in (5.44) we obtain R µ Φ( η ′ ( h, t ))( z ) R − µ = χ ( µ )Φ( η ′ ( µ ( h ) , t ))( χ ( µ ) z ) = Φ( η ′ ( R µ ( h ) , t ))( χ ( µ ) z ) . sing this we get R µ Y η ( h, x ) R − µ = R µ ( Y ( h, x ) + Φ( η ′ ( h, t ))( x )) R − µ = Y η ( R µ h, χ ( µ ) x ) . Note that { e α | α ∈ L } ∪ h is a generating subset of V ηL as a nonlocal vertex algebra.By using weak associativity (or by Lemma 2.11) it is straightforward to show that R µ Y η ( v, x ) R − µ = Y η ( R µ v, χ ( µ ) x ) for all µ ∈ G, v ∈ V L . Therefore, V ηL is a ( G, χ )-module quantum vertex algebra as desired. (cid:3)
Remark 5.12.
We here show the existence of a linear map η ( · , x ) satisfying thecondition (5.43) in Proposition 5.11 with G a finite group. Let g ( · , x ) : h → h ⊗ x C [[ x ]] be any linear map. Define a linear map e g ( · , x ) : h → h ⊗ x C [[ x ]] by e g ( α, x ) = X σ ∈ G ( σ ⊗ g ( σ − α, χ ( σ − ) x ) for α ∈ h . (5.45)Then it is straightforward to show( σ ⊗ e g ( α, x ) = e g ( σα, χ ( σ ) x ) for σ ∈ G, α ∈ h . (5.46) References [BLP] C.-M. Bai, H.-S. Li and Y.-F. Pei, φ ǫ -Coordinated modules for vertex algebras, J. Algebra (2015), 211-242.[BK] B. Bakalov and V. Kac, Field algebras,
Internat. Math. Res. Notices (2003), 123-159.[B1] R. E. Borcherds, Vertex algebars, Kac-Moody algebras, and the Monster, Proc. Natl.Acad. Sci. USA (1986), 3068-3071.[B2] R. E. Borcherds, Vertex algebars, in “Topological Field Theory, Primitive Forms andRelated Topics” (Kyoto, 1996), edited by M. Kashiwara, A. Matsuo, K. Sato and I.Satake, Progress in Math., Vol. 160, Birkh¨auser, Boston, 1998, 35-77.[D1] C. Dong, Vertex algebras associated with even lattices, J. Algebra (1993), 245-265.[D2] C. Dong, Twisted modules for vertex algebras associated with even lattices,
J. Algebra (1994), 91-112.[DLMi] B. Doyon, J. Lepowsky and A. Milas, Twisted vertex operators and Bernoulli polyno-mials,
Commun. Contemp. Math. (2006), 247-307.[EK] P. Etingof and D. Kazhdan, Quantization of Lie bialgebras, V, Selecta Math. (N.S.) (2000), 105-130.[FHL] I. B. Frenkel, Y.-Z. Huang, and J. Lepowsky, On axiomatic approaches to vertex operatoralgebras and modules , Memoirs Amer. Math. Soc. 104, 1993.[FLM] I. Frenkel, J. Lepowsky and A. Meurman,
Vertex Operator Algebras and the Monster ,Pure and Appl. Math.,
Vol. 134 , Academic Press, Boston, 1988.[H] M. Hazewinkel,
Formal Groups and Applications,
Pure and Appl. Math., Vol. 78, Lon-don: Academic Press, 1978.[JKLT] N. Jing, F. Kong, H.-S. Li, and S. Tan, (
G, χ φ )-equivariant φ -coordinated quasi modulesfor nonlocal vertex algebras, arXiv:2008.05982.[Ka] C. Kassel, Quantum Groups , GTM , Berlin-Heidelberg-New York: Springer-Verlag,1995.[L] J. Lepowsky, Calculus of twisted vertex operators,
Proc. Natl. Acad. Sci. USA (1985),8295-8299.[LL] J. Lepowsky and H.-S. Li, Introduction to Vertex Operator Algebras and Their Repre-sentations , Progress in Math. , Birkh¨auser, Boston, 2004.[Li1] H.-S. Li, Local systems of vertex operators, vertex superalgebras and modules,
J. PureAppl. Algebra (1996) 143-195. Li2] H.-S. Li, Axiomatic G -vertex algebras, Commun. Contemp. Math. (2003), 1-47.[Li3] H.-S. Li, A new construction of vertex algebras and quasi modules for vertex algebras, Adv. Math. (2006), 232-286.[Li4] H.-S. Li, Nonlocal vertex algebras generated by formal vertex operators,
Selecta Math.(N.S.) (2005), 349-397.[Li5] H.-S. Li, Constructing quantum vertex algebras, Internat. J. Math. (2006), 441-476.[Li6] H.-S. Li, Pseudoderivations, pseudoautomorphisms and simple current modules for ver-tex operator algebras, in Proceedings of the International Conference on “Infinite Di-mensional Aspects of Representation Theory and Applications,” University of Virginia,Charlottesville, May 18-22, 2004, Contemporary Math (2005), 55-65.[Li7] H.-S. Li, Twisted modules and quasi-modules for vertex operator algebras, ContemporaryMath. , Amer. Math. Soc., Providence, 2007, 389-400.[Li8] H.-S. Li, A smash product construction of nonlocal vertex algebras, Commun. Contemp.Math. (5) (2007), 605–637.[Li9] H.-S. Li, φ -coordinated quasi-modules for quantum vertex algebras, Commun. Math.Phys. (2011), 703-741.[Li10] H.-S. Li, G -equivariant φ -coordinated quasi-modules for quantum vertex algebras, J.Math. Phys. (2013), 1-26.[LTW] H.-S. Li, S. Tan and Qing Wang, Twisted modules for quantum vertex algebras, J. PureApplied Algebra (2010), 201-220.
Department of Mathematics, North Carolina State University, Raleigh, NC27695, USA
Email address : [email protected] Key Laboratory of Computing and Stochastic Mathematics (Ministry of Educa-tion), School of Mathematics and Statistics, Hunan Normal University, Changsha,China 410081
Email address : [email protected] Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102,USA
Email address : [email protected] School of Mathematical Sciences, Xiamen University, Xiamen, China 361005
Email address : [email protected]@xmu.edu.cn