On Indecomposable Vertex Algebras associated with Vertex Algebroids
aa r X i v : . [ m a t h . QA ] A ug ON INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITHVERTEX ALGEBROIDS
PHICHET JITJANKARN, AND GAYWALEE YAMSKULNA
Abstract.
Let A be a finite dimensional unital commutative associative algebra and let B be a finite dimensional vertex A -algebroid such that its Levi factor is isomorphic to sl . Undersuitable conditions, we construct an indecomposable non-simple N -graded vertex algebra V B from the N -graded vertex algebra V B associated with the vertex A -algebroid B . We showthat this indecomposable non-simple N -graded vertex algebra V B is C -cofinite and has onlytwo irreducible modules. Introduction
It is well known that the C -cofiniteness property of vertex (operator) algebras plays an impor-tant role in the study of representation theory of vertex (operator) algebras (e.g. [A2], [ABD],[Bu], [DoLiM1], [GN], [Mi1]-[Mi3], [Z]). Over the years, rational C -cofinite vertex algebrashave been studied intensively (e.g. [Bo1]-[Bo2], [Do], [DoL], [DoLiM1], [DoM1], [FrLMe1],[FZ]) . However, the literature devoted to the study of irrational C -cofinite vertex algebras issparse. In fact, there are very few known examples of families of irrational C -cofinite vertex(super)algebras (e.g. [A1], [AdM1]-[AdM4], [CF], [FFHST], [FGST1]-[FGST3]). Therefore, itis crucial to study these known examples and to seek for new models.The aim of this paper is to construct indecomposable non-simple vertex algebras that satisfythe C -condition from N -graded vertex algebras associated with vertex algebroids. For a N -graded vertex algebra V = ⊕ ∞ n =0 V ( n ) such that dim V (0) ≥
2, it is known widely that V (0) is a unital commutative associative algebra and V (1) is a vertex V (0) -algebroid. In [GMS],among other important things, Gorbounov, Malikov and Schechtman constructed a N -gradedvertex algebra V = ⊕ ∞ n =0 V ( n ) from any vertex A -algebroid, such that V (0) = A and thevertex A -algebroid V (1) is isomorphic to the given vertex A -algebroid. The classification ofgraded simple non-twisted and twisted modules for the vertex algebras associated with vertexalgebroids had been studied in [LiY1]-[LiY2] by Li and the second author of this paper.In terms of general theory of N -graded vertex algebras, Dong and Mason showed that a N -graded vertex operator algebra V is local if and only if V (0) is a local algebra. Moreover,indecomposibility of V is equivalent to V (0) being a local algebra [DoM2]. Note that in or-der to prove this statement one needs to have a Virasoro element. In [JY], we exploredcriteria for N -graded vertex algebras V = ⊕ ∞ n =0 V ( n ) such that dim V (0) ≥ Key words and phrases. C -cofinite, indecomposable, irrational vertex algebras. the algebraic structure of this type of vertex algebras. Precisely, we provided tools to charac-terize indecomposable non-simple N -graded vertex algebras. Also, we examined the algebraicstructure of N -graded vertex algebras V = ⊕ ∞ n =0 V ( n ) that is generated by V (0) and V (1) suchthat dim V (0) ≥ V (1) is a (semi)simple Leibniz algebra that has sl as its Levi fac-tor. We showed that under suitable conditions this type of vertex algebra is indecomposablenon-simple.In this paper, we continue our investigation on N -graded indecomposable non-simple vertexalgebras. First, we apply results in [JY] to show that for a given finite dimensional vertex A -algebroid B such that B is a (semi)simple Leibniz algebra that has sl as its Levi factor,the N -graded vertex algebra associated with the vertex A -algebroid B is indecomposable andnon-simple. Moreover, we establish the following results. Theorem 1.
Let A be a finite-dimensional commutative associative algebra with the identity e such that dim A ≥ . Let B be a finite-dimensional vertex A -algebroid such that A is not atrivial B -module and Leib ( B ) = { } . Let S be the Levi factor of the Leibniz algebra B suchthat S = Span { e, f, h } , e f = h , h e = 2 e , h f = − f , and e f = k e . Here, k ∈ C \{ } .Assume that one of the following statements hold.(I) B is simple Leibniz algebra;(II) B is a semisimple Leibniz algebra and Ker ( ∂ ) = { a ∈ A | b a = 0 for all b ∈ B } .We then have the following results:(i) the N -graded vertex algebra V B (= ⊕ ∞ n =0 ( V B ) ( n ) ) associated with the vertex A -algebroid B is indecomposable non-simple.(ii) The set of representatives of equivalence classes of finite-dimensional simple sl -modulesis equivalent to the set of representatives of equivalence classes of N -graded simple V B -modules N = ⊕ ∞ n =0 N ( n ) such that dim N (0) < ∞ . Next, we show that a certain quotient space of V B , constructed in Theorem 1, is an indecom-posable non-simple vertex algebra that satisfies the C -condition and has only two irreduciblemodules. Precisely, we establish the following results. Theorem 2.
Let A be a finite-dimensional commutative associative algebra with the identity e such that dim A ≥ . Let B be a finite-dimensional vertex A -algebroid that satisfies thegiven conditions in Theorem 1. Let ( e ( − e ) be an ideal of V B that is generated by e ( − e .Then(i) ( e ( − e ) ∩ A = { } , ( e ( − e ) ∩ B = { } . Moreover, the N -graded vertex algebra V B / ( e ( − e ) is indecomposable non-simple.(ii) The vertex algebra V B / ( e ( − e ) satisfies the C -condition.(iii) Let L = Z α be a rank one positive definite even lattice equipped with a Q -valued Z -bilinear form ( · , · ) such that ( α, α ) = 2 . Then V L and V L + α are the only two irreducible V B / ( e ( − e ) -modules. N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 3
This paper is organized as follows: in Section 2, we first review properties of Leibniz algebras,1-truncated conformal algebras, and vertex algebroids. We discuss about vertex algebroidsassociated with (semi) simple Leibniz algebras that have sl as their Levi factor. Also, we givenecessary background on vertex algebras and recall construction of vertex algebras associatedwith vertex algebroids, and their graded simple modules. In Section 3, we give the proof ofTheorem 1, and Theorem 2. We include some appendices in Section 4 containing backgroundon a vertex operator algebra associated with a certain type of rank one positive definite evenlattices, and vertex operator algebras associated with highest weight representations of affineLie algebras. 2. Preliminaries
Leibniz Algebras.Definition 3. ([DMS], [FM])(i) A left Leibniz algebra L is a C -vector space equipped with a bilinear map [ , ] : L × L → L satisfying the Leibniz identity [ a, [ b, c ]] = [[ a, b ] , c ] + [ b, [ a, c ]] for all a, b, c ∈ L .(ii) Let L be a left Leibniz algebra over C . Let I be a subspace of L . I is a left (respectively, right ) ideal of L if [ L , I ] ⊆ I (respectively, [ I, L ] ⊆ I ). I is an ideal of L if it is both a left anda right ideal. Example 4.
We define
Leib ( L ) = Span { [ u, u ] | u ∈ L } = Span { [ u, v ] + [ v, u ] | u, v ∈ L } . Leib ( L ) is an ideal of L . Moreover, for v, w ∈ Leib ( L ), [ v, w ] = 0. Definition 5. [DMS] Let ( L , [ , ]) be a left Leibniz algebra. The series of ideals ... ⊆ L (2) ⊆ L (1) ⊆ L where L (1) = [ L , L ], L ( i +1) = [ L ( i ) , L ( i ) ] is called the derived series of L . A left Leibniz algebra L is solvable if L ( m ) = 0 for some integer m ≥
0. As in the case of Lie algebras, any leftLeibniz algebra L contains a unique maximal solvable ideal rad ( L ) called the the radical of L which contains all solvable ideals. Example 6.
Leib ( L ) is a solvable ideal. Definition 7. [DMS](i) A left Leibniz algebra L is simple if [ L , L ] = Leib ( L ), and { } , Leib ( L ), L are the onlyideals of L .(ii) A left Leibniz algebra L is said to be semisimple if rad ( L ) = Leib ( L ). Proposition 8. ( [Ba] , [DMS] ) Let L be a left Leibniz algebra.(i) There exists a subalgebra S which is a semisimple Lie algebra of L such that L = S ˙+ rad ( L ) .As in the case of a Lie algebra, we call S a Levi subalgebra or a Levi factor of L .(ii) If L is a semisimple Leibniz algebra then L = ( S ⊕ S ⊕ ... ⊕ S k ) ˙+ Leib ( L ) , where S j isa simple Lie algebra for all ≤ j ≤ k . Moreover, [ L , L ] = L . PHICHET JITJANKARN, AND GAYWALEE YAMSKULNA (iii) If L is a simple Leibniz algebra, then there exists a simple Lie algebra S such that Leib ( L ) is an irreducible module over S and L = S ˙+ Leib ( L ) . Definition 9.
Let L be a left Leibniz algebra. A left L -module is a vector space M equippedwith a C -bilinear map L × M → M ; ( u, m ) u · m such that ([ u, v ]) · m = u · ( v · m ) − v · ( u · m )for all u, v ∈ L , m ∈ M .The usual definitions of the notions of submodule, irreducibility, complete reducibility, homo-morphism, isomorphism, etc., hold for left Leibniz modules. Remark . Leib ( L ) acts as zero on M .2.2. [GMS] A is a graded vector space C = C ⊕ C equipped with a linear map ∂ : C → C and bilinear operations ( u, v ) u i v for i = 0 , − i − C = C ⊕ C such that the following axioms hold:(Derivation) for a ∈ C , u ∈ C ,(1) ( ∂a ) = 0 , ( ∂a ) = − a , ∂ ( u a ) = u ∂a ;(Commutativity) for a ∈ C , u, v ∈ C ,(2) u a = − a u, u v = − v u + ∂ ( u v ) , u v = v u ;(Associativity) for α, β, γ ∈ C ⊕ C ,(3) α β i γ = β i α γ + ( α β ) i γ. Definition 12. ([Br1], [Br2], [GMS]) Let ( A, ∗ ) be a unital commutative associative algebraover C with the identity 1. A vertex A -algebroid is a C -vector space Γ equipped with(1) a C -bilinear map A × Γ → Γ , ( a, v ) a · v such that 1 · v = v (i.e. a nonassociativeunital A -module),(2) a structure of a Leibniz C -algebra [ , ] : Γ × Γ → Γ,(3) a homomorphism of Leibniz C -algebra π : Γ → Der ( A ),(4) a symmetric C -bilinear pairing h , i : Γ ⊗ C Γ → A ,(5) a C -linear map ∂ : A → Γ such that π ◦ ∂ = 0 which satisfying the following conditions: a · ( a ′ · v ) − ( a ∗ a ′ ) · v = π ( v )( a ) · ∂ ( a ′ ) + π ( v )( a ′ ) · ∂ ( a ) , [ u, a · v ] = π ( u )( a ) · v + a · [ u, v ] , [ u, v ] + [ v, u ] = ∂ ( h u, v i ) ,π ( a · v ) = aπ ( v ) , h a · u, v i = a ∗ h u, v i − π ( u )( π ( v )( a )) ,π ( v )( h v , v i ) = h [ v, v ] , v i + h v , [ v, v ] i ,∂ ( a ∗ a ′ ) = a · ∂ ( a ′ ) + a ′ · ∂ ( a ) , [ v, ∂ ( a )] = ∂ ( π ( v )( a )) , h v, ∂ ( a ) i = π ( v )( a )for a, a ′ ∈ A , u, v, v , v ∈ Γ. N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 5
Proposition 13. [LiY1]
Let ( A, ∗ ) be a unital commutative associative algebra and let B be amodule for A as a nonassociative algebra . Then a vertex A -algebroid structure on B exactlyamounts to a 1-truncated conformal algebra structure on C = A ⊕ B with a i a ′ = 0 ,u v = [ u, v ] , u v = h u, v i ,u a = π ( u )( a ) , a u = − u a for a, a ′ ∈ A , u, v ∈ B , i = 0 , such that a · ( a ′ · u ) − ( a ∗ a ′ ) · u = ( u a ) · ∂a ′ + ( u a ′ ) · ∂a,u ( a · v ) − a · ( u v ) = ( u a ) · v,u ( a ∗ a ′ ) = a ∗ ( u a ′ ) + ( u a ) ∗ a ′ ,a ( a ′ · v ) = a ′ ∗ ( a v ) , ( a · u ) v = a ∗ ( u v ) − u v a,∂ ( a ∗ a ′ ) = a · ∂ ( a ′ ) + a ′ · ∂ ( a ) . For the rest of this section, we assume the following:(i) ( A, ∗ ) is a unital commutative associative algebra with the identity e and dim ( A ) < ∞ .(ii) B is a vertex A -algebroid such that dim ( B ) < ∞ , and A is not a trivial B -module.Recall that a set I is called an ideal of a vertex A -algebroid B if I is a left ideal of Leibnizalgebra B and a · u ∈ I for all a ∈ A, u ∈ I . Example 14.
We set A∂ ( A ) = Span { a · ∂ ( a ′ ) | a, a ′ ∈ A } . A∂ ( A ) is an ideal of a vertex A -algebroid B . In fact, A∂ ( A ) is an abelian Lie algebra. Proposition 15. [JY]
Let B be a simple Leibniz algebra such that Leib ( B ) = { } . Assumethat its Levi factor S = Span { e, f, h } such that e f = h , h e = 2 e , h f = − f , and e f = k e ∈ ( C e ) \{ } . Then(i) e e = f f = e h = f h = 0 , k = 1 , h h = 2 e .(ii) Ker ( ∂ ) = C e (iii) Leib ( B ) is an irreducible sl -module of dimension 2. Moreover, as a sl -module, A is adirect sum of a trivial module and an irreducible sl -module of dimension 2.(iv) A is a local algebra. Let A =0 be an irreducible sl -submodule of A that has dimension2. Let a be the highest weight vector of A =0 of weight 1 and let a = f a . Hence, the set { a , a } forms a basis of A =0 , the set { e , a , a } is a basis of A , and the set { ∂ ( a ) , ∂ ( a ) } isa basis of Leib ( B ) . PHICHET JITJANKARN, AND GAYWALEE YAMSKULNA
Relationships among a , a , e, f, h, ∂ ( a ) , ∂ ( a ) are desribed below: ( ∂ ( a )) e = 0 , ( ∂ ( a )) f = a , ( ∂ ( a )) h = a , (4) ( ∂ ( a )) e = a , ( ∂ ( a )) f = 0 , ( ∂ ( a )) h = − a , (5) a · e = 0 , a · f = ∂ ( a ) , a · h = ∂ ( a ) , a · ∂ ( a i ) = 0 for i ∈ { , } , (6) a · e = ∂ ( a ) , a · f = 0 , a · h = − ∂ ( a ) , a · ∂ ( a i ) = 0 for i ∈ { , } , (7) a i ∗ a j = 0 for all i, j ∈ { , } . (8) Proposition 16. [JY]
Let B be a semisimple Leibniz algebra such that Leib ( B ) = { } , and Ker ( ∂ ) = { a ∈ A | u a = 0 for all u ∈ B } .Assume that the Levi factor S = Span { e, f, h } such that e f = h, h e = 2 e, h f = − f and e f = k e ∈ C e \{ } . We set A = C e ⊕ lj =1 N j where each N j is an irreducible sl -submoduleof A . Then(i) e e = f f = e h = f h = 0 , k = 1 , h h = 2 e ;(ii) Ker ( ∂ ) = C e ;(iii) For j ∈ { , ..., l } dim N j = 2 , and dim Leib ( B ) = 2 l ;(iv) A is a local algebra. For each j , we let a j, be a highest weight vector of N j and a j, = f ( a j, ) . Then { e , a j,i | j ∈ { , ...., l } , i ∈ { , }} is a basis of A , and { ∂ ( a j,i ) | j ∈ { , ..., l } , i ∈{ , }} is a basis of Leib ( B ) .Relations among a j,i , e, f, h, ∂ ( a j,i ) are described below: a j,i ∗ a j ′ ,i ′ = 0 , (9) a j, · e = 0 , a j, · e = ∂ ( a j, ) , (10) a j, · f = ∂ ( a j, ) , a j, · f = 0 , (11) a j, · h = ∂ ( a j, ) , a j, · h = − ∂ ( a j, ) , (12) a j,i · ∂ ( a j ′ ,i ′ ) = 0 , (13) ∂ ( a j,i ) e = e a j,i = (2 − i ) a j,i − , (14) ∂ ( a j,i ) f = f a j,i = ( i + 1) a j,i +1 , (15) ∂ ( a j,i ) h = h a j,i = (1 − i ) a j,i . (16)2.3. Vertex Algebras.Definition 17. ([Bo1], [FrLMe1], [LLi]) A vertex algebra is a vector space V equipped witha linear map Y : V → End( V )[[ x, x − ]] v Y ( v, x ) = X n ∈ Z v n x − n − where v n ∈ End( V ) N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 7 and equipped with a distinguished vector , the vacuum vector, such that for u, v ∈ V , u n v = 0 for n sufficiently large ,Y ( , x ) = 1 ,Y ( v, x ) ∈ V [[ x ]] , and lim x → Y ( v, x ) = v and 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) Y ( v, x ) Y ( u, x )= x − δ (cid:18) x − x x (cid:19) Y ( Y ( u, x ) v, x ) the Jacobi identity . From the Jacobi identity we have Borcherds’ commutator formula and iterate formula:[ u m , v n ] = X i ≥ (cid:18) mi (cid:19) ( u i v ) m + n − i (17) ( u m v ) n w = X i ≥ ( − i (cid:18) mi (cid:19) ( u m − i v n + i w − ( − m v m + n − i u i w )(18)for u, v, w ∈ V , m, n ∈ Z .We define a linear operator D on V by D ( v ) = v − for v ∈ V . Then Y ( v, x ) = e xD v for v ∈ V, and[ D, v n ] = ( Dv ) n = − nv n − for v ∈ V, n ∈ Z . Moreover, for u, v ∈ V , we have Y ( u, x ) v = e xD Y ( v, − x ) u (skew-symmetry).A vertex algebra V equipped with a Z -grading V = ⊕ n ∈ Z V ( n ) is called a Z -graded vertexalgebra if ∈ V (0) and if u ∈ V ( k ) with k ∈ Z and for m, n ∈ Z , u m V ( n ) ⊆ V ( k + n − m − .A N -graded vertex algebra is defined in the obvious way. Proposition 18. [GMS] If V = ⊕ n ∈ N V ( n ) is an N -graded vertex algebra then(i) V (0) is a commutative associative algebra with identity and V (1) is Leibniz algebra.(ii) In fact, V (0) ⊕ V (1) is a 1-truncated conformal algebra.(iii) Moreover, V (1) is a vertex V (0) -algebroid. Proposition 19. [JY]
Let V = ⊕ ∞ n =0 V ( n ) be a N -graded vertex algebra that satisfies thefollowing properties:(a) ≤ dim V (0) < ∞ , ≤ dimV (1) < ∞ , V is generated by V (0) and V (1) ;(b) V (0) is not a trivial module of a Leibniz algebra V (1) , u u = 0 for some u ∈ V (1) ; PHICHET JITJANKARN, AND GAYWALEE YAMSKULNA (c) the Levi factor of V (1) equals Span { e, f, h } , e f = h , h e = 2 e , h f = − f and e f = k .Here, k ∈ C \{ } .Assume that one of the following statements hold.(i) V (1) is a simple Leibniz algebra;(ii) V (1) is a semisimple Leibniz algebra and Ker ( D ) ∩ V (0) = { a ∈ V (0) | b a = 0 for all b ∈ V (1) } .Then V is an indecomposable non-simple vertex algebra. Definition 20. ([FrLMe1], [LLi]) An ideal of the vertex algebra V is a subspace I such that u n w ∈ I and w n u ∈ I for u ∈ V , w ∈ I and n ∈ Z .Notice that D ( w ) = w − ∈ I . Hence, under the condition that D ( I ) ⊆ I , the left idealcondition v n w ∈ I for all v ∈ V , w ∈ I , n ∈ Z is equivalent to the right ideal condition w m v ∈ I for all v ∈ V , w ∈ I , m ∈ Z .For a subset S of a vertex algebra V , we denote by ( S ) the smallest ideal of V containing S .It was shown in Corollary 4.5.10 of [LLi] that( S ) = Span { v n D i ( u ) | v ∈ V, n ∈ Z , i ≥ , u ∈ S } . Definition 21.
For a vertex algebra V , we define C ( V ) = Span { u − v | u, v ∈ V } . Thevertex algebra V is said to satisfy the C -condition if V /C ( V ) is finite dimensional Proposition 22. [Z]
For u, v ∈ V , n ≥ , D ( v ) ∈ C ( V ) and u − n v ∈ C ( V ) . Definition 23. [LLi] A V - module is a vector space W equipped with a linear map Y W from V to (End W )[[ x, x − ]] where Y W ( v, x ) = P n ∈ Z v n x − n − for v ∈ V such that for u, v ∈ V , w ∈ W , u n w = 0 for n sufficiently large ,Y W ( , x ) = 1 ,x − δ (cid:18) x − x x (cid:19) Y W ( u, x ) Y W ( v, x ) − x − δ (cid:18) x − x − x (cid:19) Y W ( v, x ) Y W ( u, x )= x − δ (cid:18) x − x x (cid:19) Y W ( Y ( u, x ) v, x ) . Definition 24.
Let V = ⊕ ∞ n =0 V ( n ) be a N -graded vertex algebra. A N -graded V -moduleis a V -module M equipped with a N -grading M = ⊕ ∞ n =0 M ( n ) such that M (0) = { } and v m M ( n ) ⊂ M ( n + p − m − for v ∈ V ( p ) , p, n ∈ N , m ∈ Z . Proposition 25. [LLi]
Let V be a vertex algebra.(i) For v ∈ V , v is weakly nilpotent if and only if ( v − ) r = 0 for some r > .(ii) Also, if u ∈ V such that u n u = 0 for all n ≥ , then Y (( u − ) r , z ) = Y ( u, z ) r for r > .(iii) Let ( W, Y W ) be a V -module. Then for any weakly nilpotent element v of V , with r > chosen so that ( v − ) r = 0 , we have Y W (( v − ) r , z ) = 0 . Also for u ∈ U such that u n u = 0 for all n ≥ , we have Y W (( u − ) r , z ) = Y W ( u, z ) r for r > . N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 9
Proposition 26. [LiY1]
Let V be a vertex algebra and let I be a (two-sided) ideal generatedby a subset S . Let ( W, Y W ) be a V -module and let U be a generating subspace of W as a V -module such that Y W ( v, x ) u = 0 for v ∈ S , u ∈ U . Then Y W ( v, x ) = 0 for v ∈ I . Next, we recall a construction of vertex algebras associated with vertex algebroids in [LiY1].Let A be a commutative associative algebra with identity e and let B be a vertex A -algebroid.We set L ( A ⊕ B ) = ( A ⊕ B ) ⊗ C [ t, t − ] . Subspaces L ( A ) and L ( B ) of L ( A ⊕ B ) are definedin the obvious way.We set ˆ ∂ = ∂ ⊗ ⊗ ddt : L ( A ) → L ( A ⊗ B ) . We define deg ( a ⊗ t n ) = − n − deg ( b ⊗ t n ) = − n for a ∈ A, b ∈ B, n ∈ Z . Then L ( A ⊕ B ) becomes a Z -graded vector space: L ( A ⊕ B ) = ⊕ n ∈ Z L ( A ⊕ B ) ( n ) where L ( A ⊕ B ) ( n ) = A ⊗ C t − n − + B ⊗ C t − n . Clearly, the subspaces L ( A ) and L ( B ) are Z -graded vector spaces as well. In addition, for n ∈ N , L ( A ) ( n ) = A ⊗ C t − n − . The linearmap ˆ ∂ : L ( A ) → L ( A ⊕ B ) is of degree 1. We define a bilinear product [ · , · ] on L ( A ⊕ B ) asfollow: [ a ⊗ t m , a ′ ⊗ t n ] = 0 , [ a ⊗ t m , b ⊗ t n ] = a b ⊗ t m + n , [ b ⊗ t n , a ⊗ t m ] = b a ⊗ t m + n , [ b ⊗ t m , b ′ ⊗ t n ] = b b ′ ⊗ t m + n + m ( b b ) ⊗ t m + n − for a, a ′ ∈ A , b, b ′ ∈ B , m, n ∈ Z . For convenience, we set L := L ( A ⊕ B ) / ˆ ∂L ( A ) . It was shown in [LiY1] that L = ⊕ n ∈ Z L ( n ) is a Z -graded Lie algebra. Here, L ( n ) = L ( A ⊕ B ) ( n ) / ˆ ∂ ( L ( A ) ( n − ) = ( A ⊗ C t − n − + B ⊗ C t − n ) / ˆ ∂ ( A ⊗ C t − n ) . In particular, L (0) = A ⊗ C t − + B/∂A .Let ρ : L ( A ⊕ B ) → L be a natural linear map defined by ρ ( u ⊗ t n ) = u ⊗ t n + ˆ ∂L ( A ) . For u ∈ A ⊕ B , n ∈ Z , we set u ( n ) = ρ ( u ⊗ t n ) and u ( z ) = P n ∈ Z u ( n ) z − n − . Let W be a L -module. We use u W ( n ) or sometimes just u ( n ) for the corresponding operator on W andwe write u W ( z ) = P n ∈ Z u ( n ) z − n − ∈ (End W )[[ z, z − ]]. The commutator relations in termsof generating functions are the following:[ a ( z ) , a ′ ( z )] = 0(19) [ a ( z ) , b ( z )] = z − δ (cid:18) z z (cid:19) ( a b )( z ) , (20) [ b ( z ) , b ′ ( z )] = z − δ (cid:18) z z (cid:19) ( b b ′ )( z ) + ( b b ′ )( z ) ∂∂z z − δ (cid:18) z z (cid:19) (21) for a, a ′ ∈ A , b, b ′ ∈ B .Next, we define L ≥ = ρ (( A ⊕ B ) ⊗ C [ t ]) ⊂ L , and L < = ρ (( A ⊕ B ) ⊗ t − C [ t − ]) ⊂ L . As avector space, L = L ≥ ⊕ L < . The subspaces L ≥ and L < are graded sub-algebras.We now consider C as the trivial L ≥ -module and form the following induced module V L = U ( L ) ⊗ U ( L ≥ ) C . In view of the Poincare-Birkhoff-Witt theorem, we have V L = U ( L < ) as a vector space. Wemay consider A ⊕ B as a subspace: A ⊕ B → V L , a + b a ( − + b ( − . We assign deg C = 0. Then V L = ⊕ n ∈ N ( V L ) ( n ) is a restricted N -graded L -module. We set = 1 ∈ V L . Proposition 27. [FKRW, MeP]
There exists a unique vertex algebra structure on V L with Y ( u, x ) = u ( x ) for u ∈ A ⊕ B .In fact, the vertex algebra V L is a N -graded vertex algebra and it is generated by A ⊕ B .Furthermore, any restricted L -module W is naturally a V L -module with Y W ( u, x ) = u W ( x ) for u ∈ A ⊕ B . Conversely, any V L -module W is naturally a restricted L -module with u W ( x ) = Y W ( u, x ) for u ∈ A ⊕ B . Now, we set E = Span { e − , a ( − a ′ − a ∗ a ′ | a, a ′ ∈ A } ⊂ ( V L ) (0) ,E = Span { a ( − b − a · b | a ∈ A, b ∈ L } ⊂ ( V L ) (1) ,E = E ⊕ E We define I B = U ( L ) C [ D ] E. The vector space I B is an L -submodule of V L . We set V B = V L /I B . Proposition 28. [GMS, LiY1] (i) V B is a N -graded vertex algebra such that ( V B ) (0) = A and ( V B ) (1) = B (under the linearmap v v ( − ) and V B as a vertex algebra is generated by A ⊕ B . Furthermore, for any n ≥ , ( V B ) ( n ) = span { b ( − n ) .....b k ( − n k ) | b i ∈ B, n ≥ ... ≥ n k ≥ , n + ... + n k = n } . (ii) A V B -module W is a restricted module for the Lie algebra L with v ( n ) acting as v n for v ∈ A ⊕ B , n ∈ Z . Furthermore, the set of V B -submodules is precisely the set of L -submodules. Next, we recall definition of Lie algebroid and its module. Also, we will review constructionof V B -modules from modules of Lie A -algebroid B/A∂ ( A ). N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 11
Definition 29.
Let A be a commutative associative algebra. A Lie A -algebroid is a Liealgebra g equipped with an A -module structure and a module action on A by derivation suchthat [ u, av ] = a [ u, v ] + ( ua ) v, a ( ub ) = ( au ) b for all u, v ∈ g , a, b ∈ A .A module for a Lie A -algebroid g is a vector space W equipped with a g -module structureand an A -module structure such that u ( aw ) − a ( uw ) = ( ua ) w, a ( uw ) = ( au ) w for a ∈ A , u ∈ g , w ∈ W . Proposition 30. [LiY1]
Let W = ⊕ n ∈ N W ( n ) be a N -graded V B -module with W (0) = { } .Then W (0) is an A -module with a · w = a − w for a ∈ A , w ∈ W (0) and W (0) is a module forthe Lie algebra B/A∂ ( A ) with b · w = b w for b ∈ B , w ∈ W (0) . Furthermore, W (0) equippedwith these module structures is a module for Lie A -algebroid B/A∂A . If W is graded simple,then W (0) is a simple module for Lie A -algebroid B/A∂A . Now, we set L ± = ⊕ n ≥ L ( ± n ) and L ≤ = L − ⊕ L (0) . Let U be a module for the Lie algebra L (0) (= A ⊕ B/∂ ( A )). Then U is an L ( ≤ -module under the following actions: a ( n − · u = δ n, au, b ( n ) · u = δ n, u for a ∈ A, b ∈ B, n ≥ . Next, we form the induced L -module M ( U ) = Ind LL ( ≤ U . Endow U with degree 0, making M ( U ) a N -graded L -module. In fact, M ( U ) is a V L -module. We set W ( U ) = span { v n u | v ∈ E, n ∈ Z , u ∈ U } ⊂ M ( U ) , and M B ( U ) = M ( U ) /U ( L ) W ( U ) . Proposition 31. [LiY1] (i) Let U be a module for the Lie algebra L (0) . Then M B ( U ) is a V B -module. If U is a modulefor the Lie A -algebroid B/A∂A then ( M B ( U )) (0) = U .(ii) Let U be a module for the Lie A -algebroid B/A∂A . Then there exists a unique maximalgraded U ( L ) -submodule J ( U ) of M ( U ) with the property that J ( U ) ∩ U = 0 . Moreover, L ( U ) = M ( U ) /J ( U ) is a N -graded V B -module such that L ( U ) (0) = U as a module for the Lie A -algebroid B/A∂A . If U is a simple B/A∂A , L ( U ) is a graded simple V B -module.(iii) Let W = ` n ∈ N W ( n ) be an N -graded simple V B -module with W (0) = 0 . Then W ∼ = L ( W (0) ) .(iv) For any complete set H of representatives of equivalence classes of simple modules for theLie A -algebroid B/A∂A , { L ( U ) | U ∈ H } is a complete set of representatives of equivalenceclasses of simple N -graded simple V B -modules. Proof of Theorem 1 and Theorem 2
Let A be a finite-dimensional commutative associative algebra with the identity e such that dim A ≥
2. Let B be a finite-dimensional vertex A -algebroid such that A is not a trivial B -module and Leib ( B ) = { } . Let S be its Levi factor such that S = Span { e, f, h } , e f = h , h e = 2 e , h f = − f , and e f = k e . Here, k ∈ C \{ } . Assume that one of the followingstatements hold.(I) B is a simple Leibniz algebra;(II) B is a semisimple Leibniz algebra and Ker ( ∂ ) = { a ∈ A | b a = 0 for all b ∈ B } .We set A = C e ⊕ lj =1 N j where each N j is an irreducible sl -submodule of A . By Proposition15 and Proposition 16, we have(i) e e = f f = e h = f h = 0, k = 1, h h = 2 e ;(ii) Ker ( ∂ ) = C e and l ≥ j ∈ { , ..., l } dim N j = 2, and dim Leib ( B ) = 2 l ;(iv) A is a local algebra. For each j , we let a j, be a highest weight vector of N j and a j, = f ( a j, ). Then { e , a j,i | j ∈ { , ...., l } , i ∈ { , }} is a basis of A , and { ∂ ( a j,i ) | j ∈{ , ..., l } , i ∈ { , }} is a basis of Leib ( B ).Relations among a j,i , e, f, h, ∂ ( a j,i ) are described below: a j,i ∗ a j ′ ,i ′ = 0 , a j, · e = 0 , a j, · e = ∂ ( a j, ) , a j, · f = ∂ ( a j, ) , a j, · f = 0 ,a j, · h = ∂ ( a j, ) , a j, · h = − ∂ ( a j, ) , a j,i · ∂ ( a j ′ ,i ′ ) = 0 ,∂ ( a j,i ) e = e a j,i = (2 − i ) a j,i − , ∂ ( a j,i ) f = f a j,i = ( i + 1) a j,i +1 ,∂ ( a j,i ) h = h a j,i = (1 − i ) a j,i . Proof of Theorem 1.
First, we will prove statement ( i ) of Theorem 1. Lemma 32. V B is an indecomposable non-simple vertex algebra.Proof. Recall that ( V B ) (0) = A , ( V B ) (1) = B and V B is generated by A and B . By Proposition19, we can conclude that V B is an indecomposable non-simple vertex algebra. (cid:3) Now, we prove statement ( ii ) of Theorem 1. First, we show in Lemma 33 that if U is anirreducible B/A∂ ( A )-module such that a j,i acts as zero for all j ∈ { , ..., l } , i ∈ { , } then U is an irreducible module for the Lie A -algebroid B/A∂ ( A ). Next, we prove in Lemma 34 thatthe converse of the previous statement holds when U has finite dimension (i.e., if U is a finitedimensional irreducible module for the Lie A -algebroid B/A∂ ( A ) then a j,i acts trivial on U for all j ∈ { , ..., l } , i ∈ { , } .) We complete the proof of the statement ( ii ) of Theorem 1 inLemma 35. N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 13
Lemma 33.
Let U be an irreducible B/A∂ ( A ) -module. If e acts as a scalar , and a j,i acts trivially on U for all j ∈ { , ..., l } , i ∈ { , } then U is an irreducible module for a Lie A -algebroid B/A∂ ( A ) .Proof. Let U be an irreducible B/A∂ ( A )-module. Assume that e acts as a scalar 1, and a j,i acts trivially on U for all j ∈ { , ..., l } , i ∈ { , } . First, we will show that U is a module forthe associative algebra A . Let a = β e + P lj =1 P i =0 λ j,i a j,i , a ′ = β ′ e + P lj =1 P i =0 λ ′ j,i a j,i ∈ A .Here, β, β ′ , λ j,i , λ ′ j,i ∈ C . Since( a ∗ a ′ ) · w = (( β e + l X j =1 1 X i =0 λ j,i a j,i ) ∗ ( β ′ e + l X j =1 1 X i =0 λ ′ j,i a j,i )) · w = ( βa ′ + β ′ l X j =1 1 X i =0 λ j,i a j,i ) · w = ( ββ ′ ) w, and a · ( a ′ · w ) = a · ( β ′ w ) = ββ ′ w for all w ∈ U, we can conclude that U is a module for the associative algebra A .Now, we will show that U is a module for the Lie A -algebroid B/A∂ ( A ). It is enough to showthat for a ∈ A , u ∈ B/A∂ ( A ), w ∈ U , u ( a · w ) − a · ( u w ) = ( u a ) · w and a · ( u w ) = ( a · u ) w .Recall that for a, a ′ ∈ A , b ∈ B , ( a · v ) a ′ = − a ′ ( a · v ) = − a ∗ ( a ′ v ) = a ∗ ( v a ′ ). Consequently,( α · ∂ ( α ′ )) a ′ = α ∗ ( ∂ ( α ′ ) a ′ ) = 0 for all α, α ′ , a ′ ∈ A. We let u = γ e e + γ f f + γ h h + A∂ ( A ) ∈ B/A∂ ( A ). Here, γ e , γ f , γ h ∈ C . Observe that u a = ( γ e e + γ f f + γ h h + A∂ ( A )) ( β e + l X j =1 1 X i =0 λ j,i a j,i )= γ e l X j =1 λ j, a j, + γ f l X j =1 λ j, a j, + γ h l X j =1 λ j, a j, + γ h l X j =1 λ j, ( − a j, ) . Hence, ( u a ) · w = 0 for all w ∈ U . Since u ( a · w ) − a · ( u w ) = u (( β e + l X j =1 1 X i =0 λ j,i a j,i ) · w ) − ( β e + l X j =1 1 X i =0 λ j,i a j,i ) · ( u w )= u ( βw ) − β ( u w )= 0we can conclude immediately that u ( a · w ) − a · ( u w ) = ( u a ) · w for all w ∈ U. Recall that for j ∈ { , ..., l } , i ∈ { , } , we have a j,i · v ∈ ∂ ( A ) for all v ∈ B . It follows that( a · u ) w = ( βu ) w = βu w for all w ∈ U. Moreover, we have a · ( u w ) = β ( u w ) = ( a · u ) w for all w ∈ U. Therefore, U is a module for the Lie A -algebroid B/A∂ ( A ).Next, we will show that U is an irreducible module for the Lie A -algebroid B/A∂ ( A ). Let N be a nonzero Lie A -algebroid B/A∂ ( A )-submodule of U . Then N is a B/A∂ ( A )-submodule of U . Since U is an irreducible B/A∂ ( A ), we can conclude that N = U and U is an irreduciblemodule for the Lie A -algebroid B/A∂ ( A ). This completes the proof of this Lemma. (cid:3) Lemma 34.
Let W be a finite dimensional irreducible module of the Lie A -algebroid B/A∂ ( A ) .Then for j ∈ { , ..., l } , i ∈ { , } , a j,i acts trivially on W . In addition, W is an irreducible sl -module.Proof. Let W be a finite dimensional irreducible module of the Lie A -algebroid B/A∂ ( A ).First, we will show that if W has dimension 1 then W is a trivial sl -module such that for j ∈ { , ..., l } , i ∈ { , } , a j,i acts as zero on W . For simplicity, we assume that W = C b forsome b ∈ W . Clearly, W is a trivial sl -module. We set a j,i · b = β j,i b . Here, β j,i ∈ C . Since h ( a j,i · b ) = ( h a j,i ) · b + ( a j,i ) · ( h b ) = ( h a j,i ) · b, and h ( a j,i · b ) = h ( β j,i b ) = 0 , we then have that 0 = ( h a j, ) · b = a j, · b = β j, b and0 = ( h a j, ) · b = − a j, · b = − β j, b. Therefore, β j,i = 0 and a j,i acts as zero on W for all j ∈ { , ..., l } and i ∈ { , } .Next, we assume that W has dimension 2. Hence, W is either a direct sum of two one-dimensional trivial sl -modules or W is a two-dimensional irreducible sl -module. Supposethat W = C b ⊕ C b where C b and C b are trivial B/A∂ ( A )-modules. For j ∈ { , ..., l } , i ∈ { , } , we set a j,i · b = β j,i, b + β j,i, b . Since h ( a j,i · b ) = ( h ( a j,i )) · b and h ( β j,i, b + β j,i, b ) = 0 , we then have that a j,i · b = 0 for all j ∈ { , ..., l } , i ∈ { , } . Consequently, C b is an irreducible A -Lie algebroid B/A∂ ( A ). This contradicts with our assumption that W is an irreducible Lie A -algebroid B/A∂ ( A ). Therefore, W is a two-dimensional irreducible sl -module. Let w bea highest weight vector of W of weight 1 and let w = f ( w ). The set { w , w } is a basis of W . For j ∈ { , ..., l } , i ∈ { , } , we set a j,i · ( w ) = α j,i, w + α j,i, w . Here, α j,i, , α j,i, ∈ C . Notice that h ( a j,i · w ) = ( h a j,i ) · w + a j,i · ( h ( w )) = ( h a j,i ) · w + a j,i · w , and h ( α j,i, w + α j,i, w ) = α j,i, w + α j,i, ( − w ) . N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 15
So, we have 2 a j, · w = ( h a j, ) · w + a j, · w = α j, , w + α j, , ( − w ) , and0 = ( h a j, ) · w + a j, · w = α j, , w + α j, , ( − w ) . Therefore, for j ∈ { , ..., l } , i ∈ { , } , we have α j,i, = α j,i, = 0 and a j,i · w = 0. Since a j, · f = ∂ ( a j, ) and a j, · f = 0, these imply that a j,i · w = a j,i · ( f w ) = ( a j,i · f ) ( w ) = 0Consequently, if dim W = 2 then W is an irreducible B/A∂ ( A )-module such that for j ∈{ , ..., l } , i ∈ { , } , a j,i acts trivially on W .Now, we study that case when dim W ≥
3. Suppose that W contains a nonzero proper B/A∂ ( A )-submodule (i.e., we consider W as a module for the Lie algebra B/A∂ ( A )). Since B/A∂ ( A ) is semisimple, this implies that there exist irreducible B/A∂ ( A )-modules U , ..., U t such that W = ⊕ ti =1 U t . For each i ∈ { , ..., t } , we let w i, be a highest weight vector of U i ofweight m i . Also, we set w i,s = s ! ( f (0)) s w i, . Clearly, { w i, , ...., w i,m i } form a basis of U i . Let j ∈ { , ..., l } , 1 ≤ s ≤ m i . Since a j, · f = ∂ ( a j, ) and a j, · f = 0, we then have that a j, · w i,s = a j, · (cid:18) s ! ( f ) s w i, (cid:19) = a j, · (cid:18) f ( 1 s ! ( f ) s − w i, ) (cid:19) = ( a j, · f ) (cid:18) s ! ( f ) s − w i, (cid:19) = 0 , and a j, · w i,s = a j, · (cid:18) s ! ( f ) s w i, (cid:19) = ( a j, · f ) (cid:18) s ! ( f ) s − w i, (cid:19) = 0 . Since h ( a j, · w i, ) = ( h a j, ) · w i, + a j, · ( h w i, ) = ( m i + 1) a j, · w i, and h ( a j, · w i, ) = e f ( a j, · w i, ) − f e ( a j, · w i, )= e ( f ( a j, ) · w i, + a j, · ( f w i, )) − f (( e a j, ) · w i, + a j, · e w i, )= e ( a j, · w i, + a j, · w i, )= e ( a j, · w i, )= ( e a j, ) · w i, + a j, · ( e w i, )= a j, · w i, , we can conclude that m i = 0 and U i is a trivial B/A∂ ( A )-module. Moreover, { w , , ..., w t, } is a basis of W . For j ∈ { , ..., l } , i ∈ { , ..., t } , we set a j, · w i, = P tp =1 α p w p, , and a j, · w i, = P tp =1 γ p w p, where α p , γ p ∈ C . Since h ( a j, · w i, ) = ( h a j, ) · w i, + a j, · h w i, = a j, · w i, ,h ( t X p =1 α p w p, ) = 0 ,h ( a j, · w i, ) = ( h a j, ) · w i, + a j, · h w i, = − a j, · w i, ,h ( t X p =1 γ p w p, ) = 0 , we can conclude that a j, · w i, = 0 = a j, · w i, for all j ∈ { , ..., l } , i ∈ { , ..., t } . Moreover,each U i is an irreducible module for the Lie A -algebroid B/A∂ ( A ). This is a contradiction.Hence, W is an irreducible B/A∂ ( A )-module.Now, we let u be the highest weight vector of W with weight m and for i ∈ { , ..., m } we let u i = i ! ( f (0)) i u . We have a j, · u i = a j, · (cid:18) i ! ( f ) i u (cid:19) = a j, · (cid:18) f ( 1 i ! ( f ) i − u ) (cid:19) = ( a j, · f ) (cid:18) i ! ( f ) i − u (cid:19) = 0 , and a j, · u i = a j, · (cid:18) i ! ( f ) i u (cid:19) = ( a j, · f ) (cid:18) i ! ( f ) i − u (cid:19) = 0 . Next, we set a j, · u = P mq =0 α q u q . Since h ( a j, · u ) = ( m + 1)( a j, · u ) = m X q =0 α q ( m + 1) u q and h ( m X q =0 α q u q ) = m X q =0 ( m − q ) α q u q , we can conclude that α q ( m + 1) = α q ( m − q ) for all 0 ≤ q ≤ m . If α q = 0, we have m + 1 = m − q which is impossible. Therefore, for all 0 ≤ q ≤ m , α q = 0. Consequently, wehave a j, · u = 0. Moreover, we have a j, · u = ( f a j, ) · u = f ( a j, · u ) − a j, · ( f u ) = − a j, · u = 0 . Hence, W is an irreducible B/A∂ ( A )-module such that for j ∈ { , ..., l } , i ∈ { , } , a j,i actstrivially on W . This completes the proof of this Lemma. (cid:3) Lemma 35.
The set of representatives of equivalence classes of finite-dimensional simple sl -modules is equivalent to the set of representatives of equivalence classes of N -graded simple V B -modules N = ⊕ ∞ n =0 N ( n ) such that dim N (0) < ∞ .Proof. By Lemma 33, every finite dimensional irreducible
B/A∂ ( A )-module is an irreduciblemodule for the Lie A -algebroid B/A∂ ( A ). By Lemma 34, the set of representatives of equiv-alence classes of finite dimensional simple modules for the Lie A -algebraoid B/A∂ ( A ) equalsthe set of representatives of equivalence classes of finite dimensional simple modules for the N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 17
Lie algebra
B/A∂ ( A ). By Proposition 31, we can conclude that the set of representativesof equivalence classes of finite-dimensional simple sl -modules is equivalent to the set of rep-resentatives of equivalence classes of N -graded simple V B -modules N = ⊕ ∞ n =0 N ( n ) such that dim N (0) < ∞ . (cid:3) This completes the proof of statment ( ii ) of Theorem 1.3.2. Proof of Theorem 2.
First, we will prove statement ( i ) of Theorem 2. Let ( e ( − e ) be an ideal of V B that isgenerated by e ( − e . Lemma 36. ( e ( − e ) ∩ A = { } and ( e ( − e ) ∩ B = { } .Proof. First, we will show that v (deg v )+1 e ( − e = 0 and v deg v e ( − e = 0 for every homoge-neous v ∈ V B . We will separate our proof into several steps. For the first step, we will showthat for a ∈ A , a ( n ) e ( − e = 0 for all n ≥ . Recall that for a ∈ A , b ∈ B , we have b ( − a = a ( − b − D ( a (0) b ) = a · b − D ( a b ). Let a = α + P lj =1 P i =0 α j,i a j,i . Here, α, α j,i ∈ C . It is straightforward to show that a (0) e ( − e = e ( − a e + ( a e )( − e = − e ( − e a − ( e a )( − e = − e ( − l X j =1 α j, a j, ) − l X j =1 α j, ( a j, )( − e = − e ( − l X j =1 α j, a j, ) − l X j =1 α j, ( a j, ) · e = − ( l X j =1 α j, a j, ) · e + D (( l X j =1 α j, a j, ) e )= 0 , and a (1) e ( − e = e ( − a (1) e + ( a e )(0) e = e ( e a ) = e (cid:16)P lj =1 α j, a j, (cid:17) = 0. Hence, a ( n ) e ( − e = 0 for all n ≥ . For the second step, we will show that for b ∈ B ,( b ( − ) n e ( − e = b ( n ) e ( − e = 0 , ( b ( − m ) ) m +1 e ( − e = 0 , and ( b ( − m ) ) m e ( − e = 0 for all m ≥ n ≥
1. Let b = β e e + β f f + β h h + P lj =1 P i =0 β j,i ∂ ( a j,i ) ∈ B . Here, β e , β f , β h , β j,i ∈ C . Since b e = ( β e e + β f f + β h h + l X j =1 1 X i =0 β j,i ∂ ( a j,i )) e = β f ( − h ) + β h (2 e ) and b e = ( β e e + β f f + β h h + l X j =1 1 X i =0 β j,i ∂ ( a j,i )) e = β f + l X j =1 β j, a j, we then have that b (1) e ( − e = e ( − b (1) e + ( b e )(0) e + ( b e )( − e = e ( − b e ) + ( β f ( − h ) + β h (2 e )) e + ( β f + l X j =1 β j, a j, )( − e = e ( − β f + l X j =1 β j, a j, ) + β f ( − e + β f e = l X j =1 β j, ( a j, · e + ∂ (( a j, ) e )= 0 , and b (2) e ( − e = e ( − b (2) e + ( b e )(1) e + 2( b e )(0) e = ( β f ( − h ) + β h (2 e )) e + 2( β f + l X j =1 β j, a j, ) e = 0 . Hence, ( b ( − ) n e ( − e = b ( n ) e ( − e = 0 for all n ≥
1. Let m ≥
2. For t ≥ b ( − m ) ) t e ( − e = X i ≥ ( − i (cid:18) − mi (cid:19) ( b ( − m − i ) ( t + i ) − ( − − m ( − m + t − i ) b ( i )) e ( − e = − ( − − m ( − m + t ) b (0) e ( − e. This implies that ( b ( − m ) ) m +1 e ( − e = 0 and ( b ( − m ) ) m e ( − e = 0.Recall that for n ≥ V B ) ( n ) = span { b ( − n ) .....b k ( − n k ) | b i ∈ B, n ≥ ... ≥ n k ≥ , n + ... + n k = n } . N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 19 If v ∈ V B is of the form b ( − n ) .....b k ( − n k ) where b i ∈ { e, f, h, ∂ ( a j,i ) | j ∈ { , ..., l } , i ∈{ , }} , we say that v is a monomial vector that has length k . For the third step, we willshow that if b ∈ B , and v is a monomial vector of length k then either b (0) v = 0 or b (0) v is asum of monomial vectors that have length k . Clearly, if b ∈ ∂ ( A ) then b v = 0. Notice that b (0) b ( − n ) = ( b b )( − n ) b (0) b ( − n ) b ( − n ) = b ( − n )( b b )( − n ) + ( b b )( − n ) b ( − n ) If b b = 0 then b (0) b ( − n ) = 0. If b b = 0 then b (0) b ( − n ) is a sum of monomialvectors of length 1. Similarly, if b b = 0 = b b then b (0) b ( − n ) b ( − n ) = 0. Otherwise, b (0) b ( − n ) b ( − n ) is a sum of monomial vectors of length 2. Now, we assume that formonomial vectors w of length t , either b (0) w = 0 or b (0) w are the sum of monomial vectorsof length t . Since b (0) b ( − n ) .....b t +1 ( − n t +1 ) = b ( − n ) b (0) b ( − n ) .....b t +1 ( − n t +1 ) +( b b )( − n ) b ( − n ) .....b t +1 ( − n t +1 ) , by induction hypothesis, we can conclude that either b (0) b ( − n ) .....b t +1 ( − n t +1 ) = 0 or b (0) b ( − n ) .....b t +1 ( − n t +1 ) is a sum of monomial vectors of length t + 1.For the fourth step, we will show that for every monomial vector v ∈ V B , v (deg v )+1 e ( − e = 0and v deg v e ( − e = 0. We will use an induction on the length of monomial vectors to prove thisstatement. By the first step and the second step, we can conclude immediately that if v is amonomial vector of length q where 0 ≤ q ≤ v (deg v )+1 e ( − e = 0 and v (deg v ) e ( − e = 0.Now, we assume that for any monomial vector v of length k ≤ t , v (deg v )+1 e ( − e = 0 and v (deg v ) e ( − e = 0. For i ∈ { , ..., t + 1 } , we let n i be a positive integer, b i ∈ B . We set n = n + n + ... + n t + n t +1 + 1. Notice that( b ( − n ) b ( − n ) ...b t ( − n t ) b t +1 ( − n t +1 ) ) n e ( − e = X i ≥ ( − i (cid:18) − n i (cid:19) ( b ( − n − i )( b ( − n ) ...b t ( − n t ) b t +1 ( − n t +1 ) ) n + i − ( − n ( b ( − n ) ...b t ( − n t ) b t +1 ( − n t +1 ) ) − n + n − i b ( i )) e ( − e = − ( − n ( b ( − n ) ...b t +1 ( − n t +1 ) ) − n + n b (0) e ( − e (by induction hypothesis)= ( − n +1 ( b (0)( b ( − n ) ...b t +1 ( − n t +1 ) ) − n + n e ( − e − ( b (0) b ( − n ) ...b t +1 ( − n t +1 ) ) − n + n e ( − e ) , For simplicity, we set u = b ( − n ) ...b t +1 ( − n t +1 ) and w = b (0) b ( − n ) ...b t +1 ( − n t +1 ) . Note that the vector u is a monomial vector of length t , the vector w is a sum of monomialvectors of length t , and − n + n = n + .... + n t +1 + 1 = (deg u ) + 1 = (deg w ) + 1 . By induction hypothesis, we can conclude that( b ( − n ) b ( − n ) ...b t ( − n t ) b t +1 ( − n t +1 ) ) n e ( − e = 0 . Next, we assume that n = n + n + ... + n t +1 . We have( b ( − n ) b ( − n ) ...b t +1 ( − n t +1 ) ) n e ( − e = X i ≥ ( − i (cid:18) − n i (cid:19) ( b ( − n − i )( b ( − n ) ...b t +1 ( − n t +1 ) ) n + i − ( − n ( b ( − n ) ...b t +1 ( − n t +1 ) ) − n + n − i b ( i )) e ( − e = δ n , b ( − n )( b ( − n ) ...b t +1 ( − n t +1 ) ) n e ( − e − ( − n ( b ( − n ) ...b t +1 ( − n t +1 ) ) − n + n b (0) e ( − e = δ n , b ( − n )( b ( − n ) ...b t +1 ( − n t +1 ) ) n e ( − e − ( − n { b (0)( b ( − n ) ...b t +1 ( − n t +1 ) ) − n + n − ( b (0) b ( − n ) ...b t +1 ( − n t +1 ) ) − n + n } e ( − e. We set p = b ( − n ) ...b t +1 ( − n t +1 ) and r = b (0) b ( − n ) ...b t +1 ( − n t +1 ) . Notice that when n = 1, p n = p (deg p )+1 . Also, p − n + n = p deg p , r is a sum of monomial vectors of length t and r − n + n = r deg r . By induction hypothesis, we can conclude that( b ( − n ) b ( − n ) ...b t +1 ( − n t +1 ) ) n e ( − e = 0 . Hence, for any homogeneous monomial vector v of length k , v (deg v )+1 e ( − e = 0 and v deg v e ( − e =0. This completes the fourth step.Because v (deg v )+1 e ( − e = 0 and v deg v e ( − e = 0 for any homogeneous monomial vector v of any length k , we can conclude further that for any homogeneous vector u ∈ V B , u (deg u )+1 e ( − e = 0 , and u deg u e ( − e = 0 . Moreover, for any homogeneous vector v ∈ V B , t ∈ Z , we have v t e ( − e ∈ ⊕ ∞ n =2 ( V B ) ( n ) . Thisimplies that for v ∈ V B , t ∈ Z , we have v t e ( − e ∈ ⊕ ∞ n =2 ( V B ) ( n ) .Next, we will show that for i ≥ v t D i e ( − e ∈ ⊕ ∞ n =2 ( V B ) ( n ) for all v ∈ V B , t ∈ Z . Clearly, v t De ( − e = Dv t e ( − e + tv t − e ( − e ∈ ⊕ ∞ n =2 ( V B ) ( n ) . Now, let us assume that v t D j e ( − e ∈ ⊕ ∞ n =2 ( V B ) ( n ) for all v ∈ V B , t ∈ Z . Since v t D j +1 e ( − e = Dv t D j e ( − e + tv t − D j e ( − e, we can conclude immediately that v t D j +1 e ( − e ∈ ⊕ ∞ n =2 ( V B ) ( n ) . Hence, v t D i e ( − e ∈ ⊕ ∞ n =2 ( V B ) ( n ) for all i ≥ , v ∈ V B , t ∈ Z . This implies that ( e ( − e ) ∩ ( A ⊕ B ) = { } . (cid:3) We set V B = V B / ( e ( − e ) . N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 21
Proposition 37. V B = ⊕ ∞ n =0 ( V B ) ( n ) is an indecomposable non-simple N -graded vertex algebrasuch that ( V B ) (0) = A and ( V B ) (1) = B .Proof. Since V B = ⊕ ∞ n =0 ( V B ) ( n ) is a N -graded vertex algebra such that ( V B ) (0) = A and( V B ) (1) = B , by Proposition 19, we can conclude that V B = ⊕ ∞ n =0 ( V B ) ( n ) is an indecomposablenon-simple N -graded vertex algebra. (cid:3) This completes the proof of statement ( i ) of Theorem 2.To prove statement ( ii ) and statement ( iii ) of Theorem 2, we need to use properties of vertexoperator algebras associated with a certain type of rank one lattices, and vertex operatoralgebras associated with highest weight representations of affince Lie algebras. We providebackground material on these topics in Appendices.Let ˆ S = S ⊗ C [ t, t − ] ⊕ C c be the affine Lie algebra where c is central and[ u ⊗ t m , v ⊗ n ] = [ u, v ] ⊗ t m + n + m hh u, v ii δ m + n, c. Here, hh , ii is a symmetric invariant bilinear form of S such that hh e, f ii = 1, hh h, h ii = 2and hh e, e ii = hh f, f ii = hh e, h ii = hh f, h ii = 0. The generalized Verma ˆ S -module M S ( k,
0) isa vertex operator algebra (see Appendices for the construction of the vertex operator algebra M S ( k,
0) and its properties).For u ∈ V B , we set Y V B ( u, z ) = P n ∈ Z u [ n ] z − n − . Since S = Span { e, f, h } is a subset of ( V B ) (1) and S is a Lie algebra with a symmetric invariant bilinear form h , i : S × S → C such that h s, s ′ i e = s [1] s ′ , the map ˆ S → End( V B ) : s ⊗ t m → s [ m ] is a representation of the affine Kac-Moody algebra ˆ S of level k where h s, s ′ i = k hh s, s ′ ii for s, s ′ ∈ S . Since h h, h i e = h [1] h = 2 e and hh h, h ii = 2, we then have that k = 1. Moreover, V B is a module of M S (1 , U be the vertex sub-algebra of V B that is generated by S . This vertex algebra U isa highest weight module for ˆ S . In fact, U is a quotient of the generalized Verma module M S (1 , U is integrable if and only if U ∼ = L (1 , e [ − l = 0 for some l ≥
0. Since ( e [ − = 0, we canconclude immediately that U is integrable. Indeed, U is isomorphic to L (1 , V B is integrable as ˆ S -module.By Proposition 47, we have f [ − f = 0. Lemma 38. (i) (( f + h − e )[ − = 0 .(ii) If ( W, Y W ) is a V B -module then Y W ( e, z ) = Y W ( f, z ) = Y W ( f + h − e, z ) = 0 .(iii) In particular, we have Y V B ( e, z ) = Y V B ( f, z ) = Y V B ( f + h − e, z ) = 0 on V B .Proof. Since 0 = e [0]( f [ − = f [ − h [ − + h [ − f [ − , f [0]( e [ − = − ( e [ − h [ − + h [ − e [ − ) and0 = ( f [0]) ( e [ − = − (2 e [ − f [ − − h [ − + 2 f [ − e [ − ) we can conclude that (( f + h − e )[ − = 0. This proves ( i ).By Proposition 25, we can conclude that if ( W, Y W ) is a V B -module then Y W ( e, z ) = Y W ( f, z ) = Y W ( f + h − e, z ) = 0. In particular, we have Y V B ( e, z ) = Y V B ( f, z ) = Y V B ( f + h − e, z ) = 0 on V B . We obtain statments ( ii ) and ( iii ) as desired. (cid:3) Lemma 39. V B satisfies the C -condition.Proof. Clearly, V B /C ( V B ) = Span { a + C ( V B ) , b + C ( V B ) , b [ − ....b k [ − + C ( V B ) | a ∈ A, b, b i ∈ S, k ≥ } . Now, we follow the proof of Proposition 12.6 in [DoLiM1]. Since { e, f, h } forms a basis of S , this implies that { e, f, f + h − e } forms a basis for S as well. Observe that for u, v ∈{ e, f, f + h − e } , w ∈ V B , u [ − v [ − w = v [ − u [ − w + ( u v )[ − w . Since e [ − e = f [ − f = ( f + h − e )[ − f + h − e ) = 0 , we can conclude that V B /C ( V B ) = Span { a + C ( V B ) , b + C ( V B ) , u [ − v + C ( V B ) ,u [ − v [ − w + C ( V B ) | a ∈ A, b, u, v, w ∈ { e, f, f + h − e }} , and V B is C -cofinite. (cid:3) This completes the proof of statement ( ii ) of Theorem 2.Next, we will study N -graded V B -modules. Observe that A ⊕ B generates V B as a vertexalgebra. Consequently, if W is a V B -module, then W is a restricted L -module with u ( n )acting as u n for u ∈ A ⊕ B , n ∈ Z . Moreover, the set of V B -submodules is the set of L -submodules. Proposition 40. [LiY1]
Let ( W, Y W ) be a V L -module. Assume that for any a, a ′ ∈ A, b ∈ B , Y W ( e , z ) u = u,Y W ( a ( − a ′ , z ) u = Y W ( a ∗ a ′ , z ) u,Y W ( a ( − b, z ) u = Y W ( a · b, z ) u, for all u ∈ U where U is a generating subspace of W as a V L -module, then W is naturally a V B -module. Lemma 41.
Let ( Q, Y Q ) be a V B -module such that Y ( e ( − e, z ) u = 0 for all u ∈ F where F is a generating subspace of Q as a V B -module. Then Q is a V B -module.Proof. By using Proposition 26, one can obtain the above statement very easily. (cid:3)
Lemma 42.
Let W = ⊕ ∞ n =0 W ( n ) be a N -graded V B -module with W (0) = { } . Then(i) W (0) is an A -module with a · w = a − w for a ∈ A , w ∈ W (0) , and W (0) is a module forthe Lie algebra B/A∂ ( A )( ∼ = sl ) with b · w = b w for b ∈ B , w ∈ W (0) . Furthermore, W (0) equipped with these module structures is a module for the Lie A -algebroid B/A∂ ( A ) (ii) Moreover, e ( e w ) = 0 , e − ( e − w ) = 0 , f ( f w ) = 0 and f − ( f − w ) = 0 for all w ∈ W (0) . N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 23 (iii) If W is simple then W (0) is an irreducible module for Lie A -algebroid B/A∂ ( A ) that hasdimension either 1 or 2. Moreover, for j ∈ { , ..., l } , i ∈ { , } , a j,i acts trivially on W (0) .Proof. By following the proof of Proposition 4.8 of [LiY1], one can shows that statements (i)and the statement “if W is simple then W (0) is a simple B/A∂ ( A )-module” hold.Now, we will prove that e e w = f f w = 0 for all w ∈ W (0) . By Proposition 38, we have Y W ( e, z ) = 0 and Y W ( f, z ) = 0 on W . Since e n u = 0 for all n ≥ u ∈ W (0) , these implythat Y ( e, z ) u = P n ≥ P m ≥ e − m e − n uz m + n − . Moreover, the coefficient of z − is e e u = 0and the constant term is e − e − u = 0 . Similarly, using the fact that Y W ( f, z ) = 0 and f n u = 0 for all n ≥ u ∈ W (0) , one can show that f f u = 0 and f − f − u = 0 . We obtain statement (ii) as desired.Next, we prove statement (iii). We only need to show that for an irreducible V B -module W , W (0) is either one dimensional or two dimensional. By statement (ii), W (0) has either onedimensional or two dimensional. Recall that { e , a j,i | j ∈ { , ..., l } , i ∈ { , }} is a basis of A , B/A∂ ( A ) ∼ = sl , and(22) v ( a − w ) − a − ( v w ) = ( v a ) − w for all a ∈ A, v ∈ B/A∂A, w ∈ W (0) , and e ( a j, ) = a j, , f ( a j, ) = a j, .If dim W (0) = 1 then W (0) is a trivial module of sl . For simplicity, we set W = C w . Byequation (22) , we have0 = e (( a j, ) − w ) − ( a j, ) − e w = ( e ( a j, )) − w = ( a j, ) − w , and0 = f (( a j, ) − w ) − ( a j, ) − f w = ( f ( a j, )) − w = ( a j, ) − w . We now assume that dim W (0) = 2. Then W (0) = Span { w , w } where w is the highestweight vector of W (0) of weight 1 and w = f w . Recall that for a ∈ A , b ∈ B/A∂ ( A ), a − ( b w ) = ( a − b ) w for all w ∈ W (0) . Hence, we have( a j, ) − w = ( a j, ) − ( e w ) = (( a j, ) − e ) w = 0 , ( a j, ) − w = ( a j, ) − ( f w ) = (( a j, ) − f ) w = ( ∂ ( a j, )) w = 0 , ( a j, ) − w = ( a j, ) − ( e w ) = (( a j, ) − e ) w = ( ∂ ( a j, )) w = 0 , ( a j, ) − w = ( a j, ) − ( f w ) = (( a j, ) − f ) w = 0 . This completes the proof of statement (iii). (cid:3)
Lemma 43.
Let W be a N -graded V B -module with W (0) = { } . If W is an irreducible modulefor the vertex algebra L (1 , , then W is an irreducible V B -module.Proof. Let W be a N -graded V B -module with W (0) = { } . Assume that W is an irreduciblemodule for the vertex algebra L (1 , G = { } be V B -submodule of W . Hence, G is a U -submodule of W . Since U is isomorphic to L (1 ,
0) as vertex algebra, we can conclude that G = W . Consequently, W is an irreducible V B -module. (cid:3) Lemma 44.
Let L = Z α be a positive definite even lattice of rank one equipped with a Q -valued Z -bilinear form ( · , · ) such that ( α, α ) = 2 . Then the vertex operator algebra V L = ⊕ ∞ n =0 ( V L ) ( n ) is an irreducible N -graded V B -module such that dim( V L ) (0) = 1 and V L + α = ⊕ ∞ n =0 ( V L + α ) ( n ) is an irreducible N -graded V B -module such that dim( V L + α ) (0) = 2 .Proof. Recall that ( V L , Y ) and ( V L + α , Y ) are irreducible modules of L (1 ,
0) (see Appendices).By Lemma 43, to show that V L and V L + α are irreducible V B -modules, we only need to showthat they are actually V B -modules. Let L ◦ be the dual lattice of L . To prove that V L and V L + α are irreducible V B -modules, we only need to show that V L ◦ = V L ⊕ V L + α is a V B -module.Recall that for γ ∈ H , β ∈ L , Y ( γ, z ) = Y ( γ ( − , z ) = X n ∈ Z γ ( n ) z − n − ,Y ( e β , z ) = X n ∈ Z ( e β ) n z − n − = exp ∞ X m =1 β ( − m ) z m m ! exp − ∞ X m =1 β ( m ) z − m m ! e β z β . For n ∈ Z , we assume that h ⊗ t n acts as α ( n ), e ⊗ t n acts as ( e α ) n , f ⊗ t n acts as ( e − α ) n , a j,i ⊗ t n acts as zero, ∂ ( a j,i ) ⊗ t n acts as zero and e ⊗ t n acts as δ n, −
1. First we claim that( V L ◦ , Y V L ◦ ) is a V L -module. Let a = λ e e + P lj =1 P i =0 λ j,i a j,i . Notice that(23) Y V L ◦ ( a, z ) = λ e Y V L ◦ ( e , z ) = λ e Id V L ◦ . Here, Id V L ◦ is the identity map on V L ◦ . The following are commutator relations among α , e ± α on V L ◦ : [ Y ( α, z ) , Y ( α, z )] = − ∂∂z z − δ (cid:18) z z (cid:19) = 2 ∂∂z z − δ (cid:18) z z (cid:19) , [ Y ( α, z ) , Y ( e ± α , z )] = ± z − δ (cid:18) z z (cid:19) Y ( e ± α , z )[ Y ( e α , z ) , Y ( e − α , z )] = z − δ (cid:18) z z (cid:19) Y ( α, z ) − ∂∂z z − δ (cid:18) z z (cid:19) , = z − δ (cid:18) z z (cid:19) Y ( α, z ) + ∂∂z z − δ (cid:18) z z (cid:19) , [ Y ( e α , z ) , Y ( e α , z )] = [ Y ( e − α , z ) , Y ( e − α , z )] = 0 . By comparing these commutator relations with commutator relations (19)-(21), we can con-clude that V L ◦ is a L -module. Hence, V L ◦ is a V L -module.Let a = λ e e + P lj =1 P i =0 λ j,i a j,i , a ′ = λ ′ e e + P lj =1 P i =0 λ ′ j,i a j,i ∈ A . Since Y V L ◦ ( a ∗ a ′ , z ) = λ e λ ′ e Id V L ◦ N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 25 and Y V L ◦ ( a ( − a ′ , z ) = Res z { z − Res z { z − δ (cid:18) z − z z (cid:19) Y V L ◦ ( a, z ) Y V L ◦ ( a ′ , z ) − z − δ (cid:18) z − z − z (cid:19) Y V L ◦ ( a ′ , z ) Y V L ◦ ( a, z ) }} = Res z z − Res z z − δ (cid:18) z − z z (cid:19) λ e λ ′ e Id V L ◦ = λ e λ ′ e Id V L ◦ , we can conclude that(24) Y V L ◦ ( a ( − a ′ , z ) = Y V L ◦ ( a ∗ a ′ , z ) . Let b = ρ e e + ρ f f + ρ h h + P lj =1 P i =0 ρ j,i ∂ ( a j,i ). Notice that Y V L ◦ ( b, z ) = ρ e Y V L ◦ ( e, z ) + ρ f Y V L ◦ ( f, z ) + ρ h Y V L ◦ ( h, z ) . Since a · b = ( λ e e + l X j =1 1 X i =0 λ j,i a j,i ) · ( ρ e e + ρ f f + ρ h h + l X j =1 1 X i =0 ρ j,i ∂ ( a j,i )= λ e b + τ where τ ∈ ∂ ( A ) , we then have that Y V L ◦ ( a · b, z ) = Y V L ◦ ( λ e b, z ) = λ e Y V L ◦ ( ρ e e + ρ f f + ρ h h, z ). Since Y V L ◦ ( a ( − b, z ) = Res z { z − Res z { z − δ (cid:18) z − z z (cid:19) Y V L ◦ ( a, z ) Y V L ◦ ( b, z ) − z − δ (cid:18) z − z − z (cid:19) Y V L ◦ ( b, z ) Y V L ◦ ( a, z ) }} = Res z z − Res z z − δ (cid:18) z − z z (cid:19) λ e Y V L ◦ ( b, z )= λ e Y V L ◦ ( b, z ) , this implies that(25) Y V L ◦ ( a ( − b, z ) = Y V L ◦ ( a · b, z ) . By (23), (24), (25), we can conclude that V L ◦ is a V B -module. Observe that Y V L ◦ ( e ( − e, z ) = Res z { z − Res z { z − δ (cid:18) z − z z (cid:19) Y V L ◦ ( e, z ) Y V L ◦ ( e, z ) − z − δ (cid:18) z − z − z (cid:19) Y V L ◦ ( e, z ) Y V L ◦ ( e, z ) }} = Res z { z − Res z { z − δ (cid:18) z − z z (cid:19) Y ( e α , z ) Y ( e α , z ) − z − δ (cid:18) z − z − z (cid:19) Y ( e α , z ) Y ( e α , z ) }} = Y ( e α − e α , z )= 0 . Hence V L ◦ is a V B -module. This completes the proof of this Lemma. (cid:3) Lemma 45.
Let W be an irreducible N -graded V B -module. Then W is either isomorphic to V L or V L + α .Proof. Let W = ⊕ ∞ n =0 W ( n ) be an irreducible N -graded V B -module. By Lemma 42, W (0) is anirreducible sl -module and the dimension of W (0) is either 1 or 2. Since W is a L (1 , W = ⊕ ti =1 U i is a direct sum of irreducible L (1 , U i where U i = ⊕ ∞ n =0 U i ( n ) is either isomorphic to V L or V L + α . Here, U i ( n ) = W ( n ) ∩ U i .If the dimension of W (0) is 1 then W is isomorphic to V L . Now, we assume that the dimensionof W (0) is 2. Then W is either isomorhic to V L ⊕ V L or V L + α . If W is isomorphic to V L ⊕ V L then W (0) is isomorphic to a sum of two trivial sl -modules which is impossible. Hence, W isisomorphic to V L + α . (cid:3) This completes the proof of statement (iii) of Theorem 2.4.
Appendices
Vertex operator algebra associated with a rank one even lattice Z α such that ( α, α ) = 2 . Let L = Z α be a positive definite even lattice of rank one, i.e., a free abelian group equippedwith a Q -valued Z -bilinear form ( · , · ) such that ( α, α ) = 2. We set H = C ⊗ Z L and extend ( · , · ) to a C -bilinear form on H . Letˆ H = H ⊗ C [ t, t − ] ⊕ C K be the affine Lie algebra associated to the abelian Lie algebra H so that[ α ( m ) , α ( n )] = 2 mδ m + n, K and [ K, ˆ H ] = 0 N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 27 for any m, n ∈ Z , where α ( m ) = α ⊗ t m . Then ˆ H ≥ = H ⊗ C [ t ] ⊕ C K is a commutativesubalgebra. For any λ ∈ H , we define a one-dimensional ˆ H ≥ -module C e λ such that α ( m ) · e λ =( λ, α ) δ m, e λ , and K · e λ = e λ for m ≥
0. We denote by M (1 , λ ) = U ( ˆ H ) ⊗ U (ˆ H ≥ ) C e λ the ˆ H -module induced from ˆ H ≥ -module C e λ . We set M (1) = M (1 , Y : M (1) → End M (1)[[ z, z − ]] such that ( M (1) , Y, , ω ) carries a simple vertexoperator algebra structure and M (1 , λ ) becomes an irreducible M (1)-module for any λ ∈ H ([FrLMe1]). The vacuum vector and the Virasoro element are given by = e and ω = α ( − , respectively.Let C [ L ] be the group algebra of L with a basis e β for β ∈ L and multiplication e β e γ = e β + γ ( β, γ ∈ L ). The lattice vertex operator algebra associated to L is given by V L = M (1) ⊗ C [ L ] . The dual lattice L ◦ of L is the set { λ ∈ H | ( α, λ ) ∈ Z } = L . Note that L ◦ = L ∪ ( L + α )is the coset decomposition of L ◦ with respect to L . Also, we set C [ L + α ] = ⊕ β ∈ L C e β + α .Then C [ L + α ] is a L -submodule of C [ L ◦ ]. We set V L + α = M (1) ⊗ C [ L + α ]. It was shownin [Bo1], [Do], [FrLMe1], [Gu] that V L is a rational vertex operator algebra. Furthermore, V L and V L + α are the only irreducible modules for V L under the following action: for β ∈ H write β ( z ) := P n ∈ Z β ( n ) z − n − , z β : e γ z ( β,γ ) e γ and set Y ( e β , z ) := exp ∞ X m =1 β ( − m ) z m m ! exp − ∞ X m =1 β ( m ) z − m m ! e β z β , and for v = α ( − n ) ...α t ( − n t ) ⊗ e β ∈ V L ( n i ≥
1) set Y ( v, z ) := n − (cid:18) ddz (cid:19) n − α ( z ) ! ... n t − (cid:18) ddz (cid:19) n t − α t ( z ) ! Y ( e α , z ) : , with the usual normal ordering conventions.4.2. Vertex operator algebras associated with highest weight representations ofaffine Lie algebras.
Let g be a simple Lie algebra over C , h its Cartan subalgebra and ∆ the corresponding rootsystem. We fix a set of positive root ∆ + and a nondegnerate symmetric invariant bilinearform h· , ·i of g such that the square length of a long root is 2. Letˆ g = g ⊗ C [ t, t − ] ⊕ C c be the affine Lie algebra with Lie bracket defined by[ u ⊗ t m , v ⊗ n ] = [ u, v ] ⊗ t m + n + m h u, v i δ m + n, c. Here, u, v ∈ g and m, n ∈ Z and c is a central element. We will write u ( n ) for u ⊗ t n . Let l be a complex number such that l = − ˇ h where ˇ h is the dual Coxeter number of g . Let C l be the one-dimensional ( g ⊗ C [ t ] + C c )-module on which c acts as scalar l and g ⊗ C [ t ] actsa zero. Form the generalized Verma ˆ g -module M g ( l,
0) = U (ˆ g ) ⊗ U ( g ⊗ C [ t ]+ C c ) C l . We define Y ( u ( − ⊗ , z ) = X n ∈ Z u ( n ) z − n − ,Y ( u ( − m − ⊗ , z ) = 1 m ! d m dz m Y ( u ( − ⊗ , z ) . In general, if Y ( v, z ) has been defined, we defined Y ( u ( − n ) v, z ) for u ∈ g and n > Y ( u ( − n ) v, z ) = Res z { ( z − z ) − n Y ( u, z ) Y ( v, z ) − ( − z + z ) − n Y ( v, z ) Y ( u, z ) } . We then get a linear map Y from M g ( l,
0) to (End M g ( l, z, z − ]]. Set = 1 ⊗ ω = l +ˇ h ) P i v i ( − ⊗ { v i } is an orthonormal basis of g with respect to h , i . Proposition 46. [FS, FZ, LLi, Li, MeP] (i) ( M g ( l, , Y, , ω ) is a vertex operator algebra. The category of weak M g ( l, -modules inthe sense that all axioms defining the notion of module except those involving grading hold iscanonically equivalent to the category of restricted ˆ g -modules of level l in the sense that forevery vector w of the module, ( g ⊗ t n C [ t ]) w = 0 for n sufficiently large.(ii) Let J l, be the maximal proper submodule of M g ( l, . Let e θ be an element in the rootspace g θ of the maximal root θ . The space J l, is generated by the vector e θ ( − l +1 , i.e.,every element in J l, can be written as a linear combination of elements of type u ( − n ) ....u m ( − n t ) e θ ( − l +1 . Note that and ω are not members of J l, . Given a g -module W , we will write u W ( n ) for the operator on W corresponding to u ⊗ t n for u ∈ g and n ∈ Z and we set u W ( z ) = P n ∈ Z u W ( n ) z − n − ∈ (End W )[[ z, z − ]]. W is arestricted module of level l if c acts as l and for every u ∈ g , w ∈ W , u ( n ) w = 0 for n sufficiently large.We set L ( l,
0) = M g ( l, /J l, . Proposition 47. [FZ, LLi, DoL] (i) L ( l, is a rational vertex operator algebra.(ii) Let α ∈ ∆ be a long root, i.e., h α, α i = 2 and let e ∈ g α . Then e ( − l +1 ∈ J l, . Inparticular, e θ ( − l +1 , f θ ( − l +1 ∈ J l, . Moreover, Y ( e, x ) l +1 = 0 on L ( l, . For any mod-ule W for L ( l, viewed as a vertex algebra, Y W ( e, z ) l +1 = 0 . In particular, Y W ( e θ , z ) l +1 = Y W ( f θ , z ) l +1 = 0 .(iii) If g = sl and l = 1 , then V L and V L + α are the only irreducible L (1 , -modules. N INDECOMPOSABLE VERTEX ALGEBRAS ASSOCIATED WITH VERTEX ALGEBROIDS 29
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School of Science, Walailak University, Nakhon Si Thammarat, Thailand
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