Module category and C_2-cofiniteness of affine vertex operator superalgebras
aa r X i v : . [ m a t h . QA ] J a n MODULE CATEGORY AND C -COFINITENESS OF AFFINEVERTEX OPERATOR SUPERALGEBRAS CHUNRUI AI AND XINGJUN LIN
Abstract.
In this paper, we investigate the Lie algebra structures of weight onesubspaces of C -cofinite vertex operator superalgebras. We also show that for anypositive integer k , vertex operator superalgebras L sl (1 | n +1) ( k,
0) and L osp (2 | n ) ( k, L g ( k,
0) is C -cofinite if and only if g is either a simple Liealgebra, or g = osp (1 | n ), and k is a nonnegative integer. As an application, we showthat L G (3) (1 ,
0) is a vertex operator superalgebra such that the category of L G (3) (1 , L G (3) (1 ,
0) is not C -cofinite. Introduction
Let g be a finite dimensional simple Lie superalgebra with a nondegenerate evensupersymmetric invariant bilinear form, L g ( k,
0) be the simple vertex operator superal-gebra associated with g [22]. In case that g is a finite dimensional simple Lie algebraand k is a positive integer, it was proved that L g ( k,
0) is C -cofinite [8], [36]. Moreover,in case that g is a finite dimensional simple Lie algebra, it was proved in [10] that k mustbe a positive integer if L g ( k,
0) is C -cofinite. It has also been known for many years that L g ( k,
0) is C -cofinite if and only if g is either a simple Lie algebra, or g = osp (1 | n ),and the b g -module L g ( k,
0) is integrable (see Section 0.4 of [17]). One of our motivationsis to give a proof of this result.Our first main result is about the Lie algebra structures of weight one subspaces of C -cofinite vertex operator superalgebras. Explicitly, let V be a vertex operator super-algebra which is of strong CFT type and C -cofinite. It is known [4] that V has a Liesuperalgebra structure. Assume that g is a subalgebra of V and that g is a basic simpleLie superalgebra, then we show that g must be isomorphic to osp (1 | n ), sl (1 | n + 1)or osp (2 | n ) , n ≥ L sl (1 | n +1) ( k,
0) and L osp (2 | n ) ( k, L sl (1 | n +1) ( k,
0) and L osp (2 | n ) ( k,
0) have infinitely many irreducible admissible mod-ules (see Theorems 4.4, 4.8). As an application of these results, we give a proof of
C. Ai was supported by China NSF grant 11701520; X. Lin was supported by China NSF grant11801419 and the starting research fund from Wuhan University (No. 413000076). the fact that L g ( k,
0) is C -cofinite if and only if g is either a simple Lie algebra, or g = osp (1 | n ), and k is a nonnegative integer (see Theorem 5.6).Another motivation of this work comes from the study of the relationship betweenrationality and C -cofiniteness of vertex operator algebras. For a vertex operator algebra V , we could consider categories of weak V -modules, admissible V -modules and ordinary V -modules [9]. A vertex operator algebra V is called regular if the category of weak V -modules is semisimple, and V is called rational if the category of admissible V -modulesis semisimple. In [36], Zhu conjectured that rational vertex operator algebras are C -cofinite. It has been proved that regular vertex operator algebras are C -cofinite [28].However, Zhu’s conjecture has not been proved. This motivates us to ask whether avertex operator superalgebra V is C -cofinite if the category of ordinary V -modulesis semisimple. Due to Kac and Wakimoto, it is known that the category of ordinary L G (3) (1 , L G (3) (1 ,
0) is a vertexoperator superalgebra such that the category of L G (3) (1 , L G (3) (1 ,
0) is not C -cofinite.Our work was also motivated by the study of coset vertex operator subalgebras ofaffine vertex operator algebras [3], [15]. It is known [33], [34] that coset vertex operatorsubalgebras of affine vertex operator algebras are closely related to W -superalgebras,which are vertex operator superalgebras obtained from affine vertex operator superalge-bras by quantum Drinfeld-Sokolov reduction [23]. We expect that our results are usefulfor studying W -superalgebras and coset vertex operator subalgebras of affine vertexoperator algebras.The paper is organized as follows: In Section 2, we recall basic definitions about ver-tex operator superalgebras and basic facts about affine vertex operator superalgebras.In Section 3, we investigate the Lie algebra structures of weight one subspaces of C -cofinite vertex operator superalgebras. In Section 4, we show that L sl (1 | n +1) ( k,
0) and L osp (2 | n ) ( k,
0) have infinitely many irreducible admissible modules. As a consequence,it is proved that L sl (1 | n +1) ( k,
0) and L osp (2 | n ) ( k,
0) are not C -cofinite. In Section 5,we prove that L osp (1 | n ) ( k,
0) is a C -cofinite vertex operator superalgebra. As a con-sequence, it is proved that L g ( k,
0) is C -cofinite if and only if g is either a simple Liealgebra, or g = osp (1 | n ), and k is a nonnegative integer. In Section 6, we show thatthe category of ordinary L G (3) (1 , ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 3 Preliminaries
Basics.
In this subsection, we recall from [14], [26], [30] basic notions about vertexoperator superalgebras. Let V = V ¯0 ⊕ V ¯1 be any Z -graded vector space, the elementin V ¯0 (resp. V ¯1 ) is called even (resp. odd ). We then define [ v ] = i for any v ∈ V ¯ i with i = 0 ,
1. A vertex superalgebra is a quadruple (
V, Y ( · , z ) , , D ) , where V = V ¯0 ⊕ V ¯1 is a Z -graded vector space, is the vacuum vector of V , D is an endomorphism of V , and Y ( · , z ) is a linear map Y ( · , z ) : V → (End V )[[ z, z − ]] ,v Y ( v, z ) = X n ∈ Z v n z − n − ( v n ∈ End V )satisfying the following axioms:(i) For any u, v ∈ V, u n v = 0 for sufficiently large n ;(ii) Y ( , z ) = id V ;(iii) Y ( v, z ) = v + P n ≥ v − n z n − , for any v ∈ V ;(iv) [ D, Y ( v, z )] = Y ( D ( v ) , z ) = ddz Y ( v, z );(v) The Jacobi identity for Z -homogeneous u, v ∈ V holds, z − δ (cid:18) z − z z (cid:19) Y ( u, z ) Y ( v, z ) − ( − [ u ][ v ] z − δ (cid:18) z − z − z (cid:19) Y ( v, z ) Y ( u, z )= z − δ (cid:18) z − z z (cid:19) Y ( Y ( u, z ) v, z ) . This completes the definition of a vertex superalgebra and we will denote the vertex su-peralgebra briefly by V . A vertex superalgebra V is called C -cofinite if dim V /C ( V ) < ∞ , where C ( V ) = h u − v | u, v ∈ V i .A vertex superalgebra V is called a vertex operator superalgebra if there is a distin-guished vector ω , which is called the conformal vector of V , such that the following twoconditions hold:(vi) The component operators of Y ( ω, z ) = P n ∈ Z L ( n ) z − n − satisfy the Virasoro algebrarelation with central charge c ∈ C :[ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + 112 ( m − m ) δ m + n, c, and L ( −
1) = D ;(vii) V is Z -graded such that V = ⊕ n ∈ Z V n , L (0) | V n = n , dim( V n ) < ∞ and V n = 0for sufficiently small n . Remark 2.1.
We adopt the definition of a vertex operator superalgebra given in [30] .The definition of a vertex operator superalgebra is different from that in [12] , [29] , [35] , CHUNRUI AI AND XINGJUN LIN where the additional condition V ¯0 = ⊕ n ∈ Z V n , V ¯1 = ⊕ n ∈ + Z V n is added. However, we willsee that affine vertex operator superalgebras do not satisfy the condition V ¯0 = ⊕ n ∈ Z V n , V ¯1 = ⊕ n ∈ + Z V n (see Theorem 2.4). A vertex operator superalgebra V is called of CFT type if V has the decomposition V = L n ≥ V n with respect to L (0) such that dim V = 1. We say that V is of strongCFT type if V satisfies the further condition L (1) V = 0. If v ∈ V n , the conformalweight wt v of v is defined to be n .2.2. Zhu’s algebra of vertex operator superalgebras.
In this subsection, we recallfrom [13] some facts about the Zhu’s algebra of vertex operator superalgebras. First,a Z -graded vertex operator superalgebra V is a vertex operator superalgebra such that V n = 0 for n ∈ + Z . For a Z -graded vertex operator superalgebra V , a weak V -module M is a vector space equipped with a linear map Y M ( · , z ) : V → (End M )[[ z, z − ]] ,v Y M ( v, z ) = X n ∈ Z v n z − n − , v n ∈ End M satisfying the following conditions: For any Z -homogeneous u ∈ V, v ∈ V, w ∈ M and n ∈ Z , u n w = 0 for sufficiently large n ; Y M ( , z ) = id M ; z − δ (cid:18) z − z z (cid:19) Y M ( u, z ) Y M ( v, z ) − ( − [ u ][ v ] z − δ (cid:18) z − z − z (cid:19) Y M ( v, z ) Y M ( u, z )= z − δ (cid:18) z − z z (cid:19) Y M ( Y ( u, z ) v, z ) . A weak V -module M is called admissible if it is Z ≥ -graded M = ⊕ n ∈ Z ≥ M ( n )such that for homogeneous v ∈ V , v m M ( n ) ⊆ M ( n + wt v − m − . In this paper, we will assume that M (0) = 0 if M = 0. A Z -graded vertex operatorsuperalgebra V is called rational if any admissible V -module is completely reducible.For a Z -graded vertex operator superalgebra V , an (ordinary) V -module is a weak V -module M which carries a C -grading induced by the spectrum of L (0), that is, M = L λ ∈ C M λ where M λ = { w ∈ M | L (0) w = λw } . Moreover, one requires that M λ is finite ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 5 dimensional and for fixed λ ∈ C , M λ + n = 0 for sufficiently small integer n . It is known[9] that a V -module is always admissible.We now recall from [13] the definition of the Zhu’s algebra A ( V ) of a Z -gradedvertex operator superalgebra V . A ( V ) is defined as A ( V ) := V /O ( V ) , where O ( V ) =span { u ◦ v | u, v ∈ V } and u ◦ v := Res z (1 + z ) wt u z Y ( u, z ) v for homogeneous u, v ∈ V and extended linearly. It is an associative algebra withmultiplication defined as u ∗ v := Res z (1 + z ) wt u z Y ( u, z ) v for homogeneous u, v ∈ V (see Theorem 3.1 of [13]).For a weak V -module M , define the space Ω( M ) as follows:Ω( M ) = { w ∈ M | u wt u − n w = 0 , u ∈ V, n > } . Then the following results have been proved in Theorem 3.2 of [13].
Theorem 2.2.
Let V be a Z -graded vertex operator superalgebra, and M be a weak V -module. Then(1) Ω( M ) is an A ( V ) -module such that v + O ( V ) acts as v wt v − .(2) If M = ⊕ n ∈ Z ≥ M ( n ) is an admissible V -module such that M (0) = 0 , then M (0) ⊆ Ω( M ) is an A ( V ) -submodule. Moreover, M is irreducible if and only if M (0) = Ω( M ) and M (0) is a simple A ( V ) -module.(3) The map M → M (0) gives a − correspondence between the irreducible admissible V -modules and simple A ( V ) -modules. We also need the following result, which can be proved by the similar argument asLemma 4.3 of [2].
Proposition 2.3.
Let V be a C -cofinite Z -graded vertex operator superalgebra. Then A ( V ) is finite dimensional. Affine vertex operator superalgebras.
In this subsection, we recall from [15],[22], [27] some facts about affine vertex operator superalgebras. Let g be a finitedimensional simple Lie superalgebra with a nondegenerate even supersymmetric in-variant bilinear form ( ·|· ). The affine Lie superalgebra associated to g is defined on CHUNRUI AI AND XINGJUN LIN b g = g ⊗ C [ t − , t ] ⊕ C K with Lie brackets[ x ( m ) , y ( n )] = [ x, y ]( m + n ) + ( x | y ) mδ m + n, K, [ K, b g ] = 0 , for x, y ∈ g and m, n ∈ Z , where x ( n ) denotes x ⊗ t n .For a complex number k , define the vacuum module of b g by V k ( g ) = Ind b gg ⊗ C [ t ] ⊕ C K C k , where C k = C is the 1-dimensional g ⊗ C [ t ] ⊕ C K -module such that g ⊗ C [ t ] acts as 0and K acts as k . Theorem 2.4 ([15, 22]) . Let g be a finite dimensional simple Lie superalgebra with anondegenerate even supersymmetric invariant bilinear form ( ·|· ) , h ∨ be the dual Coxeternumber of ( g , ( ·|· )) and k be a complex number which is not equal to − h ∨ . Then V k ( g ) is a vertex operator superalgebra such that the conformal vector ω is defined as follows: ω = 12( k + h ∨ ) X i a i ( − b i ( − , where { a i } and { b i } are dual bases of g with respect to ( ·|· ) . Moreover, the conformalweight of x ( − is for any x ∈ g . Note that V k ( g ) is a strong CFT type Z -graded vertex operator superalgebra. Itis well-known that V k ( g ) has a unique irreducible quotient module which is denotedby L g ( k,
0) (see [20]). Then L g ( k,
0) has a vertex operator superalgebra structure [27].Moreover, L g ( k,
0) is also a Z -graded vertex operator superalgebra.3. Integrability of C -cofinite vertex operator superalgebras Let V be a vertex operator superalgebra of strong CFT type. It is known [4] that theweight one subspace V of V has a Lie superalgebra structure defined by [ u, v ] = u v for any u, v ∈ V . Moreover, we have Lemma 3.1.
Let V be a vertex operator superalgebra of strong CFT type. Then V isa Lie superalgebra equipped with a supersymmetric invariant bilinear form B ( · , · ) suchthat B ( u, v ) = u v for any u, v ∈ V . Proof:
We first prove that B ( · , · ) is supersymmetric. By the skew-symmetry propertyof vertex operator superalgebras (see (2.2.5) of [29]), we have Y ( u, z ) v = ( − [ u ][ v ] e zL ( − Y ( v, − z ) u, ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 7 for any Z -homogeneous u, v ∈ V . It follows that u v = ( − [ u ][ v ] v u for any u, v ∈ V . Hence, B ( u, v ) = ( − [ u ][ v ] B ( v, u ) for any u, v ∈ V . We next prove that B ( · , · )is invariant, i.e., B ([ u, v ] , w ) = B ( u, [ v, w ]). By (2.2.6) of [29], we have for any Z -homogeneous u, v, w ∈ V , u m v n w − ( − [ u ][ v ] v m u n w = ∞ X i =0 (cid:18) mi (cid:19) ( u i v ) m + n − i w. (3.1)Hence, we have for any Z -homogeneous u, v, w ∈ V , u v w − ( − [ u ][ v ] v u w = ( u v ) w + ( u v ) w. This implies that u v w = ( u v ) w for any Z -homogeneous u, v, w ∈ V . Then we havethe invariant property. (cid:3) We now let g be a subalgebra of V . Suppose further that g is a simple Lie superalgebraequipped with a nondegenerate even supersymmetric invariant bilinear from ( ·|· ). Thenthere exists a complex number k such that B ( u, v ) = k ( u | v ) for any u, v ∈ g . By theformula (3.1) we have for any Z -homogeneous u, v ∈ V , u m v n − ( − [ u ][ v ] v m u n = ( u v ) m + n + m ( u v ) m + n − = ([ u, v ]) m + n + mB ( u, v ) δ m + n, = ([ u, v ]) m + n + mk ( u | v ) δ m + n, . Thus, V is a b g -module of level k .We next assume further that V is C -cofinite. Fix homogeneous x i ∈ V ( i ranging overan index set I ) such that the coset x i + C ( V ) , i ∈ I , span a complement to + C ( V )in V /C ( V ). Set X = { x i | i ∈ I } . Then the following property of V has been essentiallyproved in Proposition 8 of [16]. Theorem 3.2.
Let V be a vertex operator superalgebra which is of strong CFT typeand C -cofinite. Then V is spanned by together with elements of the form x − n · · · x k − n k where n > · · · > n k > , x i ∈ X . As a consequence, by the same argument as in Theorem 3.1 of [10], we have thefollowing result, which was essentially proved in Theorem 6.1 of [6].
CHUNRUI AI AND XINGJUN LIN
Theorem 3.3.
Let V be a vertex operator superalgebra which is of strong CFT type and C -cofinite, g be a Lie subalgebra of V . Assume that g is a simple Lie algebra, then V is an integrable b g -module. We are now ready to prove the main result in this section.
Theorem 3.4.
Let V be a vertex operator superalgebra which is of strong CFT type and C -cofinite, g be a Lie subalgebra of V . Assume that g is a basic simple Lie superalgebra,then g is isomorphic to osp (1 | n ) , sl (1 | n + 1) or osp (2 | n ) , n ≥ . Proof:
Let g be a basic simple Lie superalgebra. Assume that g is not isomorphic to osp (1 | n ), sl (1 | n + 1) or osp (2 | n ) , n ≥
1. We choose the nondegenerate even super-symmetric invariant bilinear from ( ·|· ) of g defined in Table 6.1 of [24]. Thus, there aresimple Lie subalgebras f and f of g ¯0 such that the restriction of ( ·|· ) on f is positivedefinite and the restriction of ( ·|· ) on f is negative definite.Consider the vertex subalgebra h g i of V generated by g , then h g i is isomorphic toa quotient module of the vacuum module V k ( g ) of b g for some complex number k . ByTheorem 3.3, V is an integrable module of b f . Hence, h g i is an integrable module of b f . Note that the vacuum vector is a highest weight vector for b f . Let W be the b f -submodule of h g i generated by . Then W is an integrable highest weight moduleof b f . As a consequence, W is an irreducible integrable highest weight module of b f .Therefore, by Lemma 10.1 of [20], k must be a nonnegative number.On the other hand, V is also an integrable module of b f . Hence, h g i is an integrablemodule of b f . Note that the restriction of ( ·|· ) on f is negative definite. By the sim-ilar argument as above, we can prove that k must be a nonpositive number. As aconsequence, k = 0. This forces that W viewed as an b f -module is isomorphic to C .However, W contains a subspace isomorphic to f , this is a contradiction. Thus, g mustbe isomorphic to osp (1 | n ), sl (1 | n + 1) or osp (2 | n ) , n ≥ (cid:3) Irreducible modules of vertex operator superalgebras L sl (1 | n ) ( k, and L osp (2 | n ) ( k, Irreducible modules of vertex operator superalgebra L sl (1 | n ) ( k, . In thissubsection, for any positive integer k , we will show that L sl (1 | n ) ( k,
0) has infinitelymany nonisomorphic irreducible admissible modules. As a result, we will show that L sl (1 | n ) ( k,
0) is not C -cofinite.Consider the simple Lie superalgebra sl (1 | n ), n ≥
2, which consists of block matricesof the form X = A BC D ! such that A is a (1 × B is a (1 × n )-matrix, C ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 9 is a ( n × D is a ( n × n )-matrix and that Str( X ) := tr( A ) − tr( D ) = 0 (cf.[31]). For any X, Y ∈ sl (1 | n ), define ( X | Y ) = − Str( XY ), then ( ·|· ) is a nondegenerateeven supersymmetric invariant bilinear form of sl (1 | n ). We use I to denote the identitymatrix and let h be the matrix n I ! . Then h is an element of sl (1 | n ) ¯0 . Let f bethe set of matrices of the form D ! such that tr D = 0, then f is a subalgebra of sl (1 | n ) ¯0 . Moreover, f is isomorphic to the simple Lie algebra sl n . Thus the even part of sl (1 | n ) is sl n ⊕ C h . Let h be the subset of sl (1 | n ) consisting of diagonal matrices. Then h is a Cartan subalgebra of sl (1 | n ). We next describe the root system of sl (1 | n ). Let ǫ i , ≤ i ≤ n, be the linear functionals on h whose values on the diagonal matrix a = diag( a , · · · , a n )are given by ǫ i ( a ) = a i . Then the root system of sl n is {± ( ǫ i − ǫ j ) | ≤ i < j ≤ n } andthe set of odd roots of sl (1 | n ) is {± ( ǫ − ǫ i ) | ≤ i ≤ n } (see subsection 2.2 of [31]).Moreover, the bilinear form ( ·|· ) is determined by ( ǫ i | ǫ j ) = 0 if i = j , ( ǫ | ǫ ) = − ǫ i | ǫ i ) = 1 for 1 ≤ i ≤ n . In the following, we choose the following subset of simpleroots of sl (1 | n ): Π = { ǫ − ǫ , ǫ − ǫ , · · · , ǫ n − − ǫ n } . Note that the highest root of sl n is θ = ǫ − ǫ n . Let e θ be a highest root vector of sl n .Then we have the following result, which was proved in Theorem 5.4.1 and Corollary5.4.3 of [18]. Theorem 4.1.
Let k be a positive integer. Then L sl (1 | n ) ( k,
0) = V k ( sl (1 | n )) /I , where I is the submodule of V k ( sl (1 | n )) generated by e θ ( − k +1 . Thus, to find irreducible admissible modules of L sl (1 | n ) ( k, V k ( sl (1 | n )) such that Y ( e θ ( − k +1 , z ) = 0. We nextshow that L sl (1 | n ) ( k,
0) has infinitely many irreducible admissible modules. Consider theaffine Kac-Moody superalgebra sl (1 | n ) (1) := \ sl (1 | n ) ⊕ C d . Set ˜ h = h ⊕ C K ⊕ C d , then ˜ h is a Cartan subalgebra of sl (1 | n ) (1) . We use δ to denote the smallest positive imaginaryroot of sl (1 | n ) (1) , and choose the following subset Σ of simple roots of sl (1 | n ) (1) :Σ = { δ − ǫ + ǫ n , ǫ − ǫ , ǫ − ǫ , · · · , ǫ n − − ǫ n } . For λ ∈ (˜ h ) ∗ , let L Σ ( λ ) be the irreducible highest weight module of sl (1 | n ) (1) of highestweight λ . Recall that L Σ ( λ ) is an integrable sl (1 | n ) (1) -module if L Σ ( λ ) is integrableover sl (1) n and locally finite over the Cartan subalgebra h (see [18], [25]). Set b = ( λ | δ − ǫ + ǫ n ) , b i = ( λ | ǫ i − − ǫ i ) , ≤ i ≤ n. Then the following result was proved in [25] (see also subsection 2.3.1 of [18]).
Theorem 4.2.
Let k be a positive integer. Then L Σ ( λ ) is an integrable sl (1 | n ) (1) -moduleof level k if and only if(i) b i ∈ Z ≥ for i ≥ .(ii) b + b ∈ Z > or b = b = 0 .(iii) b + b + · · · + b n = k . Furthermore, we have
Proposition 4.3.
Let k be a positive integer, L Σ ( λ ) be an integrable sl (1 | n ) (1) -moduleof level k . Then L Σ ( λ ) is an irreducible admissible L sl (1 | n ) ( k, -module. Proof:
First, L Σ ( λ ) is a weak V k ( sl (1 | n ))-module (see subsection 5.3 of [18]). Wenext show that Y ( e θ ( − k +1 , z ) = 0 on L Σ ( λ ). Since L Σ ( λ ) is an integrable sl (1 | n ) (1) -module of level k , it is an integrable b sl n -module of level k . By Remark 3.9 of [7], L Σ ( λ )viewed as an b sl n -module is a direct sum of irreducible highest weight integrable b sl n -modules. By Proposition 6.6.21 of [27], we have Y ( e θ ( − k +1 , z ) = 0 on L Σ ( λ ). As aconsequence, L Σ ( λ ) is a weak L sl (1 | n ) ( k, L Σ ( λ ) is an admissible L sl (1 | n ) ( k, d actsdiagonally on L Σ ( λ ) with eigenvalues bounded from above. By the discussion in Sub-section 5.6 of [18], L Σ ( λ ) is an admissible V k ( sl (1 | n ))-module (see also Remark 6.6.1 of[27]). This implies that L Σ ( λ ) is an admissible L sl (1 | n ) ( k, (cid:3) Note that by Theorem 4.2 there are infinitely many integrable sl (1 | n ) (1) -modulesof level k . Thus, by Proposition 4.3, there are infinitely many irreducible admissible L sl (1 | n ) ( k, Theorem 4.4.
Let k be a positive integer. Then the vertex operator superalgebra L sl (1 | n ) ( k, is not C -cofinite. Irreducible modules of vertex operator superalgebra L osp (2 | n ) ( k, . In thissubsection, for a positive integer k , we will show that L osp (2 | n ) ( k,
0) has infinitelymany nonisomorphic irreducible admissible modules. As a result, we will show that L osp (2 | n ) ( k,
0) is not C -cofinite.Consider the simple Lie superalgebra osp (2 | n ), n ≥
1, which consists of block ma-trices of the form X = a y y − a z z − z t − y t d ez t y t f − d t ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 11 such that a is a complex number, y, y , z, z are (1 × n )-matrices, e, f are symmetric( n × n )-matrices and d is a ( n × n )-matrix (cf. [31]). For any X, Y ∈ osp (2 | n ),define ( X | Y ) = − Str( XY ), then ( ·|· ) is a nondegenerate even supersymmetric invariantbilinear form of osp (2 | n ). Set h = − . Then h is an element of osp (2 | n ) ¯0 . Let f be the set of matrices of the form d e f − d t such that e, f are symmetric ( n × n )-matrices and d is a ( n × n )-matrix. Then f is asubalgebra of osp (2 | n ) ¯0 . Moreover, f is isomorphic to the simple Lie algebra of type C n .Thus the even part of osp (2 | n ) is f ⊕ C h . Let h be the subset of osp (2 | n ) consistingof diagonal matrices. Then h is a Cartan subalgebra of osp (2 | n ). We next describe theroot system of osp (2 | n ). Let ǫ i , ≤ i ≤ n, be the linear functionals on h whose valueson the diagonal matrix a = diag( a , · · · , a n +2 )are given by ǫ i ( a ) = a i . Then the root system of f is {± ( ǫ i ± ǫ j ) , ± ǫ i | ≤ i < j ≤ n + 2 } and the set of odd roots of osp (2 | n ) is {± ( ǫ − ǫ i ) | ≤ i ≤ n + 2 } (see Subsection2.3.3 of [31]). Moreover, the bilinear form ( ·|· ) is determined by ( ǫ i | ǫ j ) = 0 if i = j ,( ǫ | ǫ ) = − and ( ǫ i | ǫ i ) = for 3 ≤ i ≤ n + 2. In the following, we choose the followingsubset of simple roots of osp (2 | n ):Π = { ǫ − ǫ , ǫ − ǫ , · · · , ǫ n +1 − ǫ n +2 , ǫ n +2 } . Note that the highest root of f is θ = 2 ǫ . Let e θ be a highest root vector of f . Thenwe have the following result, which was proved in Theorem 5.4.1 and Corollary 5.4.3 of[18]. Theorem 4.5.
Let k be a positive integer. Then L osp (2 | n ) ( k,
0) = V k ( osp (2 | n )) /I ,where I is the submodule of V k ( osp (2 | n )) generated by e θ ( − k +1 . Thus, to find irreducible admissible modules of L osp (2 | n ) ( k, V k ( osp (2 | n )) such that Y ( e θ ( − k +1 , z ) = 0.We next show that L osp (2 | n ) ( k,
0) has infinitely many irreducible admissible modules.
Consider the affine Kac-Moody superalgebra osp (2 | n ) (1) := \ osp (2 | n ) ⊕ C d . Set ˜ h = h ⊕ C K ⊕ C d , then ˜ h is a Cartan subalgebra of osp (2 | n ) (1) . We use δ to denote thesmallest positive imaginary root of osp (2 | n ) (1) , and choose the following subset Σ ofsimple roots of osp (2 | n ) (1) :Σ = { δ − ǫ − ǫ , ǫ − ǫ , ǫ − ǫ , · · · , ǫ n +1 − ǫ n +2 , ǫ n +2 } . For λ ∈ (˜ h ) ∗ , let L Σ ( λ ) be the irreducible highest weight module of osp (2 | n ) (1) ofhighest weight λ . Recall that L Σ ( λ ) is an integrable osp (2 | n ) (1) -module if L Σ ( λ ) isintegrable over f (1) and locally finite over the Cartan subalgebra h (see [18], [25]). Set b = ( λ | δ − ǫ − ǫ ) , b = ( λ | ǫ − ǫ ) , b i = 2( λ | ǫ i − − ǫ i ) , ≤ i ≤ n + 2 , b n +3 = ( λ | ǫ n +2 ) . Then the following result was proved in Theorem 8.1 of [25].
Theorem 4.6.
Let k be a positive integer. Then L Σ ( λ ) is an integrable osp (2 | n ) (1) -module of level k if and only if(i) b i ∈ Z ≥ for i ≥ .(ii) b + b ∈ Z > or b = b = 0 .(iii) b + b + b + · · · + b n +2 + b n +3 = k . Furthermore, by the similar argument as that in Proposition 4.3, we have the followingresult.
Proposition 4.7.
Let k be a positive integer, L Σ ( λ ) be an integrable osp (2 | n ) (1) -moduleof level k . Then L Σ ( λ ) is an irreducible admissible L osp (2 | n ) ( k, -module. Note that by Theorem 4.6 there are infinitely many integrable osp (2 | n ) (1) -modulesof level k . Thus, by Proposition 4.7, there are infinitely many irreducible admissible L osp (2 | n ) ( k, Theorem 4.8.
Let k be a positive integer. Then the vertex operator superalgebra L osp (2 | n ) ( k, is not C -cofinite. C -cofiniteness of affine vertex operator superalgebras In this section, for any positive integer n , we will show that the vertex operatorsuperalgebra L osp (1 | n ) ( k,
0) is C -cofinite if k is a positive integer. This result has beenknown for many years (see Section 0.4 of [17]). For completeness, we give a proof here.We need the following result, which can be proved by the similar argument as inLemma 3.8 of [8]. ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 13 Lemma 5.1.
Let V be a vertex operator superalgebra. Then C ( V ) is closed under theoperators v and v − for any v ∈ V . We also need the following fact which was proved in Lemma 12.1 of [8].
Lemma 5.2.
Let V be a vertex operator superalgebra. Then C ( V ) contains u − m V forany u ∈ V and m ≥ . We now let g be a simple Lie superalgebra which is isomorphic to osp (1 | n ), n ≥ g ¯0 be the even part of g , then g ¯0 is isomorphic to the simple Lie algebra sp (2 n ). Inthe following, we fix a Cartan subalgebra h of g ¯0 . Note that there is a nondegeneratesupersymmetric invariant bilinear form ( ·|· ) of g such that the restriction of ( ·|· ) on g ¯0 is the normalized invariant bilinear form of g ¯0 . Then we have Proposition 5.3.
Let k be a positive integer. Then for any root vector e α of g ¯0 , Y ( e α , z ) tk +1 = 0 acting on L g ( k, , where t = 1 if α is a long root of g ¯0 and t = 2 if α is a short root of g ¯0 . Proof:
The argument of the proof is similar to that in Proposition 5.2.1 of [29]. Toprove the statement, it is enough to prove e tk +1 α = 0. From the standard semisimple Liealgebra theory (cf. [19]) we can embed sl into g ¯0 as g α linearly spanned by e α , f α , h α . Itis known [18], [21] that L g ( k,
0) is an integrable b g -module, then L g ( k,
0) is an integrable c g α -module. As a result, generates an integrable c g α -module W . In particular, W is anirreducible c g α -module. Set t = α,α ) , then ( h α , h α ) = α,α ) = 2 t . As a result, we have[( f α ) , ( e α ) − ] = h α + tk . Hence, ( f α ) ( e α ) tk +1 − = 0. Note also that ( e α ) ( e α ) tk +1 − = 0.Hence, if ( e α ) tk +1 − = 0, it generates a c g α -submodule of W , this is a contradiction. Then( e α ) tk +1 − = 0. (cid:3) Furthermore, by the similar argument as in Lemma 3.6 of [7], we have
Corollary 5.4.
Let k be a positive integer. Then there is a basis of { a , · · · , a m } of g ¯0 such that for ≤ i ≤ m we have [ Y ( a i , z ) , Y ( a i , z )] = 0 and Y ( a i , z ) k +1 = 0 as operators on L g ( k, . We are now ready to prove the following result.
Theorem 5.5.
Let k be a positive integer. Then the vertex operator superalgebra L osp (1 | n ) ( k, is C -cofinite. Proof:
The argument of the proof is similar to that in Proposition 12.6 of [8]. Bydefinition, V k ( g ) = U ( b g ) ⊗ U ( P ∞ m =0 t m ⊗ g ⊕ C K ) C ∼ = U ( ∞ X m =1 t − m ⊗ g ) (linearly) . By Lemma 5.2, C ( L g ( k, a − m L g ( k,
0) for any a ∈ g and m ≥
2. Thus, L g ( k,
0) = C ( L g ( k, U ( t − ⊗ g ) . It is enough to show that C ( L g ( k, x n ( − · · · x n s s ( − , whenever n i ≥ n + · · · + n s is large enough; here x , · · · , x s is a basis of g .First, let a be a root vector in the odd part of g . Then we have a ( − = [ a, a ]( − a ( − b is contained in C ( L g ( k, b ∈ L g ( k, { a , · · · , a m } of g ¯0 such that for 1 ≤ i ≤ m we have [ Y ( a i , z ) , Y ( a i , z )] = 0 and Y ( a i , z ) k +1 = 0as operators on L g ( k, Y ( a i , z ) k +1 is equal to a i ( − k +1 + r , where r is a sum of products of the form a i ( n ) e · · · a i ( n k +1 ) e k +1 n j ≤ −
2. Since the operators a i ( n j ) commute, we have r ∈ C ( L g ( k, a i ( − k +1 ∈ C ( L g ( k, C ( L g ( k, x n ( − · · · x n s s ( − whenever n i ≥ k + 1 for some i . This completes theproof. (cid:3) Combining with Theorems 3.4, 4.4, 4.8, we immediately obtain the following result,which has also been known for many years (see Section 0.4 of [17]).
Theorem 5.6.
Let g be a basic simple Lie superalgebra. Then L g ( k, is C -cofinite ifand only if g is isomorphic to osp (1 | n ) and k is a nonnegative integer. Category of L G (3) (1 , -modules In this section, we give a proof of the fact that the category of L G (3) (1 , L G (3) (1 ,
0) is a vertex operator superalgebrasuch that the category of L G (3) (1 , L G (3) (1 ,
0) is not C -cofinite.Let g ¯0 = sl ⊕ G and g ¯1 = V ⊗ V , where V denotes the standard two-dimensionalmodule of sl and V denotes the unique 7-dimensional irreducible G -module. Thenthere is a simple Lie superalgebra structure on G (3) := g ¯0 ⊕ g ¯1 (see Subsection 1.4.2 of[32]). We next describe the root system of G (3). First, recall [5] that G has a Cartan ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 15 subalgebra h such that we may choose the following positive root system of G :∆ +2 = { α , α , α + α , α + 2 α , α + 3 α , α + 3 α } . The bilinear form ( ·|· ) on h is determined by ( α | α ) = − , ( α | α ) = −
1, ( α | α ) = 2and ( α | α ) = . The set of weights of G -module V is as follows: { α + 2 α , α + α , α , , − α , − α − α , − α − α } . We next choose the Cartan subalgebra h of sl such that the root system of sl is { α , − α } . We now choose the following positive root system of sl :∆ +1 = { α } . The bilinear form ( ·|· ) on h is determined by ( α | α ) = − . The set of weights of sl -module V is as follows: { α , − α } . Note that h ⊕ h is a Cartan subalgebra of G (3). Then we may choose the followingpositive root system of G (3): ∆ + = ∆ +1 ∪ ∆ +0 , where ∆ +1 = { α , α ± α , α ± ( α +2 α ) , α ± ( α + α ) } and ∆ +0 = { α , α , α , α + α , α + 2 α , α + 3 α , α + 3 α } (see Subsection 10.9 of [17]).Note that θ = 2 α + 3 α is the highest root of G and that the bilinear form ( ·|· ) on G (3) is normalized such that ( θ | θ ) = 2. We now let e θ be a highest root vector G .Then the following result has been proved in Theorem 5.4.1 and Corollary 5.4.3 of [18]. Theorem 6.1.
The vertex operator superalgebra L G (3) (1 , is isomorphic to V ( G (3)) /I ,where I is the submodule generated by e θ ( − . The Zhu’s algebra A ( L G (3) (1 , L G (3) (1 ,
0) hasbeen determined in Subsection 5.6.2 of [18].
Proposition 6.2.
The algebra A ( L G (3) (1 , is isomorphic to U ( G (3)) / ( e θ ) . We now classify irreducible L G (3) (1 , Proposition 6.3. L G (3) (1 , is the only irreducible L G (3) (1 , -module. Proof:
Let M be an irreducible L G (3) (1 , M may be viewed as anirreducible [ G (3)-module of level 1. Since M is an irreducible L G (3) (1 , M hasthe decomposition M = M λ ∈ C M λ , where M λ = { w ∈ M | L (0) w = λw } and dim M λ < ∞ for any λ ∈ C . Let λ be anumber such that M λ = 0 but M λ − n = 0 for n ∈ Z > . By Theorem 2.2, M λ isan A ( L G (3) (1 , M λ may be viewed as a finitedimensional G (3)-module. This implies that M is an irreducible highest weight moduleof [ G (3).We next show that M is an integrable [ G (3)-module, that is, M is an integrable c G -module and locally finite with respect to sl (cf. [24]). By Theorem 5.3.1 of [18], M is an integrable c G -module. Note that M λ is closed under the action of G (3). Hence,the action of G (3) on M is locally finite. Thus, M is an integrable irreducible highestweight [ G (3)-module. On the other hand, by Remark 6.3 of [24], L G (3) (1 ,
0) is the onlyintegrable irreducible highest weight [ G (3)-module of level 1. Thus, M is isomorphic to L G (3) (1 , (cid:3) As a corollary, we have the following result.
Corollary 6.4.
Let M = ⊕ n ∈ Z ≥ M λ + n be an L G (3) (1 , -module, where M λ + n = { w ∈ M | L (0) w = ( λ + n ) w } and λ ∈ C . Then λ = 0 and M λ contains an A ( L G (3) (1 , -submodule isomorphic to C . Proof:
Since M is an L G (3) (1 , M λ is a finite dimensional A ( L G (3) (1 , A ( L G (3) (1 , N of M λ . Let W be the L G (3) (1 , M generated by N . Then W isalso an L G (3) (1 , U be the sum of all the L G (3) (1 , W which have zero intersection with N . It follows that W/U is an irreducible L G (3) (1 , W/U is isomorphic to L G (3) (1 , µ ∈ C , set( W/U ) µ = { w ∈ W/U | L (0) w = µw } . Note that ( W/U ) λ ∼ = N . Therefore, N viewed asan A ( L G (3) (1 , C . It follows from Theorem 2.2 that λ = 0. (cid:3) To prove that the category of L G (3) (1 , Lemma 6.5.
Let V be a Z -graded vertex operator superalgebra, v be a homogeneousvector in V , and → M → M → M → be a short exact sequence of weak V -modules. Suppose that v wt v − acts semisimply on both M and M . Then M is adirect sum of generalized eigenspaces for v wt v − . Furthermore, the eigenspace of M ofeigenvalue h is the image of the generalized eigenspace of M of eigenvalue h . We also need the following result.
ODULE CATEGORY AND C -COFINITENESS OF AFFINE VOSAS 17 Lemma 6.6.
Let → L G (3) (1 , → M → L G (3) (1 , → be a short exact sequenceof weak L G (3) (1 , -modules. Then M viewed as an L G (3) (1 , -module is isomorphic to L G (3) (1 , ⊕ L G (3) (1 , . Proof:
By Lemma 6.5, M has the following decomposition M = M λ ∈ Z ≥ M λ , where M λ = { w ∈ M | ( L (0) − λ ) k w = 0 for some k } and dim M λ < ∞ for any λ ∈ Z ≥ .By Theorem 2.2, we have M is a two-dimensional A ( L G (3) (1 , M viewed as an A ( L G (3) (1 , M = C ⊕ C . Let v be the image of the vacuum vector of L G (3) (1 ,
0) in M , and w bea vector of M such that M = C v ⊕ C w . By Proposition 6.2, M may be viewed as afinite dimensional G (3)-module. Therefore, M viewed as a G (3) ¯0 = sl ⊕ G -module iscompletely reducible. Note that C v is an A ( L G (3) (1 , M . This forcesthat the action of G (3) ¯0 = sl ⊕ G on M must be trivial. In particular, h ⊕ h actstrivially on M . As a consequence, G (3) ¯1 = V ⊗ V must act trivially on M . Thus, M viewed as an A ( L G (3) (1 , M = C ⊕ C .We now let W be the L G (3) (1 , M generated by w . Then W also hasthe following decomposition W = M λ ∈ Z ≥ W λ , where W λ = { w ∈ W | ( L (0) − λ ) k w = 0 for some k } . Since C w is an A ( L G (3) (1 , M , we have W = C w by Proposition 4.1 of [11]. Since N is irreducible,we have N ∩ W = 0. This implies that W is isomorphic to L G (3) (1 , M viewedas an L G (3) (1 , L G (3) (1 , ⊕ L G (3) (1 , (cid:3) As a consequence, by the similar argument as that in Lemma 2.6 of [1], we have thefollowing
Lemma 6.7.
Let → ⊕ i ∈ I N i → M → L G (3) (1 , → be a short exact sequence ofweak L G (3) (1 , -modules. Suppose that each N i is isomorphic to L G (3) (1 , . Then M viewed as an L G (3) (1 , -module is isomorphic to ( ⊕ i ∈ I N i ) ⊕ L G (3) (1 , . We are now ready to prove the main result in this section.
Theorem 6.8.
The category of L G (3) (1 , -modules is semisimple. Proof:
Let M be an L G (3) (1 , M has the following decomposition M = M λ ∈ C M λ , where M λ = { w ∈ M | ( L (0) − λ ) w = 0 } and dim M λ < ∞ for any λ ∈ C . Let N be thesum of all irreducible L G (3) (1 , M . If M = N , let λ be a number suchthat M λ = N λ but M λ − n = N λ − n for n ∈ Z > . Then M/N is an L G (3) (1 , λ ∈ C , set ( M/N ) λ = { w ∈ M/N | ( L (0) − λ ) w = 0 } . Then we have ( M/N ) λ = 0and ( M/N ) λ − n = 0 for n ∈ Z > . By Corollary 6.4, we have λ = 0. Moreover, ( M/N ) λ has a 1-dimensional A ( L G (3) (1 , C v be a 1-dimensional A ( L G (3) (1 , M/N ) λ , and W be an L G (3) (1 , M/N generated by v . We next prove that W is anirreducible L G (3) (1 , U be a proper L G (3) (1 , W . Note that U has the following decomposition U = M λ ∈ C U λ , where U λ = { w ∈ U | ( L (0) − λ ) w = 0 } and dim U λ < ∞ for any λ ∈ C . Let µ be anumber such that U µ = 0 but U µ − n = 0 for n ∈ Z > . By Corollary 6.4, we have µ = 0.This forces that U = W , a contradiction. Thus, W is an irreducible L G (3) (1 , W is isomorphic to L G (3) (1 , M be a submodule of M such that M contains N and M /N is isomorphicto W . By Lemma 6.7, M is isomorphic to N ⊕ L G (3) (1 , M = N and M is completely reducible. (cid:3) To summarize, by Theorems 5.6, 6.8, L G (3) (1 ,
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Email address : [email protected] Xingjun Lin, School of Mathematics and Statistics, Wuhan University, Wuhan 430072,China.
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