aa r X i v : . [ m a t h . QA ] F e b GALILEAN W ALGEBRA
GORDAN RADOBOLJA
Abstract.
Galilean W vertex operator algebra G W ( c L , c M ) is constructed as a universalenveloping vertex algebra of certain non-linear Lie conformal algebra. It is proved that thisalgebra is simple by using determinant formula of the vacuum module. Reducibility criterionfor Verma modules is given, and the existence of subsingular vectors demonstrated. Freefield realisation of G W ( c L , c M ) and its highest weight modules is obtained within a rank 4lattice VOA. Introduction
Galilean W –algebras have been studied extensively by physicists in the past decade (seefor example [1], [6], [8], [9], [13], [18], [20] and references therein). Given an infinite-dimensional W -algebra with generators W , . . . , W k of conformal weights w , . . . , w k theassociated Galilean algebra is generated by W ′ , . . . , W ′ k , W , . . . , W k of conformal weights w , . . . , w k , w , . . . , w k , such that h W , . . . , W k i is a commutative subalgebra on which all W ′ i act. Moreover, the relations between W ′ i and W ′ j as well as relations between W ′ i and W j resemble the original relations between W i and W j . This new algebra is obtained througha process called Galilean contraction . Roughly speaking, one considers a tensor product oftwo copies of the original algebra (with arbitrary central charges) and takes a non-relativisticlimit (cf. [18]).The most basic example is a Galilean conformal algebra (GCA), also known as BMS -algebra (Bondi-Metzner-Sachs) which comes from contracting the Virasoro algebra. In math-ematical literature GCA first appeared in [22] where it was called the W (2 ,
2) (Lie and vertexoperator) algebra. Here ”(2,2)” denotes conformal weights of two generators. This algebrais constructed by adjoining to the Virasoro algebra its ”commutative double”, i.e. it is adirect sum of (either Lie or vertex operator algebra) Vir and its adjoint representation. Thisis analogous to construction of Takiff algebras in finite-dimensional case: Vir ⊗ C ( C [ x ] / ( x ))with brackets [ a ⊗ x i , b ⊗ x j ] = [ a, b ] ⊗ x i + j , a, b ∈ Vir. Free field realisation of GCA wasobtained by means of βγ system in [9], albeit only for central charge c L = 26. Bosonic free Date : February 3, 2021.2010
Mathematics Subject Classification.
Primary 17B69; Secondary 17B68, 81R10.
Key words and phrases.
Galilean algebras, W algebras. W ALGEBRA field realisation for arbitrary non-zero central charge was later obtained in [2]-[4] and its rep-resentation theory has been developed in many papers. We recall the most important resultsin Subsection 4.1.Galilean W or BMS - W algebra was originally introduced in [6]. In physics sense, theconstruction of this algebra follows the same prescription as GCA (cf. [18]) – contraction of atensor product of two copies of Zamolodchikov’s W algebra. Free field realisation for centralcharge c L = 100 was obtained by double βγ system in [9]. Mathematically however, thingsare more complicated. First of all, W is not a Lie algebra (quadratic terms appear whencommuting the operators). Furthermore, from the OPE relations immediately follows thatthe Galilean W is not an extension of W . Still, there is a ”commutative double” whichmakes handling this algebra somewhat easier.The aim of this paper is to give a mathematically rigorous definition of Galilean W algebraand to initiate the study of highest weight representations. We give this definition by usingKac - De Sole language of non-linear conformal Lie algebras (NLCA) in Section 2. Theuniversal enveloping vertex algebra of presented NLCA is precisely the Galilean W VOAfrom [6] (up to normalisation). By choosing a suitably ordered basis of the Verma modulewe utilise the commutative part and reduce the problem of finding zeroes of determinantformula to a very simple matrix (27) of rank 2. This enables classification of irreducibleVerma modules. As in the case of GCA, reducibility depends only on highest weights andcentral charge corresponding to the action of commuting generators (cf. Theorem 3.4). Bycalculating singular and subsingular vectors of conformal weights 1 we give a basis of thevacuum module. Considering its determinant formula we also prove that (universal) Galilean W is a simple algebra (Theorem 3.7). The method used in this section should be easilyapplied to other Galilean algebras, and we expect that analogous results hold in general.Much like in the case of Virasoro and GCA, the structure and representation theory ofGalilean W algebra is rather different than that of W (cf. [11], [12], [16], [21]).It is well known that Quantum Drinfeld-Sokolov reduction of an affine Kac-Moody alge-bra ˆ sl N produces the W -algebra W N . The resulting algebra can, in turn, be realised as asubalgebra of M (1) N − , the Heisenberg algebra of rank N −
1, and M (1) N − is constructedover a lattice L N − with Gram matrix equal to the Cartan matrix of sl N (cf. [19]). SinceGalilean W N algebra is obtained from a tensor product of two copies of W N , it is naturalto consider its free field realisation within a product of two copies of Heisenberg algebrasused in realisation of W N . In Section 4 we start with a rank 4 lattice which is a productof two lattices whose Gram matrices are equal to Cartan matrices of sl . In the associatedrank 4 Heisenberg algebra we detect a family of subalgebras isomorphic to Galilean W al-gebras with arbitrary non-zero central charges. Furthermore by using the associated latticeVOA, we present a realisation of highest weight modules in Section 5. The highest weights ALILEAN W ALGEBRA 3 are parametrised in such a way that reducibility of Verma modules corresponds to positiveintegral values of the first parameter (Proposition 5.1). This resembles the GCA case whichis recalled in Subsection 4.1. We expect that the positive integral values of other parametersdetect subsingular vectors in general. This is verified on (sub)singular vectors at level one(Example 5.3).The author is partially supported by the QuantiXLie Centre of Excellence, a project cof-financed by the Croatian Government and European Union through the European RegionalDevelopment Fund - the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004).The author would like to thank Draˇzen Adamovi´c for useful comments and discussions andSimon Wood for bringing the OPE package for
Mathematica to my attention.2.
Definitions
We start by recalling the notion of non-linear Lie conformal algebra introduced in [15].
Definition 2.1.
A Lie conformal algebra is a C [ D ] -module R with a C -linear map [ λ ] : R ⊗ R → R [ λ ] satisfying the following axioms [ Da λ b ] = − λ [ a λ b ] [ a λ Db ] = ( λ + D )[ a λ b ] , (1) [ a λ b ] = − [ b − λ − D a ] , (2) [ a λ [ b µ c ]] − [ b µ [ a λ c ]] − [[ a λ b ] λ + µ c ] = 0 . (3)To any Lie conformal algebra R one canonically associates V ( R ), the universal envelopingvertex algebra of R . For a, b, c ∈ V ( R ) we have[ a λ b ] = Res z e zλ Y ( a, z ) b = X n ∈ Z ≥ λ n n ! a ( n ) b ∈ C [ λ ] , (4) : ab : − : ba := Z − D [ a λ b ] d λ, (5) : (: ab :) c : − : a (: bc :) :=: (cid:18)Z D a d λ (cid:19) [ b λ c ] : + : (cid:18)Z D b d λ (cid:19) [ a λ c ] :(6) [ a λ : bc :] =: [ a λ b ] c : + : b [ a λ c ] : + Z λ [[ a λ b ] µ c ] d µ, (7) [: ab : λ c ] =: ( e D∂ λ a )[ b λ c ] : + : ( e D∂ λ b )[ a λ c ] : + Z λ [ b µ [ a λ − µ c ]] d µ . (8)where a ( n ) b = Res z Y ( a, z ) b denotes the n -th product and : ab := a ( − b is a normally or-dered product of fields Y ( a, z ) and Y ( b, z ). Note that (4) encodes the commutator formula GALILEAN W ALGEBRA [ Y [ a, z ] , Y ( b, w )] also known as operator product extension (OPE) a ( z ) b ( w ) ∼ X n ∈ Z > ( a ( n ) b )( w )( z − w ) n . Infinite-dimensional Lie algebras like Virasoro, Heisenberg and affine Kac-Moody algebrasgive rise to Lie conformal algebras whose universal enveloping algebras are precisely theuniversal vertex algebras associated to starting Lie algebras.
Example 2.2. If R = C [ D ] L such that [ L λ L ] = ( D + 2 λ ) L + c λ , then V ( R ) = Vir c .Let R G = C [ D ] L ⊕ C [ D ] M such that [ L λ L ] = ( D + 2 λ ) L + c L λ , [ L λ M ] = ( D + 2 λ ) M + c M λ , [ M λ M ] = 0 . Then V ( R G ) = L W (2 , ( c L , c M ) is GCA with central charge ( c L , c M ) . However, many vertex algebras are not associated to Lie conformal algebras because the n -th products of some of their elements are nonlinear, i.e. they contain normally orderedproducts. For this reason one needs to extend the λ -bracket to T ( R ), the tensor algebra of R .For a ∈ R and B ∈ T ( R ) define : aB := a ⊗ B so that D (1) = 0, D (: AB :) =: ( DA ) B :+ : AD ( B ) : for A, B ∈ T ( R ) and then extend the λ -bracket to [ λ ] : R ⊗ R → C [ λ ] ⊗ T ( R )using (6-8). In order to deal with the Jacobi identity (3) we assume that R is Z -graded byconformal weights R = L ∆ ∈ Z ≥ R [∆] such that∆( DA ) = ∆( a ) + 1 , ∆( a ( n ) b ) = ∆( a ) + ∆( b ) − n − n ∈ Z ≥ . Extending the grading to T ( R ) define the subspace M ∆ ( R ) ⊂ T ( R ) ≤ ∆ spannedby all elements X ⊗ ( b ⊗ c − c ⊗ b ) ⊗ Y − X ⊗ (cid:18) : (cid:18)Z − D [ b λ c ] d λ (cid:19) Y : (cid:19) where b, c ∈ R , X, Y ∈ T ( R ) and ∆( X ⊗ b ⊗ c ⊗ Y ) ≤ ∆. Definition 2.3.
A non-linear Lie conformal algebra (NLCA) is a Z -graded C [ D ] -module R = L ∆ ∈ Z ≥ R [∆] with a C -linear map [ λ ] : R ⊗ R → C [ λ ] ⊗ T ( R ) satisfying (1-2), (9) and ∆([ a λ b ]) < ∆( a ) + ∆( b ) , (10) [ a λ [ b µ c ]] − [ b µ [ a λ c ]] − [[ a λ b ] λ + µ c ] ∈ C [ λ, µ ] ⊗ M ∆ ′ ( R ) , (11) for all a, b, c ∈ R , where ∆ ′ < ∆( a ) + ∆( b ) + ∆( c ) . ALILEAN W ALGEBRA 5
To each NLCA R one associates a universal enveloping vertex algebra V ( R ) = T ( R ) / M ( R )which is freely generated by R . Conversely, if V is a vertex algebra freely generated by a C [ D ]-submodule R ⊂ V , then there is a NLCA structure on R and V ∼ = V ( R ). See [15] fordetails.The first example of NLCA comes from a well known Zamolodchikov’s W algebra. VOA W ( c ) is generated by a conformal field ω ( z ) = P L ( n ) z − n − and a primary field W ( z ) = P W ( n ) z − n − satisfying the following OPE: ω ( z ) ω ( w ) ∼ c/ z − w ) + 2 ω ( w )( z − w ) + ∂ω ( w ) z − wω ( z ) W ( w ) ∼ W ( w )( z − w ) + ∂W ( w ) z − wW ( z ) W ( w ) ∼ c/ z − w ) + 2 ω ( w )( z − w ) + ∂ω ( w )( z − w ) ++ 1( z − w ) (cid:18) ∂ ω ( w ) + 2 β Λ( w ) (cid:19) + 1 z − w (cid:18) ∂ ω ( w ) + β∂ Λ( w ) (cid:19) where Λ( z ) =: ω ( z ) : − ∂ ω ( z ) and β = c .Let R = C [ D ] L ⊕ C [ D ] W be a NLCA with the following λ -brackets:[ L λ L ] = ( D + 2 λ ) L + c λ , [ L λ W ] = ( D + 3 λ ) W, [ W λ W ] = c λ + (cid:18) λ λ D + 3 λ D + 115 D (cid:19) L + 165 c + 22 ( D + 2 λ ) (cid:18) L − D L (cid:19) . Then V ( R ) is precisely W ( c ) (cf. [15]).Now we define the Galilean W algebra. Definition 2.4.
Let c L , c M ∈ C , c M = 0 . The Galilean W NLCA is defined as G W ( c L , c M ) = C [ D ] L ⊕ C [ D ] W ⊕ C [ D ] M ⊕ C [ D ] V, GALILEAN W ALGEBRA such that ∆( L ) = ∆( M ) = 2 , ∆( W ) = ∆( V ) = 3 and with the following non-trivial λ -brackets [ L λ L ] = ( D + 2 λ ) L + c L λ , (12) [ L λ M ] = ( D + 2 λ ) M + c M λ , (13) [ L λ W ] = ( D + 3 λ ) W, (14) [ L λ V ] = ( D + 3 λ ) V, (15) [ M λ W ] = ( D + 3 λ ) V, (16) [ W λ W ] = c L λ + (cid:18) λ λ D + 3 λ D + 115 D (cid:19) L +(17) + 325 c M ( D + 2 λ ) (cid:18) LM − D M (cid:19) − c M (cid:18) c L + 445 (cid:19) ( D + 2 λ ) M , [ W λ V ] = c M λ + (cid:18) λ λ D + 3 λ D + 115 D (cid:19) M + 165 c M ( D + 2 λ ) M . (18)Proving that the axioms of NLCA (in particular the Jacobi identity) hold is a straight-forward, but rather tedious task (see Appendix A.1). Another way of showing that thisdefinition is correct is by obtaining a free field realisation. This is presented in Section 4.For simplicity, we shall use the same notation G W ( c L , c M ) for the associated universalenveloping vertex algebra which is generated by fields ω ( z ) = X n ∈ Z L ( n ) z − n − ,W ( z ) = X n ∈ Z W ( n ) z − n − ,M ( z ) = X n ∈ Z M ( n ) z − n − ,V ( v ) = X n ∈ Z V ( n ) z − n − . Let Λ( k ) = X i ∈ Z ≥ : L ( − i ) M ( k + i ) : −
310 ( k + 2)( k + 3) M ( k ) , Θ( k ) = X i ∈ Z ≥ : M ( i ) M ( k − i ) : . ALILEAN W ALGEBRA 7
Then the components of these fields satisfy the following non-trivial commutation relations:[ L ( n ) , L ( m )] = ( n − m ) L ( n + m ) + δ n + m, n ( n − c L (19) [ L ( n ) , W ( m )] = (2 n − m ) W ( n + m )(20) [ L ( n ) , M ( m )] = ( n − m ) M ( n + m ) + δ n + m, n ( n − c M (21) [ L ( n ) , V ( m )] = (2 n − m ) V ( n + m )(22) [ M ( n ) , W ( m )] = (2 n − m ) V ( n + m )(23) [ W ( n ) , W ( m )] = n − m (cid:18) (2 n + 2 m − nm − L ( n + m ) + 192 c M Λ( n + m )+(24) − c M ( c L + 445 )Θ( n + m ) (cid:19) + δ n + m, n ( n − n − c L [ W ( n ) , V ( m )] = n − m (cid:18) (2 n + 2 m − nm − M ( n + m ) + 96 c M Θ( n + m ) (cid:19) +(25) + δ n + m, n ( n − n − c M . This agrees with algebra introduced in [6] up to a normalisation factor 1 /
30 (as in [9]). Noticethat ω ( z ) and M ( z ) generate a subalgebra isomorphic to the Galilean conformal algebra withcentral charge ( c L , c M ). However, ω ( z ) and W ( z ) do not generate a copy of W due to (24).A natural question arises: can we define a Galilean algebra in such a way that both Virasoro,and W are its subalgebras acting on a commutative part? It turns out that this is notpossible. Due to non-linearity, the Jacobi identity for such λ -brackets would not hold (seeAppendix A.2). Corollary 2.5.
We havechar q G W ( c L , c M ) = (1 − q ) Y n ≥ (1 − q n ) − . Proof.
We fix an ordering
V > M > W > L , and obtain a PBW basis of the universalenveloping vertex algebra G W ( c L , c M ) (cf. [15]) which consists of monomials V ( − p v ) · · · V ( − p ) M ( − r m ) · · · M ( − r ) W ( − s m ) · · · W ( − s ) L ( − t l ) · · · L ( − t ) such that p k +1 ≥ p k ≥ r k +1 ≥ r k ≥ s k +1 ≥ s k ≥ t k +1 ≥ t k ≥
2. Then the q -characterformula is char q L ( c ,
0) = (1 − q ) (1 − q ) Y n ≥ − q n ) which proves the assertion. (cid:3) GALILEAN W ALGEBRA Highest weight modules
Let M be an ordinary module over the VOA V . In particular M = L h ∈ C M h , where M h = { m ∈ M : L (0) v = hv } is a subspace of conformal weight h , dim M h < ∞ and for each u ∈ V and v ∈ M we have u ( n ) v = 0 for n >> v ∈ M h is called singular vector if for each u ∈ V and n ∈ Z ≥ wehave u ( n ) v = δ n, h u v for h u ∈ C . If M is generated by a singular vector v we say that M is ahighest weight module, and call v a highest weight vector. A homogeneous vector v ∈ M ( h )is called subsingular vector if there exists a submodule N ⊂ M such that v + N is singularin a quotient module M/N .In the following we study highest weight modules over G W ( c L , c M ), so we assume that M = L n ∈ Z ≥ M h + n for some h ∈ C . Since L (0), M (0), W (0) and V (0) are mutually com-muting operators acting on (a finite-dimensional complex space) M h , there exists a commoneigenvector which generates a submodule with one-dimensional subspace of weight h . There-fore we restrict our study to the case when M h is one-dimensional. Remark 3.1.
The construction of highest weight modules in classical case relies on Lietheory. The Verma module is defined either as a quotient of universal enveloping algebra ofa given Lie algebra g , or equivalently, as a module induced from Borel subalgebra, i.e. usinga triangular decomposition g = g − ⊕ g ⊕ g + . Since G W ( c L , c M ) is not a (linear) Lie algebraand does not have a natural triangular decomposition we do not have these tools available.However, the existence of universal highest weight modules can be obtained by applying Zhu’stheory. We sketch the idea without going into details.It is well known in vertex algebra theory that for each VOA V there exists an associativealgebra A ( V ) called Zhu’s algebra of V which controls the representation theory of V in thefollowing sense. For a V -module M = L n ∈ Z ≥ M h + n , M h is an A ( V ) -module. Conversely,every A ( V ) -module is a top level of some V -module. Obviously, one-dimensional A ( V ) -modules correspond to highest weight V -modules.It is not difficult to show that A ( G W ( c L , c M )) is a commutative algebra with 4 generators.We will show in Section 4 that G W ( c L , c M ) is a subalgebra of a rank 4 Heisenberg algebra M (1) . Highest weight M (1) -modules then provide a realisation of highest weight G W ( c L , c M ) -modules. The top levels of these modules are precisely the one-dimensional A ( G W ( c L , c M )) -modules and their existence then yields the existence of the Verma modules over G W ( c L , c M ) . Let h := ( h L , h W , h M , h V ) ∈ C be arbitrary scalars and c := ( c L , c M ). The Verma moduleof highest weight h , denoted by V ( c , h ) is a highest weight G W ( c L , c M )-module generatedby highest weight vector v h such that L (0) v h = h L v h , W (0) v h = h W v h , M (0) v h = h M v h , V (0) v h = h V v h . ALILEAN W ALGEBRA 9
Its basis consists of vectors V ( − i v ) · · · V ( − i ) M ( − j m ) · · · M ( − j ) W ( − k w ) · · · W ( − k ) L ( − n l ) · · · L ( − n ) v h (26)such that v, m, w, l ∈ Z ≥ , i v ≥ · · · ≥ i ≥ j m ≥ · · · ≥ j ≥ k w ≥ · · · ≥ k ≥ n l ≥ · · · ≥ n ≥
1. We have V ( c , h ) = M n ∈ Z ≥ V ( c , h ) n , V ( c , h ) n = { v ∈ V ( c , h ) : L (0) v = ( h L + n ) v } . Let P ( n ) denote the partition function on Z > . Thendim V ( c , h ) n = X i,j,k ≥ P ( i ) P ( j ) P ( k ) P ( n − i − j − k )so char q V ( c , h ) = q h L Y n ≥ − q n ) . In order to classify irreducible Verma modules we need to consider the determinant formulaassociated to this form. Let V be a VOA, M = ⊕ n ∈ Z ≥ M n an ordinary weight V -module,and M ∗ = ⊕ n ∈ Z ≥ M ∗ n its restricted dual. Let h· , ·i : M ∗ × M → C denote the natural pairingand Y M ∗ : V → End M ∗ [[ z, z − ]] be a linear map such that h Y M ∗ ( v, z ) w ′ , w i = h w ′ , Y M ( e zL (1) ( − z − ) L (0) v, z − ) w i for v ∈ V , w ∈ M , w ′ ∈ M ∗ . Then ( M ∗ , Y M ∗ ) is a V -module, called a contragredient of M .In case of V = G W we have L ( n ) ∗ = L ( − n ) ,W ( n ) ∗ = − W ( − n ) ,M ( n ) ∗ = M ( − n ) ,V ( n ) ∗ = − V ( − n ) , Λ( n ) ∗ = Λ( − n ) , Θ( n ) ∗ = Θ( − n ) . Lemma 3.2.
Let L ( c , h ) denote the irreducible quotient of V ( c , h ) . Then L ( c , h ) ∗ = L ( c , h ∗ ) where h ∗ = ( h L , − h W , h M , − h V ) . Natural pairing with a contragredient module induces a symmetric non-degenerate invari-ant bilinear form on V ( c , h ) such that h v h | v h i = 1 , h x.v h | y.v h i = h v h | x ∗ y.v h i . W ALGEBRA
Since h V ( c , h ) n | V ( c , h ) m i = 0 for n = m we focus on det h V ( c , h ) n | V ( c , h ) n i . We are onlyinterested in its zeros so we will not calculate exponents of all the different factors in this de-terminant. Instead we introduce the ordering on the chosen basis of V ( c , h ) n and decomposethe matrix h V ( c , h ) n | V ( c , h ) n i to a tensor product of block triangular matrices, thus reducingthe problem to finding determinant of much simpler matrices. In the following subsection weshow that this problem ultimately reduces to calculation of determinant D n := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h L ( − n ) v h | M ( − n ) v h i h L ( − n ) v h | V ( − n ) v h ih W ( − n ) v h | M ( − n ) v h i h W ( − n ) v h | V ( − n ) v h i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (27)Furthermore, we use the same method to describe the vacuum module L ( c ,
0) and provesimplicity of G W ( c L , c M ) in Subsection 3.2.3.1. Determinant formula and classification of irreducible Verma modules.
Insteadof standard PBW basis (26) we will work with the following:
Lemma 3.3. B n = n V ( − n ) v n M ( − n ) m n · · · V ( − v M ( − m W ( − n ) w n L ( − n ) l n · · · (28) · · · W ( − w L ( − l v h : n X i =1 i ( v i + m i + w i + l i ) = n ) is a basis of V ( c , h ) n .Proof. For a monomial x as in (26) we set d ( x ) = w + l . By induction on d we show that eachmonomial (26) is spanned by B n . Basis for d ( x ) = 0 is trivial since then x is a monomialwith commuting factors. Assume that d ( x ) > k < n l . Then x equals V ( − i v ) · · · V ( − i ) M ( − j m ) · · · M ( − j ) W ( − k w ) · · · L ( − n l ) W ( − k ) · · · L ( − n ) v h +(29)+(2 n l − k ) V ( − i v ) · · · V ( − i ) M ( − j m ) · · · M ( − j ) W ( − k w ) · · · W ( − k − n l ) · · · L ( − n ) v h (30)After commuting the factors of (30) we obtain a sum of monomials x ′ with d ( x ′ ) < d ( x ) andapply inductive hypothesis. By repeating the proces with (29), one obtains the proof. (cid:3) Now we introduce the ”commutative degree” of a basis monomial: for x ∈ B n letdeg c x = n X i =1 i ( v i + m i ) ,V ( c , h ) kn = span C { x ∈ B n : deg c x = k } ,B kn = B n ∩ V ( c , h ) kn . ALILEAN W ALGEBRA 11
Then V ( c , h ) n = n M k =0 V ( c , h ) kn dim V ( c , h ) kn = P ( k ) P ( n − k ) = dim V ( c , h ) n − kn , where P ( m ) = P mi =0 P ( i ) P ( m − i ).For x ∈ B n we write x = x c x nc v h , where deg c x c = deg c x , and deg c x nc = 0. In otherwords, x c is a product of factors M ( − i ) and V ( − i ), while x nc is a product of factors L ( − i )and W ( − i ). Then h x | y i = h ( y c ) ∗ x nc v h | ( x c ) ∗ y nc v h i = h ( y c ) ∗ x nc v h | v h i · h v h | ( x c ) ∗ y nc v h i = h v h | ( x nc ) ∗ y c v h i · h ( y nc ) ∗ x c v h | v h i . (31)We may order the elements of B n so that for x, y ∈ B n x ≺ y if deg c x < deg c y . From (31)follows that h V ( c , h ) kn | V ( c , h ) ln i = 0 if k + l = n so the Gram matrix of h V ( c , h ) n | V ( c , h ) n i is block triangular with the only nontrivial blocks h V ( c , h ) kn | V ( c , h ) n − kn i , k = 0 , . . . , n . Fur-thermore, (31) shows that the Gram matrix of h V ( c , h ) kn | V ( c , h ) n − kn i is a tensor product ofmatrices h V ( c , h ) n − k | V ( c , h ) n − kn − k i and h V ( c , h ) kk | V ( c , h ) k i . Therefore the problem of cal-culating (the zeroes of) det h V ( c , h ) n | V ( c , h ) n i reduces to finding determinant of matrix A = h V ( c , h ) n | V ( c , h ) nn i which represents the action of monomials in W ( i ) and L ( j ) onmonomials in V ( − k ) and M ( − l ).Let us introduce a suitable ordering on B nn and B n which makes A block-triangular. For¯ k = ( k , . . . , k n ) , ¯ l = ( l , . . . , l n ) ∈ ( Z ≥ ) n we define:¯ k ≺ ¯ l if k i = l i for i > j and k j > l j ;¯ k + ¯ l = ( k + l , . . . , k n + l n ) . We set P n = { (¯ k, ¯ l ) ∈ ( Z ≥ ) n × ( Z ≥ ) n : n X i =1 i ( k i + l i ) = n } and say that (¯ k, ¯ l ) is of type t (¯ k, ¯ l ) = ¯ k + ¯ l . Then there is a natural one to one correspondencebetween P n and B nn [ V M ](¯ v, ¯ m ) := V ( − n ) v n M ( − n ) m n · · · V ( − v M ( − m v h , i.e. between P n and B n [ W L ]( ¯ w, ¯ l ) := W ( − n ) w n L ( − n ) l n · · · W ( − w L ( − l v h . W ALGEBRA
For (¯ k, ¯ l ) , (¯ k ′ , ¯ l ′ ) ∈ P n we define order(¯ k, ¯ l ) ≺ (¯ k ′ , ¯ l ′ ) if t (¯ k, ¯ l ) ≺ t (¯ k ′ , ¯ l ′ )or t (¯ k, ¯ l ) = t (¯ k ′ , ¯ l ′ ), and ¯ k ≺ ¯ k ′ . This induces partial orders on B nn and B n . It is clear from (21)-(25) that t (¯ k, ¯ l ) ≺ t (¯ k ′ , ¯ l ′ ) ⇒ h [ W L ](¯ k, ¯ l ) | [ V M ](¯ k ′ , ¯ l ′ ) i = 0so A is block-triangular with diagonal blocks of the type h V ( c , h ) n (¯ k + ¯ l ) | V ( c , h ) nn (¯ k + ¯ l ) i where V ( c , h ) nn (¯ k, ¯ l ) = span C { [ V M ](¯ k ′ , ¯ l ′ ) ∈ B nn : t (¯ k ′ , ¯ l ′ ) = t (¯ k, ¯ l ) } ,V ( c , h ) n (¯ k, ¯ l ) = span C { [ W L ](¯ k ′ , ¯ l ′ ) ∈ B n : t (¯ k ′ , ¯ l ′ ) = t (¯ k, ¯ l ) } . However, by construction we see that h V ( c , h ) n ( k , . . . , k n ) | V ( c , h ) nn ( k , . . . , k n ) i is a tensor product of matrices of type h V ( c , h ) n (0 , . . . , , k p , , . . . , | V ( c , h ) nn (0 , . . . , , k p , , . . . , i , p = 1 , . . . , n which correspond to the action of monomials W ( p ) k p − r L ( p ) r , r = 0 , . . . , k p on monomials V ( − p ) k p − r M ( − p ) r , r = 0 , . . . , k i . Denote W ( p ) V ( − p ) v h = av h , W ( p ) M ( − p ) v h = L ( p ) V ( − p ) v h = bv h , L ( p ) M ( − p ) v h = dv h . Then the element at intersection of i th row and j th column of this matrix is L ( p ) i − W ( p ) n +1 − i V ( − p ) n +1 − j M ( − p ) j − v h == i − X k =0 a n − i − j + k +2 b i + j − k − d k ( n + 1 − j )!( j − (cid:18) n + 1 − ij − k − (cid:19)(cid:18) i − k (cid:19) . By adding a linear combination of previous rows to i th row we obtain an equivalent matrixwith elements ( n + 1 − j )!( j − (cid:18) n + 1 − ij − i (cid:19) a n − i − j +2 b j − i ( ad − b ) i − i.e. a triangular matrix with determinant equal to( ad − b ) n ( n +1)2 n Y j =0 ( n − j )! j ! ALILEAN W ALGEBRA 13
The brackets (21)-(25) yield a = p (cid:18) (5 p − h M + 96 c M h M (cid:19) + p ( p − p − c M ,b = 3 ph V ,d = 2 ph M + p ( p − c M . From these considerations follows:
Theorem 3.4.
The Verma module V ( c , h ) is reducible if and only if h V = 64 (cid:16) h M + p − c M (cid:17) (cid:16) h M + p − c M (cid:17) c M (32) for some p ∈ Z > . In that case, there is a singular vector in V ( c , h ) pp , where p ∈ Z > is thelowest such that (32) holds. Remark 3.5.
This method of finding zeros of the determinant formula and thus classifyingirreducible Verma modules relies on the fact that h M ( z ) , V ( z ) i is a commutative subalgebra of G W ( c L , c M ) . Essentially the same method was used in classification of irreducibles over theGCA ( W (2 , ) in [22] and [14] where it was shown that the Verma module V W (2 , ( h L , h M ) is reducible if and only if h L ( − p ) v | M ( − p ) v i = p (cid:18) h M + p − c M (cid:19) = 0 for some p ∈ Z > . In that case there is a singular vector in V W (2 , ( h L , h M ) pp . Free fieldrealisation of highest weight modules and formula for singular vectors was obtained in [2] , [3] and [17] .More importantly, this method can be applied to other Galilean W-algebras. Simplicity of G W ( c L , c M ) . The submodule structure of reducible Verma module canbe complicated. As an example we present formulas for (sub)singular vectors of weight h L +1.In particular, we describe the vacuum module, and prove simplicity of G W ( c L , c M ). Example 3.6. If h V = h M (32 h M − c M )45 c M (i.e. p = 1 ) the singular vector from Theorem 3.4 isgiven by s.v = V ( − − s h M − c M )5 c M M ( − v h = (cid:18) V ( − − h V h M M ( − (cid:19) v h . We have W (0) s.v h = (cid:16) h W − h V h M (cid:17) s.v h , M (0) s.v h = h M v h , V (0) .v h = h V v h . Consider thedeterminant formula of the quotient module V ( c , h ) / h s.v h i . Factor corresponding to level one W ALGEBRA is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h L ( − v h | L ( − v h i h L ( − v h | M ( − v h i h L ( − v h | W ( − v h ih M ( − v h | L ( − v h i h M ( − v h | M ( − v h i h M ( − v h | W ( − v h ih W ( − v h | L ( − v h i h W ( − v h | M ( − v h i h W ( − v h | W ( − v h i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) == h M (cid:16) h L − h M c M (3 h L + ) + 16 h M c M ( c L + ) + 3 h W q c M (32 h M − c M ) (cid:17) . a: If h M = 0 , then s.v = (cid:16) V ( − − i √ M ( − (cid:17) v h . Furthermore, s .v = M ( − v h is a subsingular vector in V ( c , h ) such that ( W (0) − h W ) s .v h = 2 (cid:18) s − i √ s (cid:19) .v h . b: If (33) h L − h M c M (3 h L + 25 ) + 16 h M c M ( c L + 445 ) + 3 h W r c M (32 h M − c M ) = 0 then s .v h = (cid:18) W ( − − h V h M L ( − − c M (cid:18) h M h V c M ( c M h L − h M ( c L − − h V h M (cid:19) M ( − (cid:19) v h is a subsingular vector in V ( c , h ) such that ( M (0) − h M ) s .v h = s.v h , ( W (0) − h W ) s .v h = − h V h M s .v h + (cid:18) c M h V h M − c M ( h L − h M c M ( c L − h M h V (cid:19) s.v h , ( V (0) − h V ) s .v h = − h V h M s.v h . ab: If h M = 0 and h L = 3i p / h W then s .v h = (cid:18) W ( −
1) + i √ L ( − (cid:19) v h = (cid:18) W ( − − h W h L L ( − (cid:19) v h is a subsingular vector such that ( W (0) − h W ) s .v h = − h W h L s .v h − h W h L c M (cid:18) h L + 25 (cid:19) s .v h ,M (0) s .v h = s.v h ,V (0) s .v h = − h W h L s.v h . ALILEAN W ALGEBRA 15 abc: If h = 0 , then s .v = L ( − v is a subsingular vector such that M (0) s .v = s .v ,V (0) s .v = 2 s.v − i r s .v ,W (0) s .v = 2 s .v − i r s .v . In particular, V ( c , ⊂ h L ( − v i . Theorem 3.7.
Vertex algebra G W ( c L , c M ) is simple for all c L , c M ∈ C , c M = 0 .Proof. First we prove that V ( c , / h L ( − v i ∼ = G W ( c L , c M ). From commutator relationsfollows that V ( − v, W ( − v ∈ h L ( − v i so the set B ′ of monomials from B n , n ∈ Z ≥ suchthat v = m = w = l = v = w = 0 spans V ( c , / h L ( − v i . However this is preciselythe PBW basis of universal enveloping VOA G W ( c L , c M ).Now we follow the same procedure as in Subsection 3.1. One of the key facts in blockdiagonalisation of the Gram matrix of V ( c , h ) is that dim V ( c , h ) kn = dim V ( c , h ) n − kn . Thebasis B ′ retains this kind of symmetry so it is easy to see that determinant formula of thequotient module reduces to product of h L ( − v | M ( − v i = c M and determinants D n , n ∈ Z > (27). However, for h = 0 we have D n = n ( n − ( n − c M / G W ( c L , c M ) ∼ = V ( c , / h L ( − v i = L ( c , (cid:3) Remark 3.8.
Proving simplicity of VOA is generally much more difficult. For examplesee [7] for treatment of affine VOA, and the W -algebras obtained by the generalised quan-tised Drinfeld-Sokolov reduction (which includes W ). Determinant formula for vacuum W -module is considered in [10] . Free field realisation
We shall first recall the free field realisation of Galilean Virasoro algebra and its highestweight modules. This realisation was obtained using a rank 2 Heisenberg algebra and asso-ciated lattice VOA. Then we use the same idea to construct Galilean W as a subalgebra ofa rank 4 lattice VOA.4.1. Realisation of Galilean conformal algebra.
Free field realisation of GCA or W (2 , L = Z c + Z d be a rank 2 lattice such that h c | d i = 2 , h c | c i = h d | d i = 0 . W ALGEBRA
Let h = C ⊗ Z L and ˆ h = h ⊗ C [ t, t − ] ⊕ C K its affinization. For any h ∈ h we write h ( n ) for h ⊗ t n and we let h ( z ) = P n ∈ Z h ( n ) z − n − .We denote by M (1 , h ) the induced ˆ h -module U (ˆ h ) ⊗ U ( C [ t ] ⊗ h ⊕ C K ) C e h such that t C [ t ] ⊗ h acts trivially on e h , k (0) e h = h k | h i e h for k ∈ h and Ke h = e h . Then M (1) := M (1 ,
0) is a rank 2 Heisenberg vertex algebra generated by the fields h ( z ) , h ∈ h and M (1 , h ) , h ∈ h are irreducible M (1)-modules. Furthermore, the fields ω ( z ) = 12 c ( z ) d ( z ) + c L − ∂c ( z ) − ∂d ( z ) ,M ( z ) = − c W (cid:0) c ( z ) − ∂c ( z ) (cid:1) generate a vertex operator subalgebra of M (1) which is isomorphic to GCA L W (2 , ( c L , c M ).Define v p,r = e − p +12 d + (cid:16) ( p +1) cL − − p − r − (cid:17) c ,h L [ p, r ] = (1 − p ) c L −
224 + p p − r − ,h M [ p ] = 1 − p c M . Then we have L (0) v p,r = h L [ p, r ] v p,r , M (0) v p,r = h M [ p ] v p,r ,h L [ − p, − r −
2] = h L [ p, r ] , h M [ − p ] = h M [ p ] ,h L [ p, r ] + p = h L [ p, r − . Denote by F p,r = M (1) .v p,r . We fix central charge ( c L , c W ) and denote by V [ p, r ] (resp. L [ p, r ]) the Verma (resp. irreducible) module of highest weight ( h L [ p, r ] , h M [ p ]). Then wehave Theorem 4.1 ([17],[14],[2],[3]) . Let p > . (1) The Verma module V [ p, r ] is reducible if and only if p ∈ Z > . In that case, there is asingular vector u ′ p ∈ V [ p, r ] such that h u ′ p i ∼ = V [ p, r − . (2) u ′ p generates the maximal submodule in V [ p, r ] if and only if r ∈ Z > . (3) If r ∈ Z > then the maximal submodule in V [ p, r ] is generated by a subsingular vector u rp of conformal weight h p,r + pr = h p, − r . (4) F p,r ∼ = V [ p, r ] and L [ p, r ] ∼ = U ( W (2 , .v p,r < F − p, − r − ∼ = F ∗ p,r . In the following we aim to obtain analogous results for G W . ALILEAN W ALGEBRA 17
Remark 4.2.
GCA is realised in [2] , [3] , [4] as a subalgebra of Heisenberg-Virasoro VOAat level zero. One may also consider the N = 1 super GCA. Realisation for central charge c L = 11 was presented in [9] . The N = 1 super Heisenberg-Virasoro VOA was introducedin a recent paper [5] , and the full treatment of level zero should appear soon as well. Thiswill provide a natural framework for studying realisation of super GCA with arbitrary centralcharge. Realisation of Galilean W algebra. Let L = Z a + Z b + Z c + Z d be a rank 4 latticesuch that h a | b i = h c | d i = − , h a | c i = h a | d i = h b | c i = h b | d i = 0 , h x | x i = 2 , x ∈ { a, b, c, d } . Fix λ, µ ∈ C such that λ + iµ = 0 and let¯ a = a + i c, ¯ b = b + i d,λ = λ + i µ. Now we define the fields in M (1) which generate the Galilean W algebra G W ( c L , c M ). Let ω ( z ) = 13 (cid:18) a ( z ) + a ( z ) b ( z ) + b ( z ) + c ( z ) + c ( z ) d ( z ) + d ( z ) (cid:19) ++ λ∂a ( z ) + λ∂b ( z ) + µ∂c ( z ) + µ∂d ( z ) W ( z ) = i27 λ √ a ( z ) − b ( z ))(¯ a ( z ) + 2¯ b ( z ))(2¯ a ( z ) + ¯ b ( z )) ++(¯ a ( z ) − ¯ b ( z ))( a ( z ) + 2 b ( z ))(2¯ a ( z ) + ¯ b ( z )) ++(¯ a ( z ) − ¯ b ( z ))(¯ a ( z ) + 2¯ b ( z ))(2 a ( z ) + b ( z ))) ++9 λ (cid:18) ∂ ¯ a ( z )(2¯ a ( z ) + ¯ b ( z )) − ∂ ¯ b ( z )(¯ a ( z ) + 2¯ b ( z )) (cid:19) ++9 λ (cid:18) ∂a ( z )(2¯ a ( z ) + ¯ b ( z )) − ∂b ( z )(¯ a ( z ) + 2¯ b ( z )) ++ ∂ ¯ a ( z )(2 a ( z ) + b ( z )) − ∂ ¯ b ( z )( a ( z ) + 2 b ( z )) (cid:19) ++18 λλ (cid:18) ∂ ¯ a ( z ) − ∂ ¯ b ( z ) (cid:19) + 9 λ (cid:18) ∂ a ( z ) − ∂ b ( z ) (cid:19)! + (cid:18) λ − λλ − (cid:19) V ( z ) W ALGEBRA M ( z ) = 13 (cid:18) ¯ a ( z ) + ¯ a ( z )¯ b ( z ) + ¯ b ( z ) (cid:19) + λ (cid:18) ∂ ¯ a ( z ) + ∂ ¯ b ( z ) (cid:19) V ( z ) = i27 λ √ a ( z ) − ¯ b ( z ))(¯ a ( z ) + 2¯ b ( z ))(2¯ a ( z ) + ¯ b ( z )) ++9 λ (cid:18) ∂ ¯ a ( z )(2¯ a ( z ) + ¯ b ( z )) − ∂ ¯ b ( z )(¯ a ( z ) + 2¯ b ( z )) (cid:19) + 9 λ (cid:18) ∂ ¯ a ( z ) − ∂ ¯ b ( z ) (cid:19)! . Direct calculation (with help of an OPE package for
Mathematica ) shows that fields definedabove satisfy relations of Galilean W algebra G W ( c L , c M ) with central charges c L = 4 − λ + µ ) ,c M = − λ , which gives us realisation for all c L , c M ∈ C , c M = 0.Notice that L is a tensor product of two sublattices whose Gram matrices equal the Cartanmatrix of sl . This kind of rank 2 lattice has been used in free field realisation of W algebra(cf. [19]).We also remark that the fields M ′ ( z ) and V ′ ( z ) obtained by substituting λ , a ( z ) and b ( z )for ¯ λ , ¯ a ( z ) and ¯ b ( z ) in M ( z ) and V ( z ) generate a copy of W ( − λ − /
5) (cf. [19]).5.
Realisation of highest weight representations
Let C [ L ] be a group algebra of L and V L = M (1) ⊗ C [ L ] associated VOA. We introducea parametrisation of highest weight vectors in V L . Let e [ p, q, r, s ] denote a highest weightvector e k ∈ V L where k = (cid:18)(cid:18) p + q (cid:19) λ + 2 − r − s λ (cid:19) a + (cid:18) (1 + q ) λ − s − λ (cid:19) b ++ (cid:18)(cid:18) p + q (cid:19) µ + i 2 − r − s λ (cid:19) c + (cid:18) (1 + q ) µ − i 1 − sλ (cid:19) d. Weights h [ p, q, r, s ] of e [ p, q, r, s ] are given by h L [ p, q, r, s ] = p (1 − r ) + 3 q (1 − s )2 + c L −
496 (4 − p − q ) , (34) h W [ p, q, r, s ] = i2 √ (cid:18) pq (1 − r ) + (1 − s )( p − q ) + q ( p − q ) 52 − c L (cid:19) , (35) h M [ p, q, r, s ] = (4 − p − q ) c M , (36) h V [ p, q, r, s ] = i c M √ q ( q − p ) . (37) ALILEAN W ALGEBRA 19
Direct calculation shows that h [ p, q, r, s ] = h [ − p, q, − r + 2 , s ](38) = h (cid:20) − p + 3 q , − p + q , − r + 3 s , − r + s − (cid:21) = h (cid:20) p + 3 q , p − q , r + 3 s − , r − s + 22 (cid:21) = h (cid:20) − p + 3 q , p − q , − r + 3 s − , r − s + 22 (cid:21) = h (cid:20) p − q , − p + q , r − s + 42 , − r + s − (cid:21) , i.e. parametrisation (34-37) is S -invariant under the action σ ( p, q, r, s ) = (cid:18) − p + 3 q , − p + q , − r + 3 s , − r + s − (cid:19) (39) τ ( p, q, r, s ) = ( − p, q, − r + 2 , s ) . (40) Proposition 5.1.
Let P = { ( p, q, r, s ) ∈ C : 0 < p < q } where ” < ” denotes a lexicographicalordering on { (Re( z ) , Im( z )) : z ∈ C } . Let h ∈ C such that h V = 64 (cid:0) h M − c M (cid:1) c M . i) There exists a unique ( p, q, r, s ) ∈ P such that h = h [ p, q, r, s ] . ii) For every r ∈ C we have h L [0 , q, r, s ] = 3 q − s ) + c L − (cid:18) − (cid:16) q (cid:17) (cid:19) , (41) h W [0 , q, r, s ] = − i r (cid:18) (cid:16) q (cid:17) (1 − s ) + (cid:16) q (cid:17) − c L (cid:19) , (42) h M [0 , q, r, s ] = c M (cid:18) − (cid:16) q (cid:17) (cid:19) , (43) h V [0 , q, r, s ] = i r (cid:16) q (cid:17) c M . (44)iii) We have h [ p, q, r, s ] ∗ = h [ p, − q, r, − s ] , and σ ( p, − q, r, − s ) ∈ P . iv) The Verma module V [ p, q, r, s ] is reducible if and only if p ∈ Z > . In that case there is asingular vector of conformal weight h L + p in V [ p, q, r, s ] .Proof. i) From the Jacobian matrix of parametrisation (34-37) follows that p = 0, and p = ± q are critical values. Consider the action of S = h σ, τ i on C defined by (39-40). Then P is a space of coinvariants ( C ) S excluding critical values.ii) and iii) direct calculation.iv) is the reducibility condition (32) stated in terms of parametrisation. (cid:3) W ALGEBRA
Remark 5.2.
Note that the weights h such that h V = ( h M − cM ) c M but h is not equal to(41-44) are not obtained by this parametrisation. This is analogous to realisation of GCApresented in Subsection 4.1 where each (0 , r ) produces weight (cid:0) c L − , c M (cid:1) . Highest weightmodules of highest weights (cid:0) h L , c M (cid:1) for h L = c L − were realised by means of deformed actionon certain Whittaker modules (cf. [4] ). Example 5.3.
Recall Example 3.6 of subsingular vectors at level 1. Then s.e [1 , q, r, s ] is asingular vector in F ,q,r,s while s.e [ − , q, − r + 2 , s ] = 0 . Furthermore a: h M [1 , q, r, s ] = 0 if and only if q ∈ {± } . s .e [1 , , r, s ] subsingular in F , ,r,s , while s .e [ − , − , r, s ] = 0 . b: (33) holds if r = 1 . s .e [1 , q, , s ] is subsingular in F ,q, ,s and s .e [ − , q, , s ] = 0 . ab: s .e [1 , , , s ] is subsingular in F , , ,s , and s .e [ − , − , , s ] = 0 . abc: s .e [1 , , , is subsingular, and s .e [ − , − , ,
1] = 0 . Remark 5.4.
As we have seen (Proposition 5.1 iv), integral values of p detect positions ofsingular vectors in reducible Verma modules. Based on Example 5.3 and on representationtheory of GCA (Theorem 4.1) we expect that integral values of each of the remaining threeparameters detect positions of subsingular vectors. Different sectors (of S action on C )should produce variant subquotients of V [ p, q, r, s ] , including the Verma module itself, and theirreducible quotient L [ p, q, r, s ] . Appendix A. λ -bracket calculation A.1.
Jacobi identity.
Recall the Jacobi identity for λ -brackets (3). The most difficultcalculation occurs in case a = b = c = W . We have (cf. [15] Lemma 3.2)[ W λ LM ] = 2( DW ) M + 2 L ( DV ) + 3 λ ( W M + LV ) + (4 λ D + 52 λ ) V [ LM λ W ] = ( D + 3 λ )( LV + W M ) + 2((
DLV + W ( DM )) + 12 ( − D − D λ + 7 Dλ + 5 λ ) V [ W λ M ] = 4 M ( DV ) + 6 λM V [ M λ W ] = 2( D + 3 λ )( M V ) + 4( DM ) V ALILEAN W ALGEBRA 21 so [ W λ [ W µ W ]] equals: (cid:18) µ µ λ + D ) + 3 µ
10 ( λ + D ) + 115 ( λ + D ) (cid:19) (2 D + 3 λ ) W ++ 325 c M (2 µ + λ + D ) (cid:18) (2 D + 3 λ )( W M + LV ) − W ( DM ) − DL ) V ++ (4 λ D + 52 λ ) V −
310 ( λ + D ) (2 D + 3 λ ) V (cid:19) + − c M (cid:18) c L + 445 (cid:19) (2 µ + λ + D )(4 M ( DV ) + 6 λM V );and [[ W λ W ] λ + µ W ] equals: (cid:18) λ − λ λ + µ ) + 3 λ
10 ( λ + µ ) −
115 ( λ + µ ) (cid:19) ( D + 3 λ + 3 µ ) W ++ 325 c M ( λ − µ ) (cid:18) ( D + 3 λ + 3 µ )( W M + LV ) + 2( DL ) V + 2 W ( DM ) ++ 12 (3 D − D ( λ + µ )7 + D ( λ + µ ) + 5( λ + µ ) ) V −
310 ( λ + µ ) ( D + 3 λ + 3 µ ) V (cid:19) + − c M (cid:18) c L + 445 (cid:19) ( λ − µ )(6( λ + µ + D ) M V − M ( DV )) . Comparing all the coefficients one sees that (3) holds. Similarly, in case a = b = W , c = V we have [ LM λ V ] = 3( D + λ )( M V ) − M ( DV )so [ W λ [ W µ V ]] = (cid:18) µ µ λ + D ) + 3 µ
10 ( λ + D ) + 115 ( λ + D ) (cid:19) (2 D + 3 λ ) V ++ 165 c M (2 µ + λ + D )(6 λM V + 4 M DV )[[ W λ W ] λ + µ V ] = (cid:18) λ − λ λ + µ ) + 3 λ
10 ( λ + µ ) −
115 ( λ + µ ) (cid:19) ( D + 3 λ + 3 µ ) V ++ 325 c M ( λ − µ )(3( D + λ + µ )( M V ) − M ( DV )) . Other cases are easier to check.A.2.
Other definitions.
Suppose we want to construct a Galilean W algebra in such a waythat W is its subalgebra, i.e.[ W λ W ] = c λ + (cid:18) λ λ D + 3 λ D + 115 D (cid:19) L + 165 c + 22 ( D + 2 λ ) (cid:18) L − D L (cid:19) . W ALGEBRA
Let us check the Jacobi identity for a triple W , W , M . We see that [ W λ [ W µ M ]] and[ W µ [ W λ M ]] produce nonlinear terms in M , while [[ W λ W ] λ + µ M ] produces LM , ( DL ) M as well, so the identity (3) can not hold.Therefore, either [ W λ V ] must contain a nonlinear term with factor L (which means W, L don’t act on a commutative subalgebra generated by M and V ), or the nonlinear terms in[ W λ W ] must contain M as a factor. References [1] T. Araujo,
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W-algebra W (2 , and the vertex operator algebra L (1 / , ⊗ L (1 / , Current address : Faculty of Science, University of Split, Ru ¯dera Boˇskovi´ca 33, 21 000 Split, Croatia
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