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Automorphism Groups and Twisted Modules for Lattice Vertex Operator Algebras

We give a complete description of the full automorphism group of a lattice vertex operator algebra, determine the twisted Zhu's algebra for the automorphism lifted from the -1 isometry of the lattice and classify the corresponding irreducible twisted modules through the twisted Zhu algebra.


Introduction
Vertex (operator) algebra V L associated with an arbitrary even lattice L is one of the important motivations for introducing the notion of vertex (operator) algebra and also provides basic examples of such algebras ( [B], [FLM]). The study of V L when L is positive has been fruitful. If L is positive definite V L is regular [DLM1] in the sense that any weak module is a direct sum of ordinary irreducible modules which have been classified (see [FLM] and [D1]). A characterization of V L was obtained in [LX]. The automorphism groups of all known vertex operator subalgebras of V L when the rank of L is one are determined in [DG] and [DGR].
The vertex operator algebra V L has a canonical automorphism θ of order 2 lifted from the −1 isometry of L. The aim of this paper is to determine the full automorphism group Aut (V L ) of V L , compute the θ-twisted Zhu's algebra [DLM3] and classify the irreducible θ-twisted modules for V L by using the twisted Zhu's algebra when L is positive definite.
The group structure of Aut (V L ) determined in this paper is important in understanding automorphism groups of vertex operator algebras in general. Let V be a vertex operator algebra without vectors of negative weights and weight zero subspace being one dimensional. Then the weight one subspace V 1 is a Lie algebra under the bracket [u, v] = u 0 v where u 0 is the component operator of vertex operator Y (u, z) = n∈Z u n z −n−1 associated to u. The exponentials e u 0 for u ∈ V 1 generate a continuous, normal subgroup N of Aut (V ). So the task is to determine the quotient group Aut (V )/N. In this paper we show that Aut (V L )/N is isomorphic to a quotient group of the finite group O(L) which is the subgroup of Aut (L) of isometries. This may suggest that Aut (V )/N is a finite group in general. There are two extremes. Namely, V 1 generates V in the sense of [FHL] or V 1 = 0. For example, the vertex operator algebras associated to affine Kac-Moody algebras (cf. [DL1], [FZ] and [Li]) are generated by weight one subspaces and Aut (V ) is isomorphic to the automorphism group of the Lie algebra V 1 ; the moonshine vertex operator algebra V ♮ [FLM] and the vertex operator algebras associated to the Virasoro algebra (cf. [FZ] and [Li]) have zero weight one subspaces and the automorphism groups are the monster simple group and the identity.
The automorphisms studied in this paper preserve the Virasoro element. But it is also an important problem to investigate the automorphisms which need not preserve the Virasoro element. Such problem was first considered in [MN] in the case of free bosonic vertex algebra, where the the group of all such automorphisms of free bosonic vertex algebra was obtained. It is still an open problem to determine all such automorphisms for V L .
In [DLM3], an associative algebra A g (V ) was introduced for a vertex operator algebra V and an automorphism g of finite order so that there is a one to one correspondence between the equivalence classes of irreducible admissible g-twisted V -modules and the equivalence classes of irreducible A g (V )-modules. In the case g = 1 this amounts Zhu's original algebra A(V ) and related results. The Zhu's algebras A(V ) have been computed for the vertex operator algebras associated to affine Lie algebras in [FZ], the vertex operator algebras associated to the Virasoro algebras in [W] and for V L in [DLM2]. Our result on A θ (V L ) in this paper is the first example of explicit calculation of twisted Zhu's algebra. It turns out the computation of A θ (V L ) is much simpler than that of A(V L ) and the structure of A θ (V L ) is also simpler.
LetL be the canonical central extension of L by a cyclic group ±1 of order 2. In [FLM] and [Le], a subgroup K ofL, which is isomorphic to 2L, was introduced and all irreducible modules forL/K on which (−1)K acts as −1 were classified and denoted by T χ . It was shown in [FLM] (also see [DL2]) each T χ gives rise an irreducible θ-twisted module. We show in this paper that A θ (V L ) is isomorphic to a quotient of the group algebra C[L/K] modulo the ideal generated by (−1)K + 1. As a result we know all the irreducible modules of A θ (V L ) are precisely these T χ . Using the theory of twisted Zhu's algebra developed in [DLM3] we give a new proof of a result in [D2] on classification of irreducible θ-twisted V L -module. Although the proof in [D2] can be easily generalized to prove that V L is θ-rational but this has not appeared in any publication. We also prove a general result which says that for an arbitrary vertex operator algebra V and an automorphism g of finite order such that A g (V ) is finite-dimensional, semisimple and the weights of top levels of the all irreducible g-twisted V -modules are the same then V is g-rational. As a corollary we show that V L is θ-rational. The most of this work was done when the second author was visiting UC Santa Cruz in academic year 1997-1998. The second author wishes to thank the Department of Mathematics here for their hospitality during his stay. The authors are grateful to Geoffrey Mason and Michael Tuite for helpful comments related to this work.

Automorphism groups
In this section we determine the automorphism group of the vertex operator algebras V L associated to even positive definite lattices L. In Subsection 2.1 we review the canonical central extensionL of L by the cyclic group of order 2 and the structure of the automorphism group ofL following [FLM]. We recall the explicit construction of the vertex operator algebra V L from [FLM] in Subsection 2.2. Subsection 2.3 is on how to lift a certain derivations of a vertex operator algebra to automorphisms. Subsection 2.4 is the core of Section 2, where the automorphism group of V L is obtained by using the automorphism group ofL and the automorphisms which are the exponentials of the weight zero components of the weight one vectors.

Lattice vertex operator algebras
Let h = C ⊗ Z L and extend the Z-form on L to h by C-linearity. The corresponding affine Lie algebra isĥ = h ⊗ C[t, t −1 ] ⊕ Cc. We also use the notation h(n) = h ⊗ t n for h ∈ h, n ∈ Z. Setĥ Thenĥ − is an abelian subalgebra ofĥ. Let U(ĥ − ) = S(ĥ − ) be the universal enveloping algebra ofĥ − . Consider the inducedĥ-module where C[t] ⊗ h acts trivially on C and c acts on C as multiplication by 1.
Recall from (2.1) thatL is the canonical central extension of L by the cyclic group ±1 . Form the inducedL-module where C[·] denotes the group algebra and −1 acts on C as multiplication by −1. For a ∈L, write ι(a) for a ⊗ 1 in C{L}. Then the action ofL on C{L} is given by: Furthermore we define an action of h on C{L} by: The untwisted space associated with L is defined to be and the vertex algebra algebra structure on V L (see [B] and [FLM]) is determined by The Virasoro element is given by Then the component operator L(0) defines the weight gradation on V L : We should point out that M(1) is a vertex operator subalgebra of V L .

Derivations and automorphisms
Let V be a vertex operator algebra and ω the Virasoro element. An automorphism of V is an isomorphism σ : V −→ V of vector spaces which preserves all the products and the Virasoro element: Note that an automorphism of V preserves each homogeneous subspace V n of V. The group of all automorphisms of the vertex operator algebra V is denoted by Aut and D(ω) = 0. In particular, D preserves the gradation of V. So the exponential e D converges on V and is well-defined. It is easy to see that e D is an automorphism. Let a ∈ V and wt (a) = 1. Then D = a 0 is a derivation of V and e a 0 is an Since σe a 0 σ −1 = e (σa) 0 and wt (σ(a)) = 1 for any σ ∈ Aut (V ), N is a normal subgroup of Aut (V ).

The automorphism group of V L
We first describe certain automorphisms of V L . Set Then O(L) is a subgroup of Aut (L) and Hom (L, Z/2Z) is a subgroup of O(L) (see (2.2)). It follows from (2.2) that we have an exact sequence Every σ ∈ O(L) induces an automorphism of V L , denoting by σ also, in the following way for α i ∈ h, n i > 0 and a ∈L. So we can regard O(L) as a subgroup of Aut (V L ). Note that such automorphisms preserve the vertex operator subalgebra M(1). The first main result is the determination of the automorphism group of V L .
Theorem 2.1. Let L be a positive definite even lattice. Then and N is a normal subgroup of Aut (V L ). Moreover, the intersection N ∩ O(L) contains a subgroup Hom (L, Z/2Z) and Aut (V L )/N is isomorphic to a quotient group of O(L).
To prove this theorem, we need the following elementary lemmas.
Proof. Note that h acts on V L semisimply, and M(1) ⊗ ι(e β ) is precisely the subspace with the eigenfunction β, that is The lemma follows immediately.
Set h(−1) = h ⊗t −1 1, a subspace of V L . Then any automorphism σ of V L preserving h(−1) can be considered as a linear automorphism of h.
Remark 2.4. Since the vectors ι(e α ) and ι(e σα ) in the proof of Lemma 2.3 have the same weight, σ induces an isometry of L.
Since the inner product is non-degenerate, there exists h ∈ h such that e h,α i = c α i for all i = 1, 2, . . . , d. Then e h(0) ι(e α i ) = c α i ι(e α i ) for all i = 1, 2, . . . , d and Since both σ and e h(0) act trivially on h(−1) and have the same action on C{L}, we conclude that σ = e h(0) .
Proof of Theorem 2.1. Let us consider the weight one space which is a finite dimensional reductive Lie algebra under the bracket [u, v] = u 0 v for u, v ∈ g with a Cartan subalgebra h = h(−1). Let σ be an automorphism of V L . Then σ induces an automorphism of the Lie algebra g. Note that the inner automorphism group of g is the restriction of N to g. The conjugacy theorem for Cartan subalgebras of finite dimensional reductive Lie algebras implies that there exists φ ∈ N such that φ −1 σ(h) = h. Then by Lemma 2.3 and Remark 2.4, µ = φ −1 σ induces an isometry of L and µι(e α ) = c α ι(e µα ) for α ∈ L and some constant c α ∈ C. Let τ ∈ O(L) such thatτ = µ. Then µτ −1 is the identity when restricted to h(−1). By Lemma 2.3 there exists ψ ∈ N such that µτ −1 = ψ. That is, σ = φψτ ∈ N · O( L) as φψ ∈ N. This shows that Aut (V L ) = N · O(L).
In order to see Hom (L, Z/2Z) is a subgroup of N we recall from [FLM] on how Hom (L, Z/2Z) is embedded into Aut (L). Let λ ∈ Hom (L, Z/2Z). The corresponding automorphism ofL is given by: a → a(−1) λ(ā) for a ∈L. As in the proof of Lemma 2.5 there exists h ∈ h such that e h,α = (−1) λ(α) for α ∈ L. It is easy to see that the λ and e h(0) give the same automorphism of V L and thus λ ∈ N.
An isomorphism from Aut (

θ-twisted modules
Let V L be as before. Fix an involution θ ofL of order 2 such thatθ = −1 on L. For example, one can define θ(a) = a −1 (−1) ā,ā /2 for a ∈L. Then θ is an automorphism of V L and M(1). We determine twisted Zhu's algebra A θ (V L ) and A θ (M(1)) explicitly in this section. We also classify the irreducible θ-twisted modules for these vertex operator algebras by using A θ (V L ) and A θ (M(1)) and prove the θ-rationality of V L .
Beginning with a arbitrary vertex operator algebra V and an automorphism g of finite order g in Subsection 3.1, we recall the definitions of admissible g-twisted modules for V and twisted Zhu's algebra A g (V ) and related results from [DLM3]. The g-rationality of V is established under certain assumption on the structure of A g (V ). In Subsection 3.2 we give the construction of θ-twisted modules for V L following [FLM]. In Subsection 3.3 we obtain a finite spanning set of A θ (V L ) indexed by L/2L and prove that A θ (M(1)) is isomorphic to the one-dimensional associative algebra C. We show in Subsection 3.4 that A θ (V L ) is isomorphic to a quotient of the group algebra C[L/K] where K is a subgroup ofL (defined in Subsection 3.2) isomorphic to 2L. Using this result we prove that every irreducible θ-twisted V L -module is isomorphic to V Tχ L for someL/K-modules T χ (see Subsection 3.3 for the definition of T χ ). The θ-rationality of V L is an easy corollary of a general result on g-rationality in Subsection 3.1.

Twisted modules and twisted Zhu algebras
Let V be a vertex operator algebra and g an automorphism of V of finite order T. Decompose V into g-eigenspaces: An admissible g-twisted V -module (cf. [DLM3], [Z]) is a 1 T Z-graded vector space where δ(z) = n∈Z z n and all binomial expressions are to be expanded in nonnegative integral powers of the second variable; if u is homogeneous. A g-twisted V -module is an admissible g-twisted V -module M which carries a Cgrading by weight. That is, we have where M λ = {w ∈ M|L(0)w = λw}. Moreover we require that dim M λ is finite and for fixed λ, M λ+ n T = 0 for all small enough integers n. V is called g-rational if any admissible g-twisted module for V is a direct sum of irreducible g-twisted V -modules.
Nest we review twisted Zhu's algebra [DLM3]. For homogeneous u ∈ V r and v ∈ V we define where δ r = 1 if r = 0 and δ r = 0 if r = 0 and where, here and below, (1 + z) α for α ∈ C is to be expanded in nonnegative integer powers of z. Let O g (V ) be the linear span of all u • g v and define the linear space A g (V ) to be the quotient V /O g (V ).
(1) Assume that u ∈ V r is homogeneous, v ∈ V and m ≥ n ≥ 0. Then Let us define the second product * g on V . For r, u and v as above, set Extend this to a bilinear product on V. Let M is an admissible g-twisted V -module. Define the "vacuum space" Ω(M) of M to be {w ∈ M|u n w = 0, n > wt (u) − 1, n ∈ 1 T Z}.
Then M(0) ⊂ Ω(M) and the weight zero operator o(v), which is equal to v wt (v)−1 when v is homogeneous, acts on Ω(M).
Theorem 3.2. [DLM3] (1) The product * g induces the structure of an associative algebra on A g (V ).
(2) For any admissible g-twisted V -module M, v + O g (V ) → End Ω(M) gives an A g (V )-module structure on Ω(M) and M(0) is a submodule of Ω(M).
(3) M → M(0) gives a bijection between the equivalence classes of irreducible gtwisted admissible V -modules and the equivalence classes of simple Ag(V )-modules.
(4) If V is g-rational, A g (V ) is a finite-dimensional semisimple algebra.
The following theorem is new.
Theorem 3.3. Let V be a vertex operator algebra and g an automorphism of order T such that A g (V ) is a finite-dimensional semisimple algebra and ω + O g (V ) acts on the all irreducible modules as the same constant λ. Then V is g-rational.
Proof. By definition of g-rationality we need to prove that any admissible g-twisted V -module M is completely reducible. First we show that if M(0) is an irreducible A g (V )-module and M is generated by is a finite-dimensional semisimple algebra, there exists an irreducible submodule of W λ+n 0 on which ω + O g (V ) acts as λ + n 0 = λ. This is a contradiction. Now we consider general M. Let X be the submodule of M generated by Ω(M). Since Ω(M) is a direct sum of irreducible A g (V )-module, X is completely reducible by the argument above. Note the X is the maximal completely reducible submodule of M. We claim that X = M. IfM = M/X = 0. Then Ω(M ) = 0. We can assume thatM is completely reducible (otherwise replaceM by the submodule generated by Ω(M )). Thus any vector of M is contained in a finite-dimensional L(0)-stable subspace and M is a direct sum of generalized eigenspaces for L(0) : where M µ is the generalized eigenspace with the eigenvalue µ. Since M λ is a completely reducible A g (V )-module, the submodule W generated by M λ is completely reducible. If M = W then Ω(M/W ) contains a A g (V )-submodule on which ω + O g (V ) acts as a constant which is greater than λ. Thus M is completely reducible and M = X. This is a contradiction.

The θ-twisted modules
We return to V L . Recall that θ is an automorphism ofL and V L such that θ(a) = a −1 (−1) ā,ā /2 for a ∈L. Define K = {θ(a)a −1 | a ∈ L}. ThenK = 2L. Also set R = {α ∈ L | α, L ⊂ 2Z} ⊃ 2L. Then the pull back R of R in L is in the center of L and K is a subgroup of R.
The following proposition can be found in [FLM] (Proposition 7.4.8).
Proposition 3.4. There are exactly |R/2L| central characters χ : R/K −→ C × of L/K such that χ((−1)K) = −1. For each such χ, there is a unique (up to equivalence) irreducible L/K-module T χ with central character χ and every irreducible L/K-module on which (−1)K acts −1 is equivalent to ones of these. In particular, viewing T χ as an L-module, θ(a) = a on T χ for a ∈ L.
For each T χ , we define a twisted space (1) is the module for the Heisenberg algebra h Z+ 1 Here h + acts trivially on C and c acts as 1.
The θ-twisted vertex operators Y θ (ι(a), z) for a ∈L are defined to be where α(z) = n∈Z+ 1 2 α(n)z −n−1 and α(n) = α ⊗ t n (see [FLM] for the detail). For v = α 1 (−n 1 ) · · · α k (−n k ) ⊗ ι(a) ∈ V L homogeneous, set n . Now we extend this to all v ∈ V L by linearity. Recall that {h 1 , . . . , h d } be an orthonormal basis of h and define constants c mn ∈ Z for m, n ≥ 0 by the formula m,n≥0 We finally define twisted vertex operator The following theorem was proved in [FLM] (see also [DL2]).
(1) For each T χ , the space (Y θ , V Tχ L ) is an irreducible θ-twisted module for the vertex operator algebra V L .
Let W be an θ-stable subspace of V L . Decompose W into eigenspaces with respect to the action of θ : Proof. Take n = 0 in Lemma 3.1 (1), then we see Substituting (1 + z) 1 2 = ∞ j=0 1 2 j z j into the above, we get the result.

A spanning set of
Lemma 3.7. Let W = M(1) or V L . Let α ∈ h and m ∈ Z ≥0 . Then for any v ∈ W , there exist scalars c m,j ∈ C for j = 0, 1, 2, . . . such that Proof. We prove the lemma by induction on m. Since α ∈ W − , setting m = 0 in Lemma 3.6 gives Namely, Then m = 0 case is done.
Let m > 0 and assume that the lemma is true for all nonnegative integers strictly less than m. By Lemma 3.6, we see Using the induction hypothesis completes the proof of the lemma.
Proof. We prove the lemma by induction on the length of v. Clearly it is enough to show the lemma for homogeneous elements. If ℓ(v) = 0, the lemma is obviously true. Suppose that the lemma is true for any homogeneous elements whose lengths are strictly less than ℓ Since ℓ(α(j)v ′ ) < t for all j ≥ 0, by induction hypothesis, we prove the lemma. If a = 1 the result is clear now.
Recall that e : L →L is a section. We may and do assume that e 2α ∈ K.
For (2) we first use Lemma 3.8 to get a constant d α such that , which finishes the proof of (2).
(2) u α + O θ (V L ) is in the center of A θ (V L ) if and only if α ∈ R.
Motivated by Proposition 3.12 we define a map f : C[L/K] → A θ (V L ) such that f (e α K) = u α and f ((−1)K) = −1. Then from Proposition 3.12, this is an onto algebra homomorphism and (−1)K + 1 is in the kernel. Let I be the two sided ideal of C[L/K] generated by (−1)K + 1. Then f induces an onto algebra homomorphism f : C[L/K]/I → A θ (V L ).
Our second main theorem in this paper is: Theorem 3.13.
(2) The irreducible θ-twisted modules up to isomorphism are V Tχ L for all T χ given in Proposition 3.4. Moreover, every irreducible admissible θ-twisted V L is an ordinary θ-twisted V L module.
(1) First note from Proposition 3.4 and Theorem 3.5 that C[L/K]/I is a finitedimensional semisimple algebra whose irreducible modules are precisely those T χ which are the top levels of the irreducible θ-twisted modules V Tχ L for V L . So every irreducible module for C[L/K]/I is an irreducible module for A θ (V L ) andf must be an isomorphism.
(3) By Theorem 3.3 it is enough to show that L(0) has the same eigenvalue on all T χ . In fact it is shown in [FLM] that L(0) acts on all T χ as 1 16 d where d is the rank of L.