Ternary Codes and Vertex Operator Algebras 1

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1 Journal of Algebra 799 do:6jabr99988 avalable onlne at on Ternary Codes and Vertex Operator Algebras Masaak Ktazume Department of Mathematcs and Informatcs Chba Unersty Chba 6-8 Japan Masahko Myamoto Insttute of Mathematcs Unersty of Tsukuba Tsukuba -87 Japan and Hromch Yamada Department of Mathematcs Htotsubash Unersty Kuntach Tokyo Japan Communcated by Geoffrey Mason Receved January 998 INTRODUCTION Vertex operator algebras have been studed from a wde varety of vew pont Ths mples rch propertes of vertex operator algebras Recently nvestgaton of vertex operator algebras as modules for ther subalgebras n somorphc to a tensor product Lc of Vrasoro vertex operator algebras was ntated by Dong et al DMZ Along ths lne Myamoto M constructed a seres of vertex operator algebras by combnng the mnmal vertex operator super algebra L L wth even bnary codes In ths artcle we construct vertex operator algebras assocated wth self orthogonal ternary codes e begn wth a lattce L ' A -lattce It s known DLMN that the vertex operator algebra VL contans a subalgebra 7 T somorphc to L L L Inspectng the acton of T we obtan a vertex operator algebra somorphc to L L and two of ts modules both of whch s somorphc to L Combnng The authors are grateful to Atsush Matsuo for nformng them of the physcs lterature The authors also would lke to thank the referee for valuable comments $ Copyrght by Academc Press All rghts of reproducton n any form reserved

2 8 KITAZUME MIYAMOTO AND YAMADA them wth a self orthogonal ternary code we construct a vertex operator algebra e also prove the ratonalty of L L and determne ts Zhu algebra Z Z The rreducble modules for the vertex operator algebra L L have been already known n physcs lterature FZa Za In ths artcle we construct them explctly n V L where L s the dual lattce of L Our notaton s standard FHL FLM If M Y M s a module for a vertex operator algebra V Y and V we shall wrte Y M z M n M Ý z wth End M to dstngush t from Y z n n n Ýn nz n wth n End V LATTICE D Let be a set of fundamental roots of type A wth an nner product ² : such that ² : and ² : Set Let L where ' For smplcty we also wrte x and y The dual lattce L of L has the dual bass x y x y and L has cosets n L Among them we choose 6 6 x y x y L L L L L L In fact L L L L and the quotent group L L L L L s of order Let D be a ternary code of length n that s a subspace of an n dmensonal vector space over GF For each codeword d d we assgn a subset L of an orthogonal sum of n copes of L n n d dn L L L L j j Then snce L L L for j where the superscrpt j s consdered to be modulo the unon D D L of all L ; D s a sublattce of L n ² : d g Note that dg f L and L for d g Let d d and g g n n be two codewords of D and d g n choose L L Denote by and the elements of L and n of L respectvely Then n ² : Ý ² : where d g d g n n

3 CODES AND VERTEX OPERATOR ALGEBRAS 8 If D s a self orthogonal ternary code of length n then D s a doubly even lattce of rank n that s ² : for D For example f n and D then ' E -lattce D 6 If n and lettng D C be the ternary tetracode then E -lattce ' D 8 VERTEX OPERATOR Y z e shall consder the Fock space V L assocated wth the lattce L n and the vertex operator Y z Ý z End V z n n L for V as n D Secton DL Chapter FLM Chapters 7 8 L e also consder the subspaces V V V V and V V of V L L L L assocated wth the cosets L L L and L of L n L Here nstead of a twsted group algebra L we use the group algebra L wth bass e ; L and multplcaton e e e j Snce ² LL : ² L L : and L L j L j we have n Ý u n z wth u n V n for u V V n Ý u n z n n j Y u z j wth u V The Jacob dentty DL for u V V gves a useful formula m umnw numw Ý u mnw for u V w V L m and n As a summary V Y s a vertex operator algebra whose Vrasoro element s 6 V Y and V Y are V -modules and j j Y z : V Hom V V z Ý n Y z j j V n V z n

4 8 KITAZUME MIYAMOTO AND YAMADA s an ntertwnng operator of type V j j V V Let D be a self orthogonal ternary code of length n For each codeword d d d d we assgn a tensor product V V V n n of vector spaces For V d defne the tensor product vertex operator Y n z Y z Y n z as n DL FHL Set V V Then V Y D D D s a vertex operator algebra In fact t s dentcal to the vertex operator algebra V D assocated wth the lattce D of Secton VERTEX OPERATOR ALGEBRA M D By DLMN the Vrasoro element of V three mutually orthogonal conformal vectors can be wrtten as a sum of 8 x x x x x x x where x e e The central charge c of the conformal vector s c c 7 c and the central charge of s c Each conformal vector generates a Vrasoro vertex operator algebra M Vr Lc and V contans a subalgebra 7 T Vr Vr Vr L L L As T-modules V s are completely reducble and each rreducble summand s of the form DMZ 7 L h L h L h

5 CODES AND VERTEX OPERATOR ALGEBRAS 8 where 7 h 6 h h 8 8 Snce V s generated by the homogeneous subspaces V and V calculatng the egenvalues of the acton of on these two homogeneous subspaces and usng the fuson rules and the dmenson of Lc h FF R we have n LEMMA V s a drect sum of the followng rreducble T-submodules each of whch s of multplcty one: Smlarly we have L L L L L L L L L L L L L L L L L L L L L L L L LEMMA V has the followng rreducble T-submodules as drect summands each of whch s of multplcty one: L L L L L L L L L The mnmal weght of any other drect summand s greater than The decomposton of V nto a drect sum of rreducble T-submodules s the same as that of V THEOREM Set M V Then M L L L L7 The weght subspace of M s spanned by v

6 8 KITAZUME MIYAMOTO AND YAMADA M s spanned by M L L7 L The weght subspace of v e xy e xy e x y M s spanned by M L L7 L The weght subspace of v e xy e x y e xy j Here Lc denotes the acuum of L c Moreoer un M for j u M and M Thus M Y s a subalgebra of V Y wth the Vrasoro element M and M are M -modules and the restrcton of to M s an ntertwnng operator of type M j j M M Proof follows from Lemma Smlarly M s a drect sum of subspaces of the form L h wth h as n L L7 The weghts of V and V are n and so and hold By see also and M Proposton 9 the last asserton holds Exchange of the fundamental roots and nduces an automorphsm of the lattce L and so t nduces a lnear automorphsm of VL of order two Snce Y u z Y u z for u V L and snce and t follows that M M and M M Also acts on the L -part of M as the dentty Now consder two homogeneous elements of weght : and ' s e e e e e e q 9 e e e e e e

7 CODES AND VERTEX OPERATOR ALGEBRAS 8 e have s q s q Hence s and q form a bass of M and q s a hghest weght vector n the L -part of M Snce s s and q q s s contaned n the L -part of M and acts as on the L -part of M Bya drect calculaton we have ' qv v qv v and qq 9 9 Let D be a self orthogonal ternary code of length n For each codeword d d d d we assgn the tensor product M M M n n and set MD D M Then by Theorem MD s a subalgebra of V DIf M contans an element u whose th entry s v and the other entres are the vacuum of M : Set ˆ u u u n Then u v THEOREM MD s aertex operator algebra wth the acuum and the Vrasoro element ˆ Moreoer M D s of dmenson one and M D Clearly the automorphsm group Aut D of the code D nduces a subgroup of Aut M Let exp ' and take j j D n GF n Then for M d n j d j n d n n defnes an automorphsm of M called a coordnate automorphsm D M Secton Ths automorphsm s trval f and only f j j n s orthogonal to D Thus the coordnate automorphsms form a subgroup of Aut M D somorphc to GF n D Snce D s self orthogonal and M D comes from M wth of weght or we may assume that D s an orthogonal sum of the examples n Secton of length or and the zero code In the case of ternary tetracode we have PROPOSITION S where S ' Let D be the ternary tetracode Then Aut M D s Aut D and s the coordnate automorphsms Proof In ths case the Gress algebra G MD has a bass u ; d d and x ; d d d d Here x x x wth x x v and x v The product defned by u u and

8 86 KITAZUME MIYAMOTO AND YAMADA the nner product ² : defned by ² u : u FLM Secton 89 are as follows: u u j u j x f d u x ½ f d 8 x f Ý u f d x x ² : j u u j ² : u x 6 x f 7 f ² x x : ½ f For u G let u be the trace of u End G where u: w u w Then we have ½ f j j u u 6 f j u x 9 6 f x x ½ f The automorphsm group Aut G of the Gress algebra G conssts of lnear automorphsms g whch preserve the product and the nner product gu g gu and ² gu g : ² u : Note that gu g u for g Aut G For convenence set w u u u u w u u w u u u and w u u u u Now ² : s a blnear form on G nvarant under Aut G Hence ts kernel spanw s nvarant under Aut G Smlarly the kernel span w w w of ² : 9 and the kernel span x ;D of 6² : 7 are nvarant under Aut G In partcular Aut G acts on span u u u u Let K Aut D and H be the group of coordnate automorphsms Snce u u j ju u are the prmtve dempotents n span u u u u and thus every automorphsm of G permutes them

9 CODES AND VERTEX OPERATOR ALGEBRAS 87 Snce K nduces S on the set u u u u t follows that Aut G acts on the set as S Let C be the kernel of the acton of Aut G on the set and take two lnearly ndependent codewords and The product u x mples that span x x s nvarant under C Moreover the product x x and the acton of H and K mply that C acts on span x x as a dhedral group of order 6 The same holds for span x x Snce x x 6 x and snce span x x does not contan x we conclude that C acts on span x ;D as and thus Aut G S Let G Aut MD and take g CG G Then g commutes wth u and u ; Now M u for D s spanned by q q q and q Moreover the egenvectors for u wth egenvalue n ths subspace are scalar multples of q Hence g maps q to ts scalar multple If d d d d D wth d then q x s a nonzero scalar multple of x Snce g fxes x ths mples that g fxes q also Lkewse we see that g fxes q q and q Snce these elements and G generate M D we have g and the asserton holds ZHU S ALGEBRA OF L L Let U Y be a vertex operator algebra such that U U U U s a subalgebra somorphc to L wth the same Vrasoro element and U s a U -module somorphc to L LEMMA One of the followng two cases occurs Yu z for all u U U s an deal of the ertex operator algebra U Y and the automorphsm group Aut U s somorphc to the multplcate group Yu z for any nonzero u U U Y s a smple ertex ² : operator algebra and Aut U s of order two where u ufuu and u ufuu Proof By our hypothess Yu z for u U and u U U are gven Moreover Yu z for u U U s determned by the zl skew symmetrcty Y zue Y u z Let : U U be the projecton Then Y z nduces an ntertwnng operator of type U U ž U U /

10 88 KITAZUME MIYAMOTO AND YAMADA for U -modules Hence the fuson rule L L L mples that Yu z U z z for u U and that for a fxed nonzero ntertwnng operator I z of type ž U / U U there s such that Yu z I u z for all u U If then holds If then all such vertex operator algebras U Y are somorphc to each other Denote by Vr the Vrasoro algebra spanned by Ln n; n and Let q be a nonzero hghest weght vector of U Then as Vr-modules U s generated by and U s generated by q Let Aut U Then commutes wth the acton of Vr and so s on U and q q for some If then we can choose arbtrarly Suppose In ths case Yu z for any nonzero u U by DL Proposton 9 Snce we have and holds Y q z q Y q z q Y q z q From now on we assume that U Y s of case of the above lemma Explctly we shall take M Y of Secton as U Y and as the Vrasoro element e shall use basc propertes of Zhu s algebras whch can be found n DMZ FZ Z Z e also follow the notaton n these references By the defnton of OU and OU of the vertex operator algebra U and ts module U we see that OU contans both of them Zhu s algebra AU U OU s a homomorphc mage of the polynomal algebra x of one varable x; x Ł x h AU ; x n n h where h runs over Also as an AU -bmodule AU s a homomorphc mage of x y and the left acton and the rght acton of AU are gven by m n m n x y q where q s a hghest weght vector of U Now snce U U U s a vertex operator algebra Z Lemma mples that the left and the

11 CODES AND VERTEX OPERATOR ALGEBRAS 89 rght acton of AU on AU are dentcal modulo OU Snce s n the center of the Zhu s algebra AU we have LEMMA A U s commutate and t s a homomorphc mage of ž Ł / ž Ł / x x h x x h h Let Y be a U-module Decompose nto a drect sum of rreducble U -submodules; wth beng rreducble U - modules Let j : j be the projecton Then j j Y z : U Hom z z s an ntertwnng operator of type ž U / j For h or consder the subspace h 8 8 spanned by the homogeneous elements of whose weghts are of the form h n wth n Then h and each h h s a U-submodule e want to show that h for h or Indeed 8 8 s the drect sum of all somorphc to L Hence the fuson rule L L L and mply that Y z for all U Let S be the set of U such that Y z Then for u U S and w the left hand sde of z z z Y u z Y z w z z z z z z Y z Y u z w z z Y Y u z z w h z s and so S s an deal of the vertex operator algebra U Y Snce U Y s smple we conclude that Smlarly h for h 8 or 8 Now suppose s an rreducble U-module Then h for some h or By Zhu s theory Z Theorem the top level of that s the homogeneous subspace of the smallest weght s an rreducble AU -module Snce AU s a commutatve algebra the top level of

12 9 KITAZUME MIYAMOTO AND YAMADA must be of dmenson one Hence f h or we have L h as U -modules If h the fuson rule and DL Proposton 9 mply that 7 s somorphc to L or L and that there are at least one 7 somorphc to L and at least one somorphc to L Snce AU acts rreducbly on the top level of there s exactly one somorphc to L Suppose there s more than one somorphc to 7 7 L e may assume that L and L Let : be an somorphsm for U -modules Then Y z : U Hom z z s an ntertwnng operator of type ž U / Snce the set of all ntertwnng operators of type L 7 L L s a vector space of dmenson one Theorem there s such that Y zwy z w for all U and w Set S w w w whch s a U -submodule somorphc to Snce Y zsfor U S s n fact a U-submodule Ths contradcts the assumpton that s an rreducble U-module Therefore s 7 somorphc to L L as a U -module If by a smlar argument we have s somorphc to L L as a U -module e have shown that LEMMA one of Any rreducble U-module s as a U -module somorphc to 7 L L L L L L e need to show the exstence of these modules and determne the 7 somorphsm classes Let h and h Suppose Y and Y are rreducble U-modules such that wth beng somorphc to L h as a U -module for both n Y and n Y Then there are constants such that Y u zw Y u zw and Y u zw Y u zw for all u U w Moreover the fuson rule L L h L h for j j

13 CODES AND VERTEX OPERATOR ALGEBRAS 9 mples that there are nonzero constants such that Y zw Y zw and Y zw Y zw for all U w These constants are not ndependent Indeed replace Y wth Y n the dentty z z z z z Y z Y z w z z z z Y z Y z w z z Y Y z z w z where U w If U and then the left hand sde s multpled by whle the rght hand sde s multpled by Hence by DL Proposton 9 t follows that Smlarly If U then the left hand sde s multpled by whle the rght hand sde remans nvarant Thus Then Y and Y are equvalent U-modules under the somorphsm w w w w Therefore all rreducble U-modules of type of Lemma are somorphc to each other The exstence of such an rreducble U-module follows from Lemma Of course U tself s an rreducble U-module whch s somorphc to L L as a U -module By a smlar argument as above we have the unqueness of such an rreducble U-module Suppose next that Y and Y are rreducble U-modules such that s somorphc to L as a U -module both n Y and n Y Then there s a nonzero constant such that Y u z Y u z for all u U Moreover the fuson rule L L L mples that there s a nonzero constant such that Y z Y z for all U Now by a smlar argument as above we have and Hence there are at most two somorphsm classes of rreducble U-modules of type On the other hand snce q q and V V Lemma shows that there are two nonsomorphc U-modules whch are as U -modules somorphc to L As for L apply a smlar argument e have classfed the rreducble modules for U Namely THEOREM The smple ertex operator algebra U has exactly sx somorphsm classes of rreducble modules whch are represented by

14 9 KITAZUME MIYAMOTO AND YAMADA These rreducble modules are as U -modules somorphc to 7 L L L L L L THEOREM Zhu s algebra A U of the ertex operator algebra U s somorphc to x x x x x x x x More precsely the mage U OUO U of U n A U s somorphc to x x x x x and the mage U OUO U of U n A U s somorphc to x x x Proof Snce the somorphsm classes of rreducble modules for U and the somorphsm classes of the rreducble modules for Zhu s algebra AU are n one-to-one correspondence Z Theorem we can determne AU by Lemma and Theorem Indeed f : AU End N s a representaton of the assocatve algebra then there exsts a module M for U such that M h of the smallest weght h of M s equal to N wth the acton gven as follows: w o w for w N and U where denotes the mage of n AU UO U and o wt s the coeffcent of z n Y z wt Let be an rreducble U-module Then we know that the smallest weght h of s one of or Now look at as a U -module The acton of AU on s so that acts as the multplcaton by h h Proposton Hence the mage of U n AU s somorphc to x x x x x Snce the mage of U n AU s generated by q as a U OU OU-module t s a homomorphc mage of U OUO U Consder 7 the case wth L and L By the fuson rule we have Y z z z for U Snce the smallest weghts of and are dfferent and snce o preserves the weghts ths mples that o w for all homogeneous elements w of weght Hence there s no factor somorphc to x x n U OUO U Smlarly there s no factor somorphc to x x In the case where h or the two rreducble U-modules h and h are somorphc as U -modules whle U acts n dfferent ways Therefore the factor somorphc to x x h must appear n U OUO U

15 CODES AND VERTEX OPERATOR ALGEBRAS 9 THEOREM 6 The ertex operator algebra U s ratonal Proof It s suffcent to show that any U-module s completely reducble By the argument before Lemma we may assume that the weghts of are of the form h n wth n and h s one of or Decompose nto a drect sum of rreducble U -submodules; Then the homogeneous subspace h of each rreducble U -submodule of weght h s or of dmenson one Denote by the set of all such that h Then h h Zhu s algebra AU acts on va w o h w for U and w h If h or then as n the proof of Theorem the mage of U n AU acts as zero on each and s n fact nvarant under AU h h If h or then all are somorphc to L h as U -modules and so any nonzero lnear combnaton of h ; generates a U -submodule somorphc to L h Snce egenvectors of the acton of AU on h are such lnear combnatons we can choose drect summands so that all h are nvarant under AU Let N be the U-submodule generated by h Then snce h s an AU -submodule the homogeneous subspace N h of weght h s equal to by Z Theorem h e want to show that N and that each N s an rreducble U-module Assume that h or Then each s somorphc to L h as a U -module Now N can be decomposed nto a drect sum of rreducble U -submodules Snce N h h we conclude that 7 and N Assume next that h or and set h or respectvely Consder the decomposton of N nto a drect sum of rreducble U -submodules Snce N h h there s only one drect summand somorphc to L h whch s of course and the other drect sumj mands say M ; j are somorphc to L h e slghtly modfy the j j argument before Lemma Take a drect summand M If Y z M for all U then M j s a submodule for U and U s equal to j u UY u z M But then U s an deal of U a contradcton Thus j Y z j M : U Hom M z z s a nonzero ntertwnng operator of type ž j U / M Suppose there s another drect summand M k Let : M j M k be an somorphsm for U -modules e can vew Y z j and Y z k M M

16 9 KITAZUME MIYAMOTO AND YAMADA as nonzero ntertwnng operators of type ž j U M / so that there s a nonzero constant satsfyng Y zw Y j z w for all U and w M Now S w ww M j s a U -submodule somorphc to L h and n fact t s a U-submodule snce Y zsfor U But then U u UY u zs s an deal of U a contradcton Therefore N L h L h as U -modules and n partcular N s an rreducble U-module Snce h N we have N as requred h REFERENCES D C Dong Vertex algebras assocated wth even lattces J Algebra DL C Dong and J Lepowsky Generalzed Vertex Algebras and Relatve Vertex Operators Progress n Math Vol Brkhauser Basel 99 DMZ C Dong G Mason and Y Zhu Dscrete seres of the Vrasoro algebra and the Moonshne module Proc Sympos Pure Math DLMN C Dong H L G Mason and S P Norton Assocatve subalgebras of the Gress algebra and related topcs n The Monster and Le Algebras J Ferrar and K Harada Eds de Gruyter Berln 998 pp 7 FZa V A Fateev and A B Zamolodchkov Conformal quantum feld theory models n two dmensons havng Z symmetry Nuclear Phys B FF B L Fegn and D B Fuchs Verma Modules over the Vrasoro Algebra Lecture Notes n Math 6 Sprnger-Verlag 98 pp FHL I B Frenkel Y-Z Huang and J Lepowsky On axomatc approaches to vertex operator algebras and modules Mem Amer Math Soc 99 FLM I B Frenkel J Lepowsky and A Meurman Vertex Operator Algebras and the Monster Pure and Appled Math Vol Academc Press San Dego 988 FZ I B Frenkel and Y-C Zhu Vertex operator algebras assocated to representatons of affne and Vrasoro algebras Duke Math J M M Myamoto Gress algebras and conformal vectors n vertex operator algebras J Algebra M M Myamoto Bnary codes and vertex operator super algebras J Algebra R A Rocha-Card Vacuum vector representatons of the Vrasoro algebra n Vertex Operators n Mathematcs and Physcs Publcatons of the Mathematcal Scences Research Insttute Vol Sprnger-Verlag BerlnNew York 98 pp 7

17 CODES AND VERTEX OPERATOR ALGEBRAS 9 Za Z Z ang Ratonalty of Vrasoro vertex operator algebras Duke Math J IMRN A B Zamolodchkov Infnte addtonal symmetres n two-dmensonal conformal quantum feld theory Theor Math Phys 6 98 Y-C Zhu Vertex operator algebras ellptc functons and modular forms PhD thess Yale Unversty 99 Y-C Zhu Modular nvarance of characters of vertex operator algebras J Amer Math Soc