Aperiodic Virasoro algebraR. Twarock Citation: Journal of Mathematical Physics 41, 5088 (2000); doi: 10.1063/1.533393 View online: http://dx.doi.org/10.1063/1.533393 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Loop Virasoro Lie conformal algebra J. Math. Phys. 55, 011706 (2014); 10.1063/1.4862683 Formal rigidity of the Witt and Virasoro algebra J. Math. Phys. 53, 073501 (2012); 10.1063/1.4731220 Whittaker modules for the twisted Heisenberg–Virasoro algebra J. Math. Phys. 51, 023524 (2010); 10.1063/1.3301129 Representations of the Schrödinger–Virasoro algebras J. Math. Phys. 49, 053512 (2008); 10.1063/1.2924216 Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations J. Math. Phys. 40, 2400 (1999); 10.1063/1.532872

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Aperiodic Virasoro algebraR. Twarocka)

Arnold Sommerfeld Institut fu¨r Mathematische Physik, Technische Universita¨t Clausthal,Leibnizstr. 10, 38678 Clausthal-Zellerfeld, Germany

~Received 14 September 1999; accepted for publication 25 February 2000!

An aperiodic analog to the Virasoro algebra is introduced and its representationtheory is investigated. In particular, highest and lowest weight representations areconstructed. An analog to the Kac determinant formula is derived and the implica-tions for unitarity are discussed. ©2000 American Institute of Physics.@S0022-2488~00!00907-5#

I. INTRODUCTION

The aperiodic Virasoro algebra is a new type of infinite dimensional Lie algebra with genera-tors in a one-to-one correspondance with points from an aperiodic point set. The structure con-stants and the central term are similar to the Virasoro algebra, so that it may be viewed as anaperiodic analog to the latter, but its structural properties and representation theory differ cruciallyfrom it due to the aperiodicity of the index set.

The point set, which serves as index set for the generators, is modeled via techniques knownfrom the field of cut-and-project quasicrystals. These are aperiodic point sets with long rangeorder, which are obtained via a projection from a higher dimensional periodic lattice. In theexample implemented here, this periodic lattice will be two-dimensional, and the projection for-malism depends on the choice of a real intervalV; the resulting aperiodic point sets depend onVand will be denoted asS(V). SinceV controls which points constituteS(V), it will be called‘‘acceptance window.’’ The dependence of the index set onV translates to the Lie algebra level,so that for different choices ofV, different Lie algebras with essentially different properties areobtained.

A related but different concept had been used1–3 to obtain an aperiodic analog to the Wittalgebra. However, a central extension for these objects does not exist, so that an analog to theVirasoro algebra as is presented in this paper is not at hand.

The basic idea in the construction of the aperiodic Virasoro algebra consists of replacing theindex setZ of the usual Virasoro algebra byS~V! and by a corresponding modification of thestructure constants such that again a Lie algebra is obtained. In this way, one obtains an analog tothe Witt algebra, called aperiodic Witt algebra, which allows for a central extension of Virasoro-type and hence leads to an aperiodic analog also of the Virasoro algebra: theAPERIODICVIRASORO

ALGEBRA.Since the reader is not assumed to be familiar with the concept of aperiodic point sets related

to cut-and-project quasicrystals, the necessary concepts are provided in a first step. In a next step,the aperiodic Witt algebra is defined and its properties are discussed. For a particular acceptancewindow V, a central extension of Virasoro-type is shown to exist and the aperiodic Virasoroalgebra is introduced. Its representation theory~highest weight representations! are discussed, ananalog to the Kac determinant formula is given and the implications for unitarity are outlined.Finally, realizations similar to the oscillator realizations for the usual Virasoro algebra are indi-cated.

a!Electronic mail: reidun.twarock@tu-clausthal.de

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 7 JULY 2000

50880022-2488/2000/41(7)/5088/19/$17.00 © 2000 American Institute of Physics

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II. APERIODIC POINT SETS VIA ONE-DIMENSIONAL CUT-AND-PROJECTQUASICRYSTALS

For the convenience of the reader, we review the construction of aperiodic point sets viaprojection from higher dimensional periodic lattices.

Given an irrationality which is a Pisot number, i.e., a real solution of an algebraic equation ofmodulus bigger than one such that all other solutions are of modulus smaller than one, we can usethe ring of integers in the algebraic extensions of the rational numbers with respect to this irra-tionality to model an aperiodic point set. Since we will use this construction here only for theirrationality t5 1

2(11A5), known as the golden mean, we will indicate the construction only forthis particular irrationality in the following. The case of a cubic irrationality has been discussedelsewhere.3

Together witht85 12(12A5), t is a solution of the equation,

x25x11. ~1!

The ring of integers in the algebraic extension of the rational numbers byA5 is given by the setZ@t#5$a1tbua,bPZ%. It is a dense set inR and in order to specify a discrete subset, a boundarycondition is needed. The latter can be obtained via the Galois automorphism which links the twosolutions of~1!,

8:a1tb→a1t8b. ~2!

In particular, we ‘‘accept’’ only those pointsx5x11tx2 in the aperiodic point set, which fallinto the so-called ‘‘acceptance window’’V in R after the8-mapping, i.e., ifx85x11t8x2PV.

This leads to the following definition of an aperiodic point set or cut-and-project quasicrystal.4

Definition II.1: In dependence onVPR we define the one-dimensional cut-and-project qua-sicrystalS~V! related to the irrationalityt as

S~V!ª$xPZ@t#ux8PV%. ~3!

SinceS~V! is used as a mathematical model for diffraction patterns of quasicrystals5 in solidstate physics, it is called quasicrystal. The name cut-and-project stems from the fact that there isa two-dimensional periodic lattice underlyingS(V), which leads to~3! via a projection formal-ism: Sincett8521, S~V! may be viewed as a point set modeled on a line with slopet in atwo-dimensional integer lattice. In this picture,V is given on the line with slopet8, perpendicularto the line modelingS(V), and ~3! says that all those points belong to the aperiodic point set,which are contained in the strip parallel to the model space and bounded byV.

It is apparent from the definition that for different choices of acceptance windowsV, differentpoint sets are obtained. For instance, the largerV, the more points are ‘‘accepted’’ as quasicrystalpoints. The behavior of the point set under variation ofV is well understood;6 in particular, thereare always either 2 or 3 distinct tiles, i.e., next nearest neighbor distances, in the quasicrystal.

Two of the aperiodic point sets will play a key role for the definition of the aperiodic Virasoroalgebra: these areS((0,1#) andS(@0,1#), which is obtained from the former under removal of thepoint 0. S(@0,1#) will be used in the following because, after removal of the point 0, it has theproperty that the second component of the quasicrystal points is isomorphic toZ.

With the notation@z# for the integer part ofzPZ, it is possible to write down an explicitexpression for the quasicrystal points, which nicely displays the long-range-order,

S~~0,1# !ªH F11n

t G1ntunPZJ , ~4!

andS(@0,1#)5S((0,1#)ø$0%. For example, one gets forS((0,1#),

...,2t,1,11t,212t,213t,314t,415t,416t,517t,518t,... . ~5!

5089J. Math. Phys., Vol. 41, No. 7, July 2000 Aperiodic Virasoro algebra

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From ~4!, one can infer the following properties:

~1! There is ax↔12x-symmetry in the quasicrystal; negative points are obtained from thepositivexPS(@0,1#) as 12x. For S((0,1#), the corresponding statement holds for all pointsexcept 1.

~2! Between adjacent points, only distancest andt2 exist in the case ofS((0,1#); in S(@0,1#) anexceptional tile of length 1 occurs in addition to these between the points 0 and 1.

III. THE APERIODIC WITT ALGEBRA AND ITS CENTRAL EXTENSION

In a first step we look for an aperiodic analog to the Witt algebra, i.e., the Virasoro algebrawithout central term. The idea is to useS~V! as index set and modify the structure constants suchthat a Lie algebra is obtained. Since we are aiming at an analog to the Virasoro algebra, thisprocedure is not always sufficient as can be seen from the deficiencies of quasicrystal Liealgebras;1 however, the algebras introduced here will have the desired property.

A. The aperiodic Witt algebra

For cut-and-project quasicrystals with connected acceptance windows not containing 0 as aninner point we define:

Definition III.1: Let F be any number field and let a,b be real numbers such that0<ab,`. Let V be one of the intervals@a, b#, ~a, b#, @a, b!, or ~a, b!. The quasicrystal Lie algebraAW(V) over F is theF-span of its basis B(AW(V)),

B~AW~V!!5$Lnun5n11tn2PS~V!%,

with the commutation relations of the basis elements given by

@Ln ,Lm#5H ~m22n2!Ln1m if n1mPS~V!

0 if n1m¹S~V!.~6!

In terms of the characteristic function ofV, this corresponds to

@Ln ,Lm#5~m22n2!xV~n81m8!Ln1m . ~7!

With the notationnªn11tn2 and m5:m11tm2 for quasicrystal pointsn,mPS(V), theaperiodic Witt algebraAW(V) may also be expressed in components via the commutation rela-tions (Ln5Ln11tn2

5Ln1

n2),

@Ln1

n2,Lm1

m2#5~m22n2!xVS n11m121

t~n21m2! DLn11m1

n21m2. ~8!

Lemma III.2: As long as the acceptance intervalV is connected and does not contain 0 as aninner point, (6) defines a Lie algebra.

Proof: Antisymmetry is obvious. The Jacobi identity follows from the fact thatxV(n81m81k8)51 implies xV(n81m8)5xV(n81k8)5xV(k81m8)51 if 0 is not an inner point in theacceptance window. h

Note that due to the fact that the index setS~V! is aperiodic,m,nPS(V) does not imply thatm1nPS(V) and thus the condition formulated in terms of the acceptance window is necessary.It leads to the fact that many commutators in~6! are equal to zero and thus the structure of theaperiodic Witt algebra differs crucially from the structure of the usual Witt algebra.

Furthermore, the aperiodic Witt algebrasAW(V) differ from quasicrystal Lie algebras1 by thefact that only the second component of the quasicrystal points enters the structure constants; thisdifference is crucial in order to obtain a central extension of Virasoro type as will be discussedbelow.

5090 J. Math. Phys., Vol. 41, No. 7, July 2000 R. Twarock

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TheAW(V) constitute a family of Lie algebras depending onV. Their properties vary withVas can be seen from the following list of properties:

~a! Any AW(@a,b#) with 2a>b is an Abelian Lie algebra. AnyAW(@a,b#) with 2a,b isnon-Abelian.For the non-Abelian algebras we have furthermore:

~b! The Lie algebraAW(@a,b#) has a nontrivial center precisely ifa.0. Its center is the LiealgebraAW((b2a,b#).

~c! The Lie algebrasAW(@a,b#) and AW(@c,d#) are isomorphic ifa5t2kc and b5t2kd forkPZ.

~d! The algebraAW(@c,b#) is an ideal ofAW(@a,b#) provided 0<a,c,b. Consequentlythere are no semisimple quasicrystal Lie algebras.

~e! The derived algebra ofAW(@a,b#) is AW((2a,b#). Hence, only the algebrasAW((0,b#)andAW((0,b)) are perfect, i.e.,@AW(V),AW(V)#5AW(V) for V5(0,b#,(0,b).

~f! The algebrasAW(@a,b#) are nilpotent ifa.0.~g! In contrast to the usual Witt algebra, where indecomposable subalgebras are formed by

triples, theAW(@a,b#) allow for indecomposable finite dimensional subalgebras of anydimension: The closure of any finite set of generators under the commutation relations is afinite dimensional subalgebra ofAW(@a,b#). In particular, the closure of two generatorsLc

and Ld with a<c8,d8<b leads to a subalgebra of dimension 21g, wheregª]$(l,m)PN3Nulc81md8<b%.

B. The aperiodic Virasoro algebra

The structure and existence of a central extension forAW(V) depends crucially on the choiceof V. In view of potential applications in physical models, we are interested in a type of algebrawhich—although being aperiodic—resembles as much as possible the original Virasoro algebra.Since the second component of the point setS((0,1#) is isomorphic toZ, i.e., the index set of theusual Virasoro algebra, it is a canonical choice for the construction of an aperiodic Virasoroalgebra. SinceS(@0,1#) differs from S((0,1#) only by the point 0 and thus on the algebra level,AV(@0,1#) differs from AV((0,1#) only by the generatorL0 , we will focus onV5@0,1# in thefollowing and comment on the corresponding results for~0,1#. When results apply to both alge-bras, they will be stated simultaneously. For this purpose, for the reminder of this paper, we willuse the notationsV1ª@0,1# andV2ª(0,1#.

Definition III.3: The Lie algebra AV(V i), defined for n5n11tn2 and m5m11tm2

PS(V i) ( i 51,2), via the commutation relations,

@Ln ,Lm#5~m22n2!xV i~n81m8!Ln1m1

c

12n2~n2

221!dn2 ,2m2,

@Ln ,c#50 for all LnPAV~V i ! ~9!

is called aperiodic Virasoro algebra.In components (Ln5Ln11tn2

5Ln1

n2),

@Ln1

n2,Lm1

m2#5~m22n2!xV iS n11m121

t~n21m2! DLn11m1

n21m21c

12n2~n2

221!dn2 ,2m2. ~10!

Theorem III.4: The aperiodic Virasoro algebra AV(@0,1#) is the unique central extension ofthe algebra AW(@0,1#).

Proof: In order to find a cocyclec(n,m) with

@Ln ,Lm#5~m22n2!x@0,1#~n81m8!Ln1m1c~n,m!, ~11!

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notice that apart from antisymmetry we have the following condition implied by the Jacobiidentity,

~m22n2!x@0,1#~m81n8!c~n1m,k!1cyclic permutations50, ~12!

wheren1m is short hand notation forn11m11t(n21m2). For the choicem50 one obtains

~k21n2!c~k,n!1~n22k2!x@0,1#~k81n8!c~n1k,0!50. ~13!

Suppose first thatn1kÞ1. Since we have@L0 ,Lr #5r 2Lr1c(0,r ) we may choose a gaugewherec(0,r )50, i.e.,L r5Lr1(1/r 2)c(0,r ) and rename it again asLr by an abuse of notation. Itthus follows from~13!, sincen21k2Þ0, that in this case we havec(k,n)50.

Suppose now thatn1k51. We haven21k250 and since (n22k2)x@0,1#(k81n8)5(n2

2k2)Þ0 it follows, again from~13!, that c(1,0)50. Furthermore, we deducec(k,n);dk2 ,2n2

5dk,12n .As a next step, choosen,mPS(@0,1#) such thatn1mPS(@0,1#) and let k512(m1n),

which is possible since this is inS(@0,1#). With this we obtain from~12! that

~m22n2!c~n1m,12~n1m!!2~2m21n2!c~12n,n!1~2n21m2!c~12m,m!50. ~14!

Now we choose a gauge forL1 . Because of@L11t ,L2t#522L11c(11t,2t) we can impose

c~11t,2t!50 ~15!

via L15L12 12c(11t,2t).

Now, a case study based on~14! is needed.Suppose first thatmPS is such thatm2tPS. One gets

~m211!c~m2t,12~m2t!!2~2m221!c~11t,2t!1~m222!c~12m,m!50, ~16!

and thus with~15!,

~m211!c~m2t,12~m2t!!1~m222!c~12m,m!50. ~17!

Suppose next thatmPS is such thatm1t25m1t11PS. One gets

~m221!c~11t1m,2~t1m!!2~2m211!c~2t,11t!1~m212!c~12m,m!50, ~18!

and thus with~15!

~m221!c~m1t11,2~m1t!!1~m212!c~12m,m!50. ~19!

Suppose furthermPS is such thatm2t25m2(t11)PS. One gets

~m222!c~m,12m!)2~2m213!c~12t2,t2!1~m211!c~12~m2t2!,m2t2!50, ~20!

and thus due toc(12t2,t2)5c(2t,11t) with ~15!

~m222!c~m,12m!1~m211!c~12~m2t2!,m2t2!50. ~21!

We have thus covered all pointsmPS(@0,1#), since any of them fulfills at least one of theconditions,m1t2PS(@0,1#), m2tPS(@0,1#) or m2t2PS(@0,1#).

Now it remains to show that there is a unique solutionc(n,m) which fulfills simultaneously~17!, ~19!, and ~21!. Define c(n1 ,n2)ªc(n,12n) in order to display the dependence of thecocycle on the two distinct components. Then the above conditions give~renaming againc by c byan abuse of notation!,

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~m211!c~m1 ,m221!2~m222!c~m1 ,m2!50, ~22!

~m221!c~m111,m211!2~m212!c~m1 ,m2!50, ~23!

~m222!c~m1 ,m2!2~m211!c~m121,m221!50. ~24!

For each fixedm1-component, the unique solution is

d~m1!m2~m2221!, ~25!

like in the case of the Virasoro algebra, but now with a functiond(m1). Suppose nowd(m1) isnot constant and insert~25! into ~23! and ~24!. It follows d(m111)5d(m1). Since any pointmPS(@0,1#) fulfills either m1t2PS(@0,1#) or m2t2PS(@0,1#) and thus either~23! or ~24!, theclaim of the theorem is proven. h

Remarks:

~1! SinceL0 is not contained in the commutator algebra, butL1 is, L1 plays the role ofL0 in thecase of the usual Virasoro algebra. In particular, we are here interested in the casen1kÞ1instead ofn1kÞ0.

~2! The argument does not apply forV5(0,1# becauseL0¹AV((0,1#). Since ~14! does notinvolve L0 , it is clear that the cocycle found above is also a solution here, but it is not clearif this solution is unique, because the uniqueness argument involvesL0 ~L0 being in thealgebra, but not in the derived algebra!. There are presently only partial answers to thequestion of uniqueness of the central extension forAV((0,1#) and for the case of generalV.The corresponding results are stated in Appendix A.

~3! Finally we remark that the aperiodic Virasoro algebra is a member of the following moregeneral class of algebras parametrized bylPR,

@Ln ,Lm#5l~m22n2!xVi~n81m8!Ln1m1

c

12~~ln2!32~ln2!!dn2 ,2m2

,

@Ln ,c#50 for all LnPAV~V i !. ~26!

We consider the casel51 because we want to be as close as possible to the usual Virasoroalgebra.

IV. STRUCTURAL PECULIARITIES OF AV „†0,1‡… AND AV „„0,1‡…

As a first step towards a representation theory forAV(@0,1#) andAV((0,1#), we investigatetheir structure in more detail. WhereasAV((0,1#) is perfect as the usual Virasoro algebra,AV(@0,1#) is not. Both are not semisimple. The maximal nontrivial Abelian ideal is given by

AV(@ 12,1#) % Cc and contains all other Abelian ideals. This maximal Abelian ideal is a special

feature of the aperiodic Virasoro algebra.

Recall that by definitionAV(@ 12,1#) % Cc contains those generatorsLn , wherenPS(@ 1

2,1#). Inparticular, this corresponds to thoseLn , wheren5n11tn2 fulfills 1

2<n12(1/t)n2<1.We need the following definitions, where as beforeV1ª@0,1# andV2ª(0,1#:

Definition IV.1: LetV i#V i . Then,

AV~V i !5$LnPAV~V i !unPS~V i !%, ~27!

and the set ofLOWERING, respectively, RAISING generators in AV(V i), i 51,2, is denoted by

AV~V i !25$LnPAV~V i !un,0%,

~28!AV~V i !

15$LnPAV~V i !un.1%.

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Definition IV.2: TheDEGREEof a generator Ln is given by the second component of n,

deg~Ln!5n2 . ~29!

Furthermore, the degree of a product of generators is given by the sum of the degrees of theindividual generators,

deg~LnLk!5deg~Ln!1deg~Lk!. ~30!

Notice that the elementsL0 and L1 are the only generators of degree 0. Thus, the sets

AV(V i)2 andAV(V i)

1 consist of all elements of strictly negative, respectively, strictly positive

degree inAV(V i).The algebrasAV(@0,1#) andAV((0,1#) may be split into a direct sum of the maximal Abelian

ideal and its complement,

AV~@0,1# !5$AV~@ 12,1# ! % Cc% % AV~@0,1

2# !,

~31!

AV~~0,1# !5$AV~@ 12,1# ! % Cc% % AV~~0,1

2# !.

Both contain parts from the rising as well as lowering generators and we have the following:Lemma IV.3: For AV(@0,1#), to each lowering generator of a particular degree in

AV(@ 12,1#)2 corresponds exactly one raising generator of opposite degree in AV(@0,1

2#)1, and to

each rising generator of a particular degree in AV(@ 12,1#)1 corresponds exactly one lowering

generator in AV(@0,12#)

2 of opposite degree.

Proof: SupposeLnPAV(@ 12,1#)2 with deg(Ln)5n2 . Then n8P@ 1

2,1#, whence (12n)851

2n8P@0,12#. This impliesL12nPAV(@0,1

2#)2 and deg(L12n)52n2 . h

We remark that a similar statement holds also forAV((0,1#).For convenience of the reader who is not familiar with cut-and-project quasicrystals, we

indicate the beginning of the one-sided infinite sequences corresponding toAV(@ 12,1#)2 and

AV(@ 12,1#)1 as well asAV(@0,1

2#)2 andAV(@0,1

2#)1 explicitly. The corresponding sequences for

AV((0,12#)

2 andAV((0,12#)

1 differ from those with closed intervals only by the generatorL0 .

AV(@ 12,1#)2,

L2t ,L2123t ,L2326t ,L2428t ,L2529t ,..., ~32!

AV(@ 12,1#)1,

L212t ,L314t ,L415t ,L517t ,..., ~33!

AV(@0,12#)

2,

L2122t ,L2224t ,L2325t ,L2427t ,..., ~34!

AV(@0,12#)

1,

L11t ,L213t ,L416t ,L518t ,L619t ,... . ~35!

Further, we indicate the commutation relations between generators fromAV(@ 12,1#) and from

AV(@0,12#), as well as between generators inAV(@0,1

2#). We obtain

For LnPAV(@ 12,1#) andLmPAV(@0,1

2#) we have

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@Ln ,Lm#5~m22n2!Ln1mx@1/2,1#~m81n8!1c

12~n2

32n2!dn2 ,2m2.

Here,Ln1mPAV(@ 12,1#), i.e.,Ln1m is contained in the maximal Abelian ideal if the commu-

tator is nonvanishing. Ifn1m51, the central term appears.

For Ln ,LmPAV(@0,12#),

@Ln ,Lm#5~m22n2!Ln1m ,

whereLn1m may be either inAV(@0,12#) or in AV(@ 1

2,1#), but the cental term cannot appear. Atrivial commutator cannot occur, because the addition of such points is always in the aperiodicpoint set sincen81m8,1.

Finally, notice that the central terms are contained in the maximal Abelian idealAV(@ 12,1#)

% Cc. Thus, factoring out this set would lead to the loss of the central terms, i.e., both ofL1 andof the central term of Virasoro typec.

V. REPRESENTATION THEORY FOR AV „†0,1‡… AND AV „„0,1‡…

Before constructing highest weight representations, we need a triangular decomposition. WithAV(@0,1#)6 andAV((0,1#)6 as defined above we obtain

Lemma V.1:

AV~@0,1# !5AV~@0,1# !2% C$L0 ,L1 ,c% % AV~@0,1# !1,

AV~~0,1# !5AV~~0,1# !2% C$L1 ,c% % AV~~0,1# !1, ~36!

is a triangular decomposition of AV(@0,1#) and AV((0,1#), respectively.Proof: Observe that

@L1 ,Lr #50 for all LrPAV~~0,1# ! and AV~@0,1# !. ~37!

It follows from the fact that the tiles~distances between adjacent points! are eithert or t2 inS((0,1#). The same holds forS(@0,1#) with exception of the tile of length 1 between the points0 and 1. But in the latter case, the structure constants of the aperiodic Virasoro algebra imply@L1 ,L0#50, because the second component of these two points is zero.

All Cartan elements are of degree 0. Furthermore, the grading~29! is such that if XPAV(@0,1#)2 with deg(X)5 i andYPAV(@0,1#)1 with deg(Y)5 j then deg(@X,Y#)5 i 1 j . h

It is important to notice that forAV((0,1#) the Cartan elements are central. Notice, that

AV(@0,1#)65AV(@0,12#)

6% AV(@ 1

2,1#)6, so that the corresponding generators may be obtainedfrom ~32!–~35!.

A. Highest weight representation

In order to determine a highest weight representation ofAV(@0,1#) we need a highest weightvector ~or vacuum! v0 which is an eigenvector of the Cartan elements and is annihilated by theraising generators, i.e.,

L1v05hv0 ,cv05cv0 , L0v05gv0

Lnv050 for LnPAV~@0,1# !1. ~38!

For AV((0,1#) we use the same definition only dropping the condition withL0 .As a basis for the universal enveloping algebraU(AV(V i)

2) of AV(V i)2 we use the mono-

mials built with the following ordering:

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Ln1¯Lnk

, deg~Ln1!<¯<deg~Lnk

!. ~39!

It is easy to check~using the commutation relations! that indeed any monomial ofU(AV(V i)2)

can be transformed into a linear combination of monomials of the form~39!.We shall use induced highest weight representations~or Verma modules! for which the rep-

resentation space is formed by such basis monomials applied to the vacuumv0 , i.e.,

Ln1¯Lnk

v0 , deg~Ln1!<¯<deg~Lnk

!. ~40!

~Here we do not adhere to strict mathematical exposition, e.g., we omit a tensor sign over a Borelsubalgebra.!

The action of anyLnPAV(@0,1#)1 on a vectorvªLn1¯Lnk

v0 can be calculated by means ofthe commutation relations of the aperiodic Virasoro algebra, whereLn is commuted to the rightuntil it hits and annihilates the vacuum. The action ofLnPAV(@0,1#)2 on v leads to a linearcombination of vectors in the representation space. Further, forL0 we obtain

L0v5~deg~Ln1¯Lnk

!1g!v. ~41!

~Notice that by construction deg(Ln1¯Lnk

),0.) Hence,L0 acts as a grading operator. It splits thevectorsvªLn1

¯Lnkv0 of the representation space into subsets of the same degree and due to~41!

we say thatv is of degree deg(Ln1¯Lnk

). Finally,

L1v5hv, cv5cv. ~42!

Definition V.2: We say that theC-linear combinations of the vectorsv5Ln1¯Lnk

v0 with l

ª2deg(Ln1¯Lnk

) constitute the level l.SinceL0 is not contained in the commutator algebra ofAV(@0,1#), its only function is to

assign a grading or level to the vectors. Thus, even ifL0 is not contained in the algebra as in thecase ofAV((0,1#), one may define the level of a vectorv independently ofL0 according to thedegree of the basis monomials by which it is formed, i.e., we can define

deg~v !ªdeg~Ln1¯Lnk

!52 l , deg~v0!50. ~43!

Hence, the casesAV((0,1#) and AV(@0,1#) may be treated in parallel. Actually, we see thatrepresentations ofAV((0,1#) may be obtained from those ofAV(@0,1#) by settingg50.

Remark V.3: The situation with the generator L0 is similar to the situation with the gradinggenerator d in affine Kac–Moody algebras. In many applications, d is dropped or replaced by agrading operator~incidentally denoted by2L0! obtained via the Sugawara construction.7

This representation of the algebra is not irreducible if the representation space contains asingular vectorvs , i.e., if there is a level which contains a vectorvs¹Cv0

with

Lnvs50 for LnPAV~@0,1# !1,

L0vs5~g2 l !vs , ~44!

L1vs5hvs ,cvs5cvs .

For AV((0,1#) the definition is similar replacing the condition withL0 by

deg~vs!52 lvs . ~45!

It is sufficient to test the first condition in~44! on a minimal set of generating elements ofAV(V i)

1. In contrast to the usual Virasoro algebra, where the corresponding set is formed only

5096 J. Math. Phys., Vol. 41, No. 7, July 2000 R. Twarock

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by the two generatorsL1 andL2 , the minimal set of generating elements ofAV(V i)1 is infinite

because of the aperiodicity of the index set. We give the generating elements of smallest degreeexplicitly up to degree 21,

L11t ,L212t ,L213t ,L416t ,L518t ,L10116t ,L13121t , . . . . ~46!

We give the singular vectors for the first eight levels explicitly in Appendix B. From theseresults, we obtain the followingCONSTRUCTION PATTERN:

Proposition V.4: On the first eight levels, a singular vector for h5c(n221)/24 appears forthe first time on level n and a power of it reappears on level kn with kPN for the same value ofh.

The fact that powers of all previous singular vectors reappear for a givenh is due to the factthat L1 acts as a central charge. The latter accounts also for the rather simple form of thedeterminant formula~see below!.

Furthermore, we have the following:Lemma V.5: The representation with h50 appears on any level. The corresponding singular

vector on level n is given by L2tn .

Proof: The only elements in the linear combination of basis elements describing a generalvector of leveln in the enveloping algebra, which may result inL2t

n21, areL2tn andL2122tL2t

n22

under the action ofL11t . The condition implied byL11t(aL2tn 1bL2122tL2t

n22)50 is 2anh13b50. As a consequenceh50 impliesb50, so thatL2t

n is singular forh50. h

B. The determinant formula

We start by defining an operation which replaces conjugation. Since an↔2n-symmetry doesnot exist for cut-and-project quasicrystals where 0 is not an inner point,~i.e., those which arerelevant for the aperiodic Witt and Virasoro algebra!, the conjugation known for the usual Vira-soro algebra cannot be used here.

Recall, however, that there is an↔12n-symmetry forS(@0,1#), which corresponds to an2↔2n2-symmetry for the second component. ForS((0,1#), this symmetry holds for all pointsexcept 1.

We defineDefinition V.6: TheCONJUGATION v of the generators Ln is given as

v~Ln!5L12n . ~47!

Notice however thatv(@Ln ,Lm#)Þ@v(Ln),v(Lm)#.Implementing this definition of conjugation we can derive an analog to the Kac determinant

formula. An explicit calculation for the first eight levels shows that the determinant formula isgiven as a product of the diagonal elements of the corresponding matrix~see~48!!. It is a conjec-ture that also for higher levels, this statement is valid. We formulate it as

Conjecture V.7: The analog to theKac determinant formula for the aperiodic Virasoroalgebra is given as

detk~h,c!5a~k!)n51

k

~h~n,c!2h!r n~k! ~48!

with

h~n,c!ªc~n221!

24, ~49!

a(k)PR for each kPN and rn(k)5](nPN appearing in p(k)).

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Notice thatr n(k) is, in words, the number of occurrence ofnPN in all possible partitions intonatural numbers ofkPN.

A comparison with the results for singular vectors shows thath(n,c) corresponds to thosevalues ofh where singular vectors have been found on the first eight levels.

In order to make the formula more transparent for the reader, we indicate details about itsderivation for the first levels in Appendix C and point out the construction pattern, from which weconclude the conjecture for higher levels.

C. Some remarks about lowest weight representations

It is possible to define lowest weight representations in a similar way via

L0v05gv0 , L1v05hv0 , cv05cv0 ,

Lnv050 for LnPAV~@0,1# !2, ~50!

where again forAV((0,1#) the condition withL0 drops out, or is replaced by deg(v0)50.A calculation of the first levels shows that singular vectors occur for the same values ofh as

described in Proposition V.4 for highest weight representations. However, the singular vectors areof a different structure. For instance, it happens that for some values ofh the singular vectorcorresponding to the highest weight representation is given in terms of a single basis element ofthe universal enveloping algebra, e.g., forh5c/3 on level 3 byL2123t , whereas the singularvector corresponding to the lowest weight representation at that level and for this value ofh isgiven as a linear combination of basis vectors, e.g.,hL213t22L212tL11t for h5c/3 on level 3.Similarly, one may find a singular vector in a highest weight representation, e.g., on level 2 forh5c/8 ~see Appendix B!, which is given as a linear combination of basis vectors, whereas thecorresponding singular vector for the lowest weight representations on the same level and for thesameh is just a single element, e.g.,L212t on level 2 forh5c/8.

D. Implications for unitarity

The vanishing curves in the determinant formula~48! ~see Fig. 1! allow to determine theregions of unitarity, which are the regions where the determinant is positive semidefinite. From thecondition^huL12n ,Lnuh&5iLnuh&i2>0 it is clear that a necessary condition for unitarity ish<0andc>0.

Notice, that this deviates from the standard exposition of results about the Virasoro algebra,where the corresponding necessary condition leads to a region of unitarity given by the positivequadranth,c>0. The reason for this difference stems from the fact that usually@Ln ,Lm#5(n2m)Ln1m is used, whereas here we consider@Ln ,Lm#5(m22n2)x@0,1#Ln1m , i.e. the structureconstants in reverse order. Changing our structure constants from@Ln ,Lm#5(m22n2)xV i

Ln1m

into @Ln ,Lm#5(n22m2)xV iLn1m would not change the qualitative picture: we would just obtain

Fig. 1 reflected at thec-axis.The vanishing curves of the determinant formula impose further restrictions on unitarity. In

the case of the usual Virasoro algebra, they form parabola in the first quadrant the intersectionpoints of which determine the discrete series corresponding to minimal models.

In the case ofAV(V i), the vanishing curves of~48! are given by straight lines which do notimpose any further conditions on unitarity, because they do not cross the quadrant of unitarity.

5098 J. Math. Phys., Vol. 41, No. 7, July 2000 R. Twarock

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VI. OSCILLATOR REPRESENTATIONS

It is possible to find representations forAV(V i), i 51,2, with V15@0,1# and V25(0,1# asbefore, which are of a similar type as the oscillator representations known for the usual Virasoroalgebra. In order to define them, we need the following operators:

with

j l~k1 ,...,kl !5H 1 if k11¯1klPS~V!

0 otherwise~51!

together with a conjunction* ,

j l 1~k!* j l 2~m!5j l 11 l 2~k,m!. ~52!

The operatorsj l will play the role of a ‘‘bookkeeping device’’ for the dependencies of thepoints of the aperiodic index set on each other.

With

@am2,an2

#ªm2dm2 ,2n2~53!

for m2 , n2PZ, we haveProposition VI.1: The generators,

Lkª1

2 (j 2PZ

:a2 j 2aj 21k2

:j1~k!5..Lk2j1~k! ~54!

FIG. 1. Region of unitarity and vanishing curves of the determinant formula.

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with LkLmª(Lk2Lm2

)j1(k)* j1(m)5(Lk2Lm2

)j2(k,m) and where the dots denote normal order-

ing, fulfill the commutation relations

@Ln ,Lm#5~m22n2!xV i~n81m8!Ln1m1 1

12n2~n2221!dn2 ,2m2

~55!

which corresponds to the aperiodic Virasoro algebra with c51.The proof follows the standard proof.8 For convenience of the reader, we outline the major

steps.Proof: We use the standard cutoff procedure in order to replace infinite sums by finite ones.

For this, we use a functionc:R→$0,1% with

c~x!5H 1 if ixi<1

0 otherwise

and considerLk(e)ª 12S j 2PZ :a2 j 2

aj 21k2:j1(k)c(e j 2), whereLn(e)→Ln ase→0.

One obtains

@ak2,Ln~e!#5

1

2 (j 2PZ

@ak2,a2 j 2

aj 21n2#j1~n!c~e j 2!5

1

2k2an21k2

~c~ek2!1c~e~k21n2!!!j1~n!.

~56!

Then,

@Lm~e!,Ln#51

2 (j 2PZ

@a2 j 2aj 21m2

,Ln#j1~m!c~e j 2!

51

2 (j 2PZ

~2 j 2!:an22 j 2aj 21m2

:c~e j 2!j2~m,n!

11

2 (j 2PZ

~ j 21m2!:a2 j 2aj 21m21n2

:c~e j 2!j2~m,n!

21

2dm2 ,2n2 (

j 2521

2m2

j 2~m21 j 2!c~e j 2!j2~m,n!. ~57!

With the transformationj 2→ j 21n2 and2 12S j 2521

2m2 j 2(m21 j 2)5(m232m2)/12 the claim follows

in the limit e→0, becausedm2 ,2n2implies j2(m,n)5j2(12n,n)51. h

Similarly, for cÞ1 one obtains via the Fairlie construction,8,9 a corresponding result for theaperiodic Virasoro algebra for generalc. Since the proof follows the usual one if the operatorsjare implemented as above, the corresponding proof is omitted here.

VII. CONCLUSION

An aperiodic analog to the Virasoro algebra has been presented, which links Lie algebras withaperiodic point sets and cut-and-project quasicrystals, respectively. The latter serve as an index setfor the generators of the algebra. The structure constants of the aperiodic Virasoro algebra reflectthe aperiodicity of the index set and take into account—via the appearence of the characteristicfunction of the acceptance window—that the aperiodic point set is not invariant under addition.This modification of the structure constants with respect to the usual Virasoro algebra leads tocrucial differences in structure and representation theory.

The representation theory and in particular the determinant formula derived in the frameworkof highest weight representations suggest that minimal models corresponding to a discrete seriesdo not occur in the scope of the aperiodic Virasoro algebra; the setting rather corresponds to the

5100 J. Math. Phys., Vol. 41, No. 7, July 2000 R. Twarock

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h>0 andc.1-case of the usual Virasoro algebra. It is interesting to investigate the application ofthe aperiodic Virasoro algebra in physical models where the Virasoro algebra corresponding tothis parameter range is relevant.

It would also be interesting to find invariant differential operators using the constructedsingular vectors in combination with the oscillator representation above or with some analog of theFeigin–Fuks approach.10

ACKNOWLEDGMENTS

I am very grateful to Professor H.-D. Doebner, Professor V. K. Dobrev, Professor J. Hennig,and Professor K.-H. Neeb for many helpful suggestions and discussions. My special thanks go tothe Rotary Club Goslar-Nordharz for their financial support of my stay at the ICTP in Trieste,where part of this work has been accomplished. Also, the financial support of the ministry ofLower Saxony in the framework of the Dorothea-Erxleben program is gratefully acknowledged.

APPENDIX A: CENTRAL EXTENSION FOR GENERAL ACCEPTANCE WINDOWS

For the acceptance window@0, 1# it has been shown that a unique central term of Virasorotype exists. Concerning the central extension of the aperiodic Witt algebra for generalV, onlypartial answers exist, which are summarized in this appendix.

If L0¹AW(V) then it is not possible to find a gauge such thatc(r ,0)50 for a cocyclec(n,m) leading to a central extension ofAW(V). However, if we assume thatc(n,m) dependsonly on the second components as in the case of@0, 1#, i.e., if we assumec(r ,k)5c(r 2 ,k2) alsohere, then we can prove the following:

Lemma I.1: The cocyle corresponding to a central extension of AW(V) for generalV fulfillsc(r 2,0)50 for all r 5r 11tr 2PS.

Proof: The Jacobi identity implies the condition,

~m22n2!xV~m81n8!c~m21n2 ,k2!1~k22m2!xV~m81k8!c~m21k2 ,n2!

1~n22k2!xV~k81n8!c~k21n2 ,m2!50. ~A1!

With m51, i.e.,m250, we obtain

~n22k2!xV~k81n8!c~k21n2,0!50, ~A2!

and thus conclude thatc(r 2,0)Þ0 is only possible ifr 252n2 , i.e.,r 52n and 2nPS. Since thereare pointsnPS with 2nPS, we have to rule out this option explicitly.

To do this, use the fact that bothn and 2n are quasicrystal points, so that we may setm512n andk52n in ~12!. We obtain

2n2c~0,2n2!1n2xV~3n8!c~3n2 ,2n2!50 ~A3!

and thus find thatc(0,2n2)Þ0 iff 3nPS and c(3n2 ,2n2)Þ0. The idea is to show that theconditionc(3n2 ,2n2)Þ0 would imply 4nPS and so forth. For this, we proceed by induction.

SupposelnPS for l51,...,N. Then we can choosek5ln andm512(l21)n in ~12! toobtain

2ln2c~~22l!n2 ,ln2!1~12l!n2xV~~l11!n8!c~~l11!n2 ,~2l11!n2!50. ~A4!

In particular, it follows that (l11)nPS, otherwise the entire bulk would be zero. However,this contradicts the aperiodicity of the set and hence proves the claim. h

Furthermore, one has for the special caseV5(0,1#,Lemma I.2: Let m,n,kPS((0,1#). Then we have

c~k22m2 ,m2!50 for 0,12m8,k8<m8<1 ~A5!

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and

c~k21n2 ,m2!50 for 0,12m8,k8,12n8,m8<1. ~A6!

In particular, the restriction on the parameter range impliesmPS(@ 12,1#). Notice that with

sª12m similar conditions can be formulated for points inS~~0, 12#!.

Proof: Let n512m in ~A1! and use the previous Lemma to obtain

~k22m2!xV~m81k8!c~m21k2 ,2m2!2~m21k2!xV~12m81k8!c~k22m2 ,m2!50.~A7!

Suppose first thatxV(m81k8)50 andxV(12m81k8)51 ; it follows that c(k22m2 ,m2)50 for 12m8,k8<m8, which corresponds to the first claim.

Note that forxV(m81k8)51 and xV(12m81k8)50 one deduces thatc(m21k2 ,2m2)50 for 12m8>k8.m8, which can be seen to be equivalent to the above statement by taking

instead ofmPAV((0,12#)12mPAV(@ 1

2,1#).The other claim is a direct consequence of the cocycle condition together with the assumption

that k81n8<1 andk81m8.1 as well asn81m8.1. h

As a direct consequence of Lemma 2.2 we formulate the following:

Theorem I.3: Let mPS(( 12,1#) and lPS((2(12m8),1#). Then c( l 2 ,m2)50.

It is interesting to notice that forLl andLm with l andm as given in the Theorem, we alwayshave@Ll ,Lm#50, becausel 81m8.2(12m8)1m8522m8>1, so thatl 1m¹S.

Furthermore, note that if we restrict ourselves toAV((0,12#) and commute two such generators

Ln andLm , then we obtain (m22n2)Ln1m1c(n2 ,m2) where~A7! does not imply any restriction.In particular, suppose that in~A1! we choose the coefficients such that the sum of any two of the

three elements is inAV((0,12#), so that we have

~m22n2!c~m21n2 ,k2!1~k22m2!c~m21k2 ,n2!1~n22k2!c~k21n2 ,m2!50. ~A8!

Then this is solved byc(s2 ,r 2)5s22r 2 ~can be gauged away! as well ass222r 2

2 and s232r 2

3

1s22r 22r 2

2s2 . Hence, ifL0 is absent, there may be other central extensions different from thecentral term of Virasoro type derived forV5@0,1#.

APPENDIX B: SINGULAR VECTORS

Explicit results for the singular vectors on the first eight levels in the highest weight module.Level 1:

h50: L2t ;

Level 2:

h5c/8: v2ª4h/3L2122t2L2t2 ,

h50: L2t2 ;

Level 3:

h5c/3: v3¯L2123t ,

h50: L2t3 ;

Level 4:

h5c/8: v ~4,1!ª8h/3L2123tL2t116h2/9L2122t2 1L2t

4 28h/3L2122tL2t2 5v2

2,

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h55c

8: v ~4,2!ª2hL2224t25L2123tL2t ,

h50: L2t4 ;

Level 5:

h5c: v5ª4h2L2325t212hL2224tL2t28hL2123tL2122t121L2123tL2t2 ,

h50:L2t5 for cÞ0; L2123tL2t

2 for c50.

Level 6:

h535

2c: v ~6,1!ª2

206

9cL2326t1L2123t

2 ,

h5c

3: v ~6,2!ªL2123t

2 5v32,

h5c

8: v~6,1!ª

4h2

9L2123t

2 24h2

3L2123tL2122tL2t1hL2123tL2t

3 28h3

27L2122t

3 ,

12h2

3L2122t

2 L2t2 2

h

2L2122tL2t

4 11

8L2t

6 521

8v2

3,

h50: L2t6 for cÞ0; L2123t

2 and L2123tL2t3 for c50.

Level 7:

h52c: v7ªh2L2427t24hL2326tL2t22hL2224t L2123t15L2123t2 L2t ,

h50: L2t7 for cÞ0; L2123t

2 L2t and L2123tL2t4 for c50.

Level 8:

h5c

8: v ~8;1!ªv2

45v ~4,1!2 ,

h55

8c: v ~8,2!ª4h2L2224t

2 220hL2224tL2123tL2t125L2123t2 L2t

2 5v ~4,2!2 ,

h521

8c: v ~8,3!ªL2428t ,

h50: L2t8 for cÞ0,

L2428t ,L2326tL2122t ,L2326tL2t2 2 9

7L2224tL2123tL2t1 157 L2123t

2 L2122t ,

L2123t2 L2t

2 ,L2123tL2t5 ,L2122t

4 ,L2122t3 L2t

2 ,L2122t2 L2t

4 ,L2122tL2t6 for c50.

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APPENDIX C: DETAILS ABOUT THE DETERMINANT FORMULA

In this appendix, we use the notationLn1¯Lnk

uv0& for the vector Ln1¯Lnk

v0 and

^v0uL12n1¯L12nk

for its conjugate.

~1! On level 1, one obtainsv0uL11tL2tuv0&522h.~2! On level 2, one gets

^v0uL212tL2122tuv0&524h1c

2, ^v0uL212tL2t

2 uv0&50,

^v0uL11t2 L2122tuv0&56h, ^v0uL11t

2 L2t2 uv0&54h2. ~C1!

Thus, det25216h2(h2(c/8)).

~3! Since the determinant formula relies on the observation that the determinant is given by theproduct of the diagonal elements, it is sufficient to indicate where nontrivial entries aresituated. The exact knowledge of the entries is not necessary. Thus, we will indicate nontrivialentries only by a* in the following.

On level 3, the determinant formula is based on the following matrix, where columns arelabeled by v0uL213t , ^v0uL11tL212t , and^v0uL11t

3 , and rows byL2123tuv0&, L2122tL2tuv0&,andL2t

3 uv0&, respectively, to give

S* 0 0

* * *0 0 *

D• ~C2!

The determinant is thus the product of the entries on the diagonal, i.e., the product of^v0uL213tL2123tuv0&, ^v0uL11tL212tL2122tL2tuv0&, and ^v0uL11t

3 L2t3 uv0&. One obtains det3

5(26h12c)(24h1(c/2))(22h))45384(h2(c/3))(h2(c/8))h4.

The principle is similar for higher levels. Also here, we only indicate the nontrivial entries andshow that the determinant is indeed the product of the diagonal elements. Then the claim isimmediate.

~4! For level 4, columns are labeled byv0uL314t , ^v0uL11tL213t , ^v0uL212tL212t ,^v0uL11t

2 L212t , and ^v0uL11t4 , and rows by L2224tuv0&, L2123tL2tuv0&,

L2122tL2122tuv0&, L2122tL2t2 uv0&, andL2t

4 uv0&, respectively, and we obtain

M4ªS*1 *2 0 0 0

0 *3 0 0 0

* * * * *0 * 0 * *0 0 0 0 *

D• ~C3!

When calculating the determinant, we encounter the following phenomenon: whenever there aretwo nontrivial entries in one principal row or column according to which the general determinantis developed, then the subdeterminant corresponding to the nondiagonal element vanishes. This isa general pattern appearing also on higher levels, so we indicate it here once explicitly,

det~M4!5*1 detS*3 0 0 0

* * * *

* 0 * *0 0 0 *

D1*2 detS0 0 0 0

* * * *0 0 * *0 0 0 *

D5*1*3 detS* * *0 * *0 0 *

D. ~C4!

5104 J. Math. Phys., Vol. 41, No. 7, July 2000 R. Twarock

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~5! On level 5 we order columns and rows such that the dependence on level 4 is displayed.This type of order will be used also on higher levels. The entries of the submatrix, whichcan be infered directly from the previous level are denoted by ‘‘4.’’With the order,

^v0uL415t ,^v0uL212tL213t ,^v0uL11tL314t ,^v0uL11t2 L213t ,^v0uL11tL212t

2 ,^v0uL11t3 L212t ,^v0uL11t

5

for columns, and L2325tuv0&, L2123tL2122tuv0&,L2224tL2tuv0&, L2123tL2t2 uv0&,

L2122t2 L2tuv0&, L2122tL2t

3 uv0&, L2t5 uv0&

for rows, we find

M5ªS* * * * 0 0 0

0 * 0 * 0 0 0

0 0 4 4 4 4 4

0 0 4 4 4 4 4

0 * 4 4 4 4 4

0 0 4 4 4 4 4

0 0 4 4 4 4 4

D• ~C5!

~6! Columns in consecutive order for level 6,

^v0uL416t ,^v0uL212tL314t ,^v0uL213t2 ,^v0uL212t

3 ,^v0uL11tL415t ,^v0uL11tL212tL213t ,

^v0uL11t2 L314t ,^v0uL11t

3 L213t ,^v0uL11t2 L212t

2 ,^v0uL11t4 L212t , and ^v0uL11t

6 .

Rows in consecutive order,

L2326tuv0&,L2224tL2122tuv0&,L2123t2 uv0&,L2122t

3 uv0&,L2123tL2122tL2tuv0&,

L2224tL2t2 uv0&,L2123tL2t

3 uv0&,L2122t2 L2t

2 uv0&,L2122tL2t4 uv0&, and L2t

6 uv0&.

Again the order of the elements is chosen such that the dependence on the previous level isdisplayed. The entries of the submatrix, which can be infered directly from the previous level aredenoted by ‘‘5,’’

M6ª1* 0 * 0 0 0 0 0 0 0 0

* * * 0 0 * * * 0 0 0

0 0 * 0 0 0 0 0 0 0 0

* * * * 0 * * * * * *

* 0 * 0 5 5 5 5 5 5 5

0 0 * 0 5 5 5 5 5 5 5

0 0 0 0 5 5 5 5 5 5 5

0 0 0 0 5 5 5 5 5 5 5

0 0 * 0 5 5 5 5 5 5 5

0 0 0 0 5 5 5 5 5 5 5

0 0 0 0 5 5 5 5 5 5 5

2• ~C6!

~7! The calculations for level 7 and level 8 have been carried out and display an analoguousconstruction pattern which again shows that the determinant is the product of the entrieson the diagonal.

Under the sum, the calculation of the first eight levels suggests that the determinant formulais always given as the product of the diagonal entries. This leads to the conjecture formulatedabove.

5105J. Math. Phys., Vol. 41, No. 7, July 2000 Aperiodic Virasoro algebra

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1J. Patera, E. Pelantova, and R. Twarock, ‘‘Quasicrystal Lie algebras,’’ Phys. Lett. A246, 209–213~1998!.2J. Patera and R. Twarock, ‘‘Quasicrystal Lie algebras and their generalizations,’’ Phys. Rep.315, 241–256~1999!.3R. Twarock, ‘‘Lie algebras associated to one-dimensional cut-and-project quasicrystals,’’ Seminaires et Congress, 1999.4J. Patera, ‘‘Noncrystallographic root systems and quasicrystals,’’ inMathematics of Long Range Aperiodic Order,Proceedings of the NATO ASI, Waterloo, 1995, edited by R. V. Moody~Kluwer, Dordrecht, 1997!.

5D. Shechtman, I. Blech, D. Gratias, and J. Cahn, ‘‘Metallic phase with long-range order and no translational symmetry,’’Phys. Rev. Lett.53, 1951–1953~1984!.

6Z. Masakova, J. Patera, and E. Pelantova´, ‘‘Minimal distances in quasicrystals,’’ J. Phys. A31, 1539–1552~1998!.7P. Goddard and D. Oliver, ‘‘Kac–Moody and Virasoro algebras in relation to quantum physics,’’ Int. J. Mod. Phys. A1,303–414~1986!.

8V. G. Kac and A. K. Raina,Highest Weight Representations of Infinite Dimensional Lie Algebras~World Scientific,Singapore, 1987!.

9A. Chodos and C. Thorn, ‘‘Making the massless string massive,’’ Nucl. Phys. B72, 509–522~1974!.10B. L. Feigin and D. B. Fuks, ‘‘Invariant skew-symmetric differential operators on the line and Verma modules over the

Virasoro algebra,’’ Funct. Anal. Appl.16, 114–126~1982!.

5106 J. Math. Phys., Vol. 41, No. 7, July 2000 R. Twarock

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