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Leech örgüsünün bir E8 x E8 x E8 kuruluşu

An E8 x E8 x E8 construction for the leech lattice

  1. Tez No: 39262
  2. Yazar: BAHRİ GÜLDOĞAN
  3. Danışmanlar: PROF.DR. HASAN R. KARADAYI
  4. Tez Türü: Doktora
  5. Konular: Matematik, Mathematics
  6. Anahtar Kelimeler: Leech örgüsü, Leech lattice
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET Bu çalışmada, 24 boyutlu çift ve self-dual bir örgü olan Leech örgü sünün herhangi bir normdaki bir vektörünün, seçilmiş bir Es x Es X Es bazındaki yazılımları elde edilmiştir. Bilindiği gibi kaynaklarda böyle bir lineer bağıntı bulunmamaktadır. Bu yazılımlar çerçevesinde, aynı norma sahip Leech örgüsü vektörlerinin 30 ayrık Cx alt-uzayma ayrılmış olduğu, Cx ve C30-X uzayları arasındaki eşdeğerlik ilişkisinin varlığı gösterilmiştir. Bu arada Cı elemanlarının tüm örgünün basit köklerini teşkil ettiği ve bu basit köklerin sayısının aynı norma sahip vektörler için sonlu olduğu gösterilmiştir. Ayrıca Leech örgüsü için öngördüğümüz gösterim; Leech örgüsünü ayrık noktalarla bir çeşit metrik uzay olarak tanımlamaya da imkan vermektedir.

Özet (Çeviri)

AN E8xE8x E8 CONSTRUCTION FOR THE LEECH LATTICE SUMMARY The lattice theory begins to play an important role in theoretical physics as well as in mathematics. For instance, E% and Leech lattices are frequently used in Dual or Superstring models. Leech lattice A24 has also mysterious connections with hyperbolic geometry, Lie algebras, and the Monster simple group. We can define a lattice in a real vector space V of finite dimension N as a set of points of the form N A = (y^njej : m G Z} (1) i=l where e* (i = 1, 2,..., N) forms a basis for V; it is also called a basis for A. The lattice A will be Euclidean or Lorentzian in the cases that V is Euclidean or Minkowski space respectively. A lattice is said to be unimodular if |dei(ei,ei)| = l; (İJ = 1,2,...,İV). (2) We can also consider the situation where A spans a subspace rather than the whole of V. For any lattice A C V, we define the dual of A, denoted A*, to be set of points y G V for which the inner product (x, y) is integral for all x ? A. if A spans V and the inner product is non-singular, A* is also a lattice which is called the dual lattice of A. In that case we can form a basis for A* by taking the basis e*, (i = 1, 2,..., AT), for V dual to e,-, (i = 1, 2,..., N), so that (e,, e|) = 6ij. The lattice A is integral if (x, y) is an integer for every x, y 6 A. This is equivalent to the condition A C A*. The condition that A be both integral and unimodular is equivalent to the condition that it is self-dual, i.e A = A*. The type of a lattice A is even (or II) if the norm x2 = (a;, a;) of every element x of A is even integer, and odd (or I) otherwise. Even unimodular lattices are especially interesting. E& and the Leech lattice A24 are even unimodular lattices, while Z, Z2, Z3,..., are odd unimodular lattices.The classification of odd and even unimodular lattices is an important problem in number theory and in the other parts of mathematics. Even unimodular lattices exist if and only if the dimension is a multiple of 8, while odd unimodular lattices exist in all dimensions. E$ is the unique even unimodular 8-dimensional lattice, and E%®E% and D\q are the only two such 16-dimensional lattices. The even unimodular 24-dimensional lattices were enumareted by Niemeier [2], who found that there axe 24 such lattices, 23 of them with minimal norm 2 and remaining one is the Leech lattice A24, with minimal norm 4. His proof was simplified by Venkov [3], who used modular forms to restrict the possible root system of such lattices. There are at least 80000000 distinct even unimodular 32-dimensional lattices [20]. The lattice of J. Leech is really a striking structure in many respects. It is an even self-dual lattice in 24-dimension and among 24 Niemeier lat tices it is only one which defeats an explicit Lie algebraic construction. It is interesting to note that its first construction [4,5] is due to completely different reasons, the densest packing of spheres in 24-dimensions. Con way gave a short proof that Leech lattice is characterised by some of its simple properties [6]. When he found his lattice, Leech conjectured that it had the covering radius y/2 because there were several known holes of this radius. Parker later noticed that the known holes of radius \/2 seemed to correspond to some of Niemeier lattices, and inspired by this Conway et a! [7] found all the holes of this radius. There turned out to be 23 classes of holes which were observed to correspond in a natural way with the 23 Niemeier lattices other than the Leech lattice. Conway and Sloane presented [8] 23 constructions for the Leech lattice, one for each class of hole or Niemeier lattice. Two of these are the usual constructions of the Leech lattice from the Golay codes over JF2 and F3. Conway [9] later used the fact that the Leech lattice had a covering radius y/2 to prove that the 26- dimensional even Lorentzian lattice 1/25,1 has a Weyl vector, and that its Dynkin diagram can be identified with the Leech lattice A24. There is also related explicit presentation [10] of some Leech lattice elements. Second important motivation which arouses the mathematical interest to large extent comes from the intriguing relation [12] of the Leech lattice with the Monster group [11]. Conway [13,14] has used the Leech lattice in a straight forward manner in his simple construction for the Monster group. It is also proposed [15] that there could be an infinite dimensional algebra of infinite rank acting for the Monster group just like the Lie algebras as its simple roots. We will turn back to this point in later. On the other hand, one of the deep holes is related with E&x E$x E$ algebra in the Niemeier list. There are several approaches [17,18] which viuse this Lie algebra in order to expose the interplay between the Leech lattice and the Monster group. We will also use here this Eg x Eg x Eg structure but in a quite different manner. Beside all these mathematical interest, the Leech lattice has an in evitable attraction also for physicists. Chapline [16] has pointed out us a direct relationship of the Monster group with a string model which is realized in vertex operators construction with which he compactified the model on the Leech lattice. There are also other contributions which leave us with the exciting possibility that our true physical universe could be that of a 26-dimensional space-time with the discrete points and Eg x Eg x Eg structure will be of great importance for carrying out this, especially in order to relate our 10 dimensional space time with the E$ x Eg gauge symmetry. With all these motivations in mind, we want to make a different Eg X Eg x E& appoach to the Leech lattice. In the framework of this approach we have the possibility to represent any Leech lattice vector expilicitly in terms of a definite weight system of Ag Lie algebra which is an Eg sub-algebra of maximal rank. We hope that this proves useful in vertex operator construction of the Monster group, in understanding of the Monster Lie algebra and its simple root system and also relating isometries of the Leech lattice metrics with its automorphism group -0 [19]. It is useful to divide the whole Eg lattice to disjoint sets A(2m) of points a which are defined for m = 0, 1, 2,..., by A(2m) = {a ? Eg : (ex, a) = 2m} (3) The Eg lattice is to be considered in the form of oo £8=£A(2m) (4) TO=0 for which every A(2m) has the decomposition A(2m)= J2 WW ^ A£A(2m) in terms of Ag Weyl orbits W(A) or a filtered decomposition A(2m) = Ko + Ki + K2 (6) in terms of the equivalance classes K^, (fi =0,1,2), K0 = {(*Ai -/M2,---}> (Ai ^ M = 1,2,..., 9) Kx = {Juv4l+^A2+A'A3,...}, {At < A2< A3 = 1,2,...,9) (7) K2 = {pai + Va2 +... + ft As,...}, (Ai <...<A6 = 1,2,...,9) viiIt is now easily seen that the whole E% lattice will be generated by the relations KM + K“ = K”+“ {mod 3). (8) The sum here is in the sense of the stun of two weight vectors and A$ tracelessness condition 1*1 + A«2 + ^3 + £*4 + /*5 + IK + ^7 + A*8 + p% = 0 (9) acts here just like a modularity condition. Leech lattice A24 has a de composition Leech = ^ A24(2m) (10) m=0 similar to (4) with._ 65520.,.,.>,”“,. where r(m) is Ramanujan function and <ru(m) is the sum of 11-th powers of the divisors of integer m. What it means, on the other hand, by the fact that it defeats a Lie algebraic treatment is the lack of a decomposition as in (5). We have shown that it has a decomposition 30 A24(2m) = J]Cx (12) x=l just as in (6). We presuppose a generic form for a Leech vector «(a, /?, 7) v(a, Ş, 7) = (« ”^«) 9 09 - ^Sfi) © (7 - ^) (13) which shows us that the Leech lattice continues to present us some new unexpected features. The vectors a,fi,y participate in a vector of the E$ weight lattice on condition that each vector is orthogonal to the others and 8a, 8p, £7 are Weyl vectors corresponding to E$ vectors. We will mainly concerned with A24(4), A24(6),A24(8) and specified its elements ar,/3,7 for each class Cx explicitly. A first glance, we only have the requirement 30 lui2 t(a) + i(Ş) + 1(7) = -(na + np + nJ + x2 - *-£-) + x (14) viiiwhere (a, a) = 2na, (0, p) = 2n^, (7, 7) = 2n7 (a,*a) = t'(a),0M/O = W,(tA) = «'(7) axe possible solutions for (14) and they axe restricted by i(a) + i(0) + »(7) <! »T**(tt) + İmax(f3) +imax(rr)- (15) We propose a decomposition Cx = J3v,(n). (16) re Every element of the subclasses Vx(n) is then expressed as the sum of x number of elements of Vs(ni) for i = 1, 2,..., x. This simply means for the allowed values of n in (16) that n = ±-((2na + 2np + 2n7) - \v\2(l - x2)) (17) Ax It is now seen that the vectors (13) together with the conditon (14) and (15) provide us a fundamental setting in which the Leech lattice is embedded via the self-duality condition (17). In the tables (7.1), (7.2) and (7.3) we explicitly displayed that dimCx = dimCso-x. (18) is valid for x = 1,2,...,29, which reflects the existence of a kind of duality relation C* = C8o-x(* = l,2,...,29) This however appears in a more detailed form as Vx(n) = y3o-x(31 + 30m - n) (19) in the sense that a + a* + fi + Ş* + 'Y + 'Y*^Sa + Sfi + Sy (20) with an apparent notation. We note that the sub-classes V15 and V30 are self-dual in the sense of (20). As a final remark, We stress that this work describes the Leech lattice as a modular Z-graded structure which is composed out of 30 disjoint classes of points. This shows a striking similarity with the Lie algebras for which we know they also have modular Z-graded formulations. IXBOLUM I GİRİŞ Örgü (lattice) teorisi matematiksel açıdan başlı başına bir çalışma alanı oluşturduğu gibi, son yıllarda özellikle dual ve sicim (string) teori lerinin ele alınmasıyla fizik açısından da ilgi çekici olmuştur. Örgülerin genel yapıları dışında, tamsayı örgü, çift örgü ve self dual örgü olma gibi bazı özellikleri taşıması istenmektedir. Sicim teorileri açısından da bazı fiziksel koşullar, örgüler arasında seçim yapmamızı sağ lamaktadır. Örneğin modüler invaryansm çift ve self dual örgüler istemesi gibi. ilgi çekici olan noktalardan biri de örgüler üzerinde bu şekilde konu lan koşulların, o örgünün boyutlarını kısıtlamasıdır. Örneğin bir çift örgünün yahut self dual örgünün, her boyutta olamayacağını biliyoruz. Bu konularda bize örgü sınıflandırma teoremleri yardımcı olmaktadır. Esasında tamsayı örgülerin (ya da kuadratik formların) sınıflandırma çalışmaları, Siegel ve Witt 'den beri yapıla gelmektedir [1]. Bu çalış malarla ilgili olarak; self dual örgülerin sımflandırma teoremleri, bize şu sonucu vermektedir: çift ve self dual örgüler, sırasıyla E$, Eğ® Es ve D\§ Lie cebirlerinin kök sistemleriyle ilişkilidir. Bu tür örgülere Lie cebirsel örgüler diyoruz. Diğer taraftan 24 boyut için, 24 tane çift ve self dual örgü bulunduğu bilinmektedir. Witt tarafından başlatılan bu çalışmalar Niemeier [2] tarafından ta mamlanmıştır. Bu örgülerin ispatı daha sonra Venkov [3] tarafından

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