Leech örgüsünün bir E8 x E8 x E8 kuruluşu
An E8 x E8 x E8 construction for the leech lattice
- Tez No: 39262
- Danışmanlar: PROF.DR. HASAN R. KARADAYI
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 107
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 ,“”,. where r(m) is Ramanujan function and
Benzer Tezler
- Etkin yükseltgenler varlığında kalkopiritin basınç liçinin incelenmesi
Investigating the pressure leaching of chalcopyrite in the presence of effective oxidazing agents
MEHMET DENİZ TURAN
Doktora
Türkçe
2010
Kimya MühendisliğiFırat ÜniversitesiKimya Mühendisliği Ana Bilim Dalı
DOÇ. DR. H. SONER ALTUNDOĞAN
- Türkiye'deki myotis myotis (borkhausen, 1797) ve myotis blyti (tomes, 1857) türlerinin karyotipleri ve taksonomik durumları (mammali chiroptera)
Karyology and taxonomi status of myotis myotis (borkhausen,1797) and mytis blythi (tomes, 1857) in Turkey (mammali chiroptera)
NURSEL AŞAN
- Molecular description of medicinal leech (Hirudo verbana carena, 1820) based on DNA sequences
DNA dizilerine dayalı olarak tıbbi sülük (Hirudo verbana carena, 1820)'ün moleküler tanımlaması
AL-HUSSEIN HAMEED HUSSEIN OKBI
Yüksek Lisans
İngilizce
2023
BiyolojiAfyon Kocatepe ÜniversitesiMoleküler Biyoloji ve Genetik Ana Bilim Dalı
PROF. DR. MEHMET OĞUZ ÖZTÜRK
- Ratlarda tıbbi sülük ve tıbbi sülük salgı ekstraktının insizyonel yara iyileşmesi üzerine etkisi
Effect of medicinal leech and medicinal leech secretion extract on incisional wound healing in rats
NURMUKHAMAD KHUSAINOV
Yüksek Lisans
Türkçe
2024
BiyokimyaGazi ÜniversitesiTıbbi Biyokimya Ana Bilim Dalı
DOÇ. DR. KÜBRANUR ÜNAL