Leech örgüsünün açık bir kuruluşu

Abstract

ÖZET Bu çalışmada Leech örgüsü vektörlerini E8 xE 8xE8 Lie cebri cinsinden açık bir şekilde elde edebilmek maksadi ile bir yöntem geliştirdik. Böylece Leech örgüsü vektörlerini bu Lie cebrinin Vfeyl örgüsündeki hangi noktalara tekabül edeceğini gösterdik. Geliştirdiğimiz bu yöntem SUC9D weyl yörüngelerini kullanarak Leech örgüsü vektörlerinin sayısını açık olarak bize verir. Bu sonuçlar SUC 9 0xSUC 90 xSUC 90 ağırlıkları formunda verilmiştir.

A FUNDAMENTAL SETTING FOR THE LEECH LATTICE SUMMARY The theory of lattices begin to play a fundamental role in theoretical physics as well as in mathematics. A lattice can be considered to be a system of points which are embedded in a linear vector space. The lattice point X will be chosen by the constraint D X = / Xe., X. <= Z (1) = Vxe, X <= where Z is the set of integer numbers and D is the dimension of underlying linear vector space. The basis vectors e. are naturally taken to be an appropriate basis of this space. Let us note here that the condition d < D is in general valid for dimension d of the lattice - v-The scalar product Ce., e D of this under lyina linear space can also be adopted for the lattice and this can be used to defined a dual lattice. Let us take the lattice A is the set of points C1D and the dual lattice A is the set of points X which are specified by CX*/XD e Z C8D it is now seen in general that A c A* C3D and in case of A = A* C4Z> the lattice called self -dual. An even lattice is defined by the condition that -vi-(X, X) = 2n (5) for all points X e A and n=0,l,2,. Main problem here is to classify and explicit construction of lattices. It is an interesting fact even self- dual lattices can be constructed only for dimensions d=8n. This is one of the main results in clasif ication of Euclidean even self- dual lattices. There is only one even self-dual lattice in d=8 and it is directly the root lattice of E` exceptional algebra in d=16 the even self-dual lattices be which belong to E xE and D Lie a a itf algebras. Neimeier has shown that thera are 24 even self-dual lattices in d=24. Twenty three of Neimeier lattices are to be characterised by the existence of vector lengths (X5X) = 0,2,4,6,8,... (6) -vi x -whereas only one is characterized by (X,X) = 4,6,8,... (7) This is the so-called Leech lattice and it is the condition (7) which prevents this lattice from being a Lie algebraic lattice. Although it is non- Lie algebraic, we will show in this work that there is a way to contruct the Leech lattice explicitly in terms of the E xE xE algebraic lattice. CI 8 8 Our main proposition is to consider the points a + ft + r ~ -30` A (8) to be the point in Leech lattice where A = 6 + 6` + 6 (9) a ft y -viii-and 6 is an E Meyi vector. The point here is to choose appropriately the vector a from among the weight lattice of E a algebra in such a way that (8) belongs to the Leech lattice. The main result is that the parameter X takes values 30. ri algebra. Let us recall here that 30 is the Coseter number of E exeptional algebra. -