Afin kac-moody cebirleri
Affine kac-moody algebras and their representations
- Tez No: 14360
- Danışmanlar: DOÇ.DR. JAN KALAYCI
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1990
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 32
Özet
ÖZEE Tez giriş niteliğindeki ilk bölüm dahil 5 bölümden oluşmaktadır. İkinci.; bölümde kesikli toplanabilir kuantum sayılanım oluşturduğu örgü incelendi. Bu örgünün içinde bulunduğu uzay Cartan altcebiri uzayının dualidir. Bu iki uzay arasındaki ilişkiler incelendi ve skaler çarpımlar tanımlandı. 3\\' Bölüm 3' de sonlu boyutlu ve Afin cebirlerinin yapısı göste rildi. Dördüncü bölümde yine sonlu boyutlu ve Afin uzaylarındaki weyl yansımaları ve temsillerin sınıflandırılması yapıldı. Beşinci bölümde SU(3) ve Afin SU(3)'ün temsilleri incelendi. Enyüksek ağırlık vektörleri ile merkezli eleman arasındaki ilişki incelendi. -iv-
Özet (Çeviri)
AFFINE KAC-MQODY ALGEBRAS AND THFTTR REPRESENTATIONS SUMMARY The purpose of this study is to obtain highest weight representation theory of finite dimensional and affine Kac-moody, algebras. In section II, we contact with the quantum mechanics through the quantum numbers that label the field operators and states of the system. The primitive concept here is the discrete additive quantum numbers that are the eigenvalues of a maximal set of independent, simultaneously diagonalizable elements of a Lie algebra. The physically allowed values of d independent discrete additive quantum numbers form a d-dimensional lattice p with veetors &i^aiWi -.:. EqCl) where the w ; are basis vectors on the lattice and the coefficients a. are integers if & is on the lattice. The lattice is embedded in a a dimensional real space with all points coordinatized by Eq(l), but H^ being real numbers* This space is called t*; it is shown that t* is dual to the Cartan subalgebra of g over the reals; the real Cartan Subalgebra is called I.. A Cartan subalgebra is a maximal set of commuting elements in the Lie algebra.. If the coeffients av in Eq(l) are taken to be complex numbers, the space is h*, and iiis dual to the complexified Cartan subalgebra ft of g. Our starting point is the lattice of quantum numbers, rather than the diagonalizable elenrens of Lie algebra c that are constructed from the quantum fields. The diagonal operators belong to a linear space that is dual to the space spanned by the quantum numbers. There is a basis of the Catan subalgebra t(or h) hjl, i=i,...,d, such that each state in finite dimensionaj. representation is labelled by a weight &=(o.,.,.,a,).. The space t * containing the weights is dual to t. It is (by aefinition) the space of linear functionals on t, which leads to the notation, &(h )^a,...,&(h,)=ad, with a. the components of &. There exists a basis of ^.t*(or h*7vwioi elements o, Thus, tt.attice in t* and the set of operators fchA in t another. In a lojically different sence of tne word dual, bat not very different in a practical sense, (w/J. and [pli} are dual bases of t*. The states ; [ |&^}are labeled by the quantum numbers &(h.): Eq(l), it follows that w.(h.)= (w.|o(j) = 6^, Thus, the basis {w/J of the lattice in t* and the set of operators ThA in t are dual tö one ^l^&Ch.)! fy* C&|flÇ)| ^= ^l*)» i=i,...,d -v-The f h.l are set of simultaneously diagonalizable operators spanning a linear space t, whose simultaneous; eigenvalues lie on a d-dimensional lattice p in its dual space t * The situation is not very different for affine and even more general classes of Lie algebras. A basis of g is obtained by choosing a vector e^ from each subspace £*, >* * Because of the duality,- between 11 and Ti, there exists a vector in ft for each h£n. Aş stated ^earlier, h. of ft is defined to correspond the co-roots oC. in ft. A consistent choice for all Kac-Moody algebras is, *. v oC{= x°= &i^m~ wov)-w«l«)e, and the value of the central term k. There is a simple set of necessary and sufficient conditions far..there tn. be- -unitary -highest weight representation of f in which -vii-k takes a particular value and the vacuum representation of g has highest weight VQ ; it is and where f is the highest root of g. The non-negative integer Ikfy1 is called the level of the representation of ğ. The correspondence oÇ^ «- * h, i=o,,...,d betwen Pandft ia* now extendend to nf andfct- '(h is extended Cartan subalgebra and fi its dual). E was extended by L^ so that and uniguely detrmine the weight &. To correspondence °^ , u ö,...-^ This is a convention, and other choices can be useful, but this freedom doeg not change the general results below. The scalar product on nv is synietric., since
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