Afin kac-moody cebirleri

Abstract

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-

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<Y,i=i,...,d, where h. corresponds to the co-root, oÇ. so that all of their eigenvalues are integers. For a^ &(h^) to~hold in i, it follows that w.(h.)= (w.o(j) = £{>, 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 £*,<*& O, and a basis for the Cartan subalgebra ft. Thus, the roots label a basis of g. The elanent e^ has the effect of shifting the weight of a state:, Where N is a constant depending on &, c* and the particular representation space, and may even be zero. The Cartan subalgebra acts on the basis vectors of the root space decomposition of Lie algebra as _. /. « » IK,vtf = o Each root B is a linear combination of the simple roots, Where the integers v, have same sign. and d is the rank of the group. > >* * 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°<iA«iK) and w,u>= &i^m~ wov)-w</^^/) The geometry of the root system is encoded in the Cartan matrix A; A has integer matrix elemnets» The Cartan matrix A of a simple Lie algebra of rank d is an integral dxd matrix* It is satisfied for the different conditions. N A finite dimensional simple Lie algebra has d simple roots, whenoc *s adjMsöd-to*,.. PC of a finite dimensional algebra, yields a (d+l)x(d+l) Cartan` matrix satisfying an affine algebra. To picture the affine root system, imagine an infinite number of sheets of paper in a stack extending * in both directions with, finite dimensional Lie algebra root system drawn on each page. The n-th page in the stack contains the set of roots «x+n£, wherec»(istan finite dimensional Lie algebra root. The scalar product is degenerate in the sense that it does not depend on the page number the scalar product ofot on page zero with the <X on page n, where -vl-Adding or subtracting £ performs a vertical shift form one page of the stack to. the next. In analyses of Kac-Moody root and weight systems, there is often opportunity to exploit their symetries. If c* is a nonzero root, then the Euclidean reflection r acts on a weight & as r«W *. & -H*l°0/t>«l«)<* with r^CS) a weight. Each r^ is called a weyl reflection, and the set of Weyl reflections generates a group called the Weyl group. Weyl1 s theory of finite dimensional representations is similar, affine representation theory and with that of all Kac-Moody algebras. Weyl* s classification of finite dimensional irreducible representations of g is conveniently stated inj'terms of dominant weights. Any weight with non-negative components &'(h.) is called a dominant weight. Each dominant weight is a highest weight of one and only one finite dimensional irreducible representation. A highest weight vector is and IA.A) or I IV) e J A, A) = O for any positive root. The state of highest weight j//»A7 » which is the ground state of a physical system in simple models, is a vacuum '} state. of lattice P:-' is on a rather different footing then the sublattice `p associated with finite dimensional algebra g. The weyJL group on p is finite dimensional, iu,~. contrast to the weyl group on P*, which is infinite dimensional. The weyl reflections can be used to organize the weight systems, and are a major factor in determining the `shape` of the affine weight systems. The Weyl reflections of weights in affine lattice are defined in the same way as in the finite case. The weyl group action on hc is generated by the reflections weyl reflections do not change level. In physical application of affine Kac-Moody algebras g, the operator -d, or equivaletly L0 is identified with a physical quantity such as the en&gy, whose spectrum must be bounded below for physical reasons. A representation in which the spectrum of -d is bounded below is called a highest weight representation. The whole of a unitary highest weight representation can be built up from vacuum states A$3atisfying If the representation is irreducible, there states form an irreducible representation of the finite dimensional algebra g and the irreducible representation of g is characterised by this vacuum representation of g, or, equivalently its highest weight />e, and the value of the central term k. There is a simple set of necessary and sufficient conditions far..there tn. be- -unitary -highest weight representation of f in which -vii-k takes a particular value and the vacuum representation of g has highest weight VQ ; it is and where f is the highest root of g. The non-negative integer Ikfy1 is called the level of the representation of ğ. The correspondence oÇ^ «- * h, i=o,,...,d betwen Pandft ia* now extendend to nf andfct- '(h is extended Cartan subalgebra and fi its dual). E was extended by L^ so that and uniguely detrmine the weight &. To correspondence °^ <A `^Wi i^o, <a between n and h is now« extended to if* and ne. AThe problem is to find a wector in w that corresponds to Le in t'. Since h. corresponds to °${, i=o,...,d, and L6 is linearly independent of h., a linearly independent vector f^mast be added to nc. The simplest choice for the correspondence is A0« - » -Lo. The minus sign is inserted for consistency of with the duality relations and, i A basis for d+2 dimensional h is (A^,, o^,.... t/p, and its dual basis is (&,w,w,...,w,). Let tıs further define he by o x d A.lV/;Ww>, 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 <[+.)* WMa VAo/<)c AAKV-^o This leaves the evaluation of (A./Ao) = AA-V..) for which the above considerations give no constraints. Its value has no effect on the general structure of the representation theory, because det (DA)=0 for an affine Cartan matrix. The bilinear form for ft**' in the basis, [ A,,«K*,...., °0*'}is then ^O`.! 0 0` G l= i: (DA)X is the dxd Cartan matrix of g, where bars are placed over quantities refering to the finite dimensional algebra from which £ is constructed. G is inverted to give the. matrix of the metric relative to the dual basis ^-1 £& v wdV where (DA) x is the metric of weights of g* as comput in The solution is f^)]y Sinre DA is invertable for the finite dimentional case. G= 0 C! o 0 -d 0 (DÂ)`1 For the direct affine case treated Cq=1 The choire (A A)=0 has no effect of making wQ=A0, since (yyJoÇ )=4i=(wol^P «* (A0I Where A 0 is a highest weight and ^»....^ axe roots. -ix-