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İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Mühendisliği Anabilim Dalı

2012

A_n^d cebirinden elde edilen S_d-1 simetrik grubu

Symmetric group S_d-1 from an algebra A_n^d

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Bu tezde A_n^d cebirinden S_(d-1) simetrik grubuna izomorf olan S-grubu elde edilmiştir. Tezimizin ilk bölümünde bazı temel tanımlar verilmiş ve ileriki bölümlerde genelleştirilmesiyle yeni bir cebirsel yapı oluşturacak Fermion ve Boson cebirleri ve bu cebirlerin sayı operatörleri tanımlanmıştır. Bu sayı operatörlerinin özvektörleri yardımıyla temsil uzayları kurulmuştur. Fermion cebirinin özdeğerleri iki tane, Boson cebirinin ise sonsuz sayıdadır. Özdeğeri bu iki değer arasında olan A_d cebiri tanımlanmış ve yapısı incelenmiştir. Bu cebir Orthofermion cebirine izomorf olduğundan önemlidir. Tek Fermion ve tek Boson cebirlerinin daha genel hale getirilmeleri q-deforme Boson cebiri CBY (Coon-Baker-Yu) modelidir. Bu modelin içerdiği reel değerli q-parametresinin limiti sıfıra giderken bize Cuntz cebirini verir. Buradan sonlu boyutlu Cuntz cebirinin bir genelleştirilmesi olan ve A_n^d cebirinin n=1 durumuna karşılık gelen cebir elde edilir. Tüm bu durumları içeren A_n^d cebiri ise tezina_(?_1 ) a_(?_2 )? a_(?_d )=0a_(?_1)^* a_(?_2)^*? a_(?_d)^*=0a_? a_?^*=?_?? (1- ?_(d-1) ),?,??{1,2,?,n}?_(d-1)= a_(?_1)^* a_(?_2)^*?a_(?_(d-1))^* a_(?_(d-1) )? a_(?_1 ),?_i=1,2,?,n ,i=1,2,?,d-1bağıntıları ile verilir.Tezimizin ikinci bölümünde A_1^d cebirinin n-sayıda Fermion için genelleştirmesi olan A_n^d cebiri incelenmiştir. Bu cebirin izdüşüm operatörleri tanımlanmış, bu operatörlerle cebirin üreteçleri arasındaki ilişkiler incelenmiştir. Yine bu bölümde A_n^d cebirinin sonlu boyutlu temsilleri sayı operatörünün özvektörleri üzerine etkisiyle elde edilmiştir.Üçüncü bölümde L_? diye adlandırılan operatörler tanımlanmıştır. Bu operatörlerin özvektörler üzerine etkisi incelenmiş ve L_? temsilleri bir örnek üzerinde verilmiştir. L_? temsillerinin indirgenemez kısımlarının multinomial formül yardımıyla sayılabileceği açıklanmıştır. L_? operatörleri temsil uzayının bir kısıtlanması altında tamamen tersinir operatörlere dönüşmektedir. Bu durumdaki L_?-operatörlerinin kümesinin matris çarpımı altında bir grup oluşturduğu ve bu grubun S_(d-1) simetrik grubuna izomorf olduğu gösterilmiştir.Son bölümde ise genel olarak elde edilen bulgular ve sonuçlar kısaca verilmiştir.

Summary:

In this thesis, a group S that is isomorphic to the symmetric group S_(d-1) is obtained by the elements of an algebra A_n^d. We present some basic definitions in the first chapter to get the meaning of the algebraic constructure of A_n^d and the relations between defined operators and its generators further. In the introduction part, a general definiton of the Fermion algebraaa^*+a^* a=1, a^2=0and Boson algebraaa^*-a^* a=1are given as well and their representations are obtained by means of eigenvectors of the number operator which counts the possible particles of the given state. Number operator is crucial in respect to obtaining matrix representations of the algebra. So we define appropriate number operator at first to obtain representations. Fermion algebra has two eigenvalues as for bosons have infinite number of eigenvalues. In this chapter, we define the algebra A_d whose number of eigenvalues are a positive integer bigger than two. The algebra A_d is defined as follows:aa^*-a^* a=1-d/(d-1)! ? a^* ? ^(d-1) a^(d-1),a^d= ? a^* ? ^d=0This algebra is important because it is isomorphic to Orthofermion algebra which is defined as followsc_i c_j^*+?_ij ?_(k=1)^(d-1)? ? c_k^* c_k=?_ij ? ,c_i c_j=0,c_i^* c_j^*=0,i,j ? {1,2,?,d-1}and in which has d-dimensional representation. We also present the algebra A_d?s matrix representations in the same chapter. Unique Fermion and unique Bosons are generalized as q-deformed Boson algebra which is defined as follows,aa^*-qa^* a=1In this deformed model gives us Cuntz algebra when limit goes to zero. Thus it is obtained an algebra which is a generalization of finite dimensional Cuntz algebra and corresponding to A_n^d when n=1. This algebra is indicated A_1^d and defined as follows:aa^*=1- ? a^* ? ^(d-1) a^(d-1),a^d= ? a^* ? ^d=0As we can see in the definition of A_1^d, the algebra gives Fermion when d=2 and for the whole other d?s it gives finite dimensional Cuntz algebra. Then as a previous research is mentioned in the last passage of the first chapter that the tensor product of A_1^(d_1 )?A_1^(d_2 ) is isomorphic to the algebra A_(d_1 d_2 ).In chapter two, a generalization of A_1^d for n-number of fermions is given as the algebra A_n^d. The new discovery algebra A_n^d is defined as:a_(?_1 ) a_(?_2 )? a_(?_d )=0a_(?_1)^* a_(?_2)^*? a_(?_d)^*=0a_? a_?^*=?_?? (1- ?_(d-1) ),?,??{1,2,?,n}?_(d-1)= a_(?_1)^* a_(?_2)^*?a_(?_(d-1))^* a_(?_(d-1) )? a_(?_1 ),?_i=1,2,?,n ,i=1,2,?,d-1The projection operators of this algebra defines as follows:?_m=a_(?_1)^* a_(?_2)^*?a_(?_m)^* a_(?_m ) a_(?_(m-1) )? a_(?_1 ),m=1,2,?,d-1There is given a lemma to prove that ?_i?s are projection operators. This lemma helps us to proceed more easily on finding the representations of the algebra and the relations between number operator and generators. Projection operators are essential to define number operator. In this chapter it is also shown that each element of the algebra can be written as normal form which means the generators with stars are on the left and the rest on the right. We need to define the number operator of the algebra A_n^d in order to obtain its representations. The representations of the algebra obtain by means of the action of the eigenvectors on number operator. Thus after proving the ??s are projection operators by given lemma, the number operator of the algebra is defined as the sum of all possible projection operators. There is also investigated the action of eigenvectors on the number operator. Then we compose the finite dimensional representations of A_n^d by the action on eigenvectors of number operators and give an example on finding its representations. In the example it is seen that each generators can be represented in the matrix form and stars of generators can be directly found. We may understand how to find representations of the algebra by looking at the example.In chapter three, new discovery L_?,??S_(d-1) operators are defined by using the inverse of ? and expressed how to write by means of generators. Then we investigate their action on eigenvectors by proving a lemma. This lemma helps us to find L_? matrix representations easily. Then we find its representations by using proved lemma. With this proof we simply find L_?-representations by giving an example. As we can see from the example, the matrices of L_? are not invertible and it is shown that irreducible part of L_?-representations can be counted by using appropriate multinomial formula in the same chapter. Then to make L_? matrix representations invertible we use an appropriate restriction on defined set S which consist of all possible L_? operators and write its representation space V as V=kerL_??W since W is a subspace of V. This appropriate restriction which is mentioned above is on the subspace W to seperate zero rows in the representations which cause the matrices non-invertible. If we restrict the set S on the subspace W, then we can show that S is isomorphic to S_(d-1). After making L_? matrix representations invertible, it is first shown that S constitute a group under multiplication by proving group axioms one by one and this group is isomorphic to the symmetric group S_(d-1) by defining a map ?: S|_W?S_(d-1) which is ?(L_(?_i ) )=?_i. Thus it is proved that the set S is a symmetric group.In conclusion part, it is summarized what we have done and briefly given obtaining results throughout thesis.