Analytic Representations Based on SU(1,1) Lie Algebra Coherent States for Squeezed Displaced Fock States

The coherent states (CSs) of the SU(1,1) group can be divided into two broad categories: (a) the Barut-Girardello coherent states (BGCSs) and (b) the Perelomov coherent states (PCSs). Some definitions for the squeezed displaced Fock states (SDFSs) are given. The hyperbolic analytic representation in the complex plane is considered. An analytic representation of the SU(1,1) Lie group is given and the representation in the unit disk based on the SU(1,1) PCSs for SDFSs is considered.      

SL(3,3~n)和SU(3,3~n)的第一Cartan不变量

确定Cartan不变量是代数群与相关的李型有限群的模表示理论中的一个重要方面.作者利用代数群模表示理论中的一系列结果,计算了3~n个元素的有限域上特殊线性群SL(3,3~n)和特殊酉群SU(3,3~n)的第一Cartan不变量,得到如下结论:当G=SL(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n;而当G=SU(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n+2·(1+(-1)~n),其中a,b是多项式x~2-20x+48的两个根.另外,作者也得到了射影不可分解模U_n(0,0)的维数公式:dim U_n(0,0)=(12~n-6~n+∈)·3~(3n),其中,当G=SL(3,3~n)时,∈=1;而当G=SU(3,3~n)时,∈=-1.     

Factorization of Cosine Functions on Compact Connected Groups

Based on the structure theorem of compact connected groups, the following factorization theorem is obtained in this paper: if a compact connected group G has direct factors isomorphic to SU(2), then non-classical cosine functions on G exist and each of them factors through one of these direct factors; otherwise, all cosine functions on G are classical. On the other hand, all cosine functions on SU(2) are obtained using some techniques on functional equations. Our main tools are from the structure theory of semisimple Lie groups.     

On Hardy’s Theorem on SU(1, 1)

The classical Hardy theorem asserts that ■ and its Fourier transform ■ can not both be very rapidly decreasing.This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform.However,on SU(1,1)there are infinitely many"good"functions in the sense that ■ and its spherical Fourier transform ■ both have good decay. In this paper,we shall characterize such functions on SU(1,1).     

Defining relations on the Hamiltonians of <Emphasis Type="Italic">XXX</Emphasis> and <Emphasis Type="Italic">XXZ R</Emphasis>-matrices and new integrable spin-orbital chains

We present several complete systems of integrability conditions on the density of the Hamiltonian of a spin chain matrix. The corresponding formulas for R-matrices are also given. The latter are expressed via the local Hamiltonian density in a form similar to spin one half XXX and XXZ models. The result is applied to the problem of integrability of SU(2) × SU(2)-and SU(2) × U(1)-invariant spin-orbital chains (the Kugel-Homskii-Inagaki model). Eight new integrable cases are found. One of these cases corresponds to the Temperley-Lieb algebra, three cases correspond to the algebra associated with the XXX model, one case corresponds to the algebra associated with the XXZ model, and one case corresponds to the algebra associated with the graded XXZ model. The remaining two R-matrices are also presented. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 50–58.     

Some explicit bounds in groups with a finite number of non-normal subgroups

The number of conjugacy classes of non-normal subgroups is an invariant of a group G denoted by v{G). In this paper explicit upper bounds for the order of the commutator subgroup G' and for the index of the centre in G, [G : Z(G)] are given for a group G with only a finite number of non-normal subgroups. The bounds provided are functions of v(G) and the primes appearing as orders of elements of G.     

(a,b,n)-临界图的几个充分条件

设G是一个图 .设g和f是两个定义在V(G)上的整值函数使得对V(G)所有顶点x有g(x) ≤f(x) .图G被称为 (g ,f,n) 临界图 ,如果删去G的任意n个顶点后的子图都含有G的 (g ,f) 因子 .本文给出了图是 (a ,b ,n) 临界图几个充分条件 ,即度和邻域条件 .进一步指出这些条件是最佳的 .     

二分(mg,mf)-图中的(g,f)-因子

设G是一个二分图具有顶点集V（G）和边集E（G）。设g和f是定义在V（G）上的两个正整值函数使对任意的x∈V（G）有g(x)≤f(x)，G的一个（g,f）－因子H是G的一个生成子图满足g(x)≤dH(x)≤f(x）。若图G本身是一个（g,f）－因子，则称G是一个（g,f）－图。本文得到一个（mg,mf）－图具有特殊性质的（g,f）－因子的充分条件，从而推广了文献[6]中的一个结果。     

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of \((\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})\) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.     

关于分数可消去图的若干结果

令G=(V(G),E(G))是一个图,并令9和f是两个定义在V(G)上的整数值函数且对所有的x∈V(G)有g(x)≤f(z)成立．若对G的每一条边e都存在G的一个分数(g,f)-因子G_h使得h(e)=0,其中h是G_h的示性函数,则称G是一个分数(g,f)-消去图,若在G中删去E′■E(G),|E′|=k后,所得图有分数完美匹配,则称G是分数k-边-可消去的。本文给出了图是1-可消去,2-可消去和k-边-可消去的与韧度和孤立韧度相关的充分条件。证明了这些结果在一定意义上是最好可能的．     

Canonical solution of the SU(3) ↓ SO(3) reduction problem from the SU(3) pattern calculus

Because of its numerous applications to physics, there have been many solutions published on the problem of reducing a given irreducible representation (irrep) of the unitary unimodular group SU(3) into irreps of the proper orthogonal subgroup SO(3). Such solutions are generally based on an arbitrary construction of a nonorthogonal basis of the highest weight space for an irrep of SO(3), followed by an equally arbitrary orthonormalization procedure. This paper presents a unique solution of this problem based on the intrinsic structure of the multiplicity function, which is a function M _L(p, q) giving the number of times irrep[L] of SO(3) is contained in irrep[pq0] of SU(3). This structure is implemented uniquely into the reduction problem through the use of the SU(3) pattern calculus.Work performed under the auspices of the U.S. Department of Energy.     

(g,f) -因子和可扩图

设G是一个图,并设g和f是定义在V(G)上的整值函数使得对所有的点x∈ V(G)均有g(x)≤ f(x).称一个图G是(g,f,H) -可扩的,如果在删除了任意一个同构于H的子图中所有点后,剩下G的子图有一个(g,f) -因子.该文给出了(g,f,H) -可扩图的特征.进一步,研究了(g,f,H) -可扩(H=nK₁)的性质.     

CLASSFICATION OF HOMOGENEOUS TWO-SPHERES IN G(2, 5; C)

In this article, we determine all homogeneous two-spheres in the complex Grassmann manifold G(2, 5; C) by theory of unitary representations of the 3-dimensional special unitary group SU(2).     

A polynomial algorithm for finding (g,f)-colorings orthogonal to stars in bipartite graphs

Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x)≤ f(x) for every vertex x of V(G). A (g. f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f(x) times. In this paper a polynomial algorithm to find a (g. f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.     

迷向表示分为6个不可约直和的旗流形上不变爱因斯坦度量

众所周知,计算广义旗流形G/K上不变爱因斯坦度量存在两个困难:(1)如何计算旗流形的非零结构常数;(2)如何计算旗流形爱因斯坦方程组的Grobner基.在这篇文章中用定理2.1来计算旗流形的非零结构常数,用Maple软件来计算旗流形爱因斯坦方程组的Gr?bnexr基.最后得到旗流形F_4/U~2(1)×SU(3),E_6/U~2(1)×SU(3)×SU(3),E_7/U~2(1)×SU(2)×SU(5),E_7/U~2(1)×SU(6),E_7/U~2(1)×SU(2)×SO(8)与E_8/U~2(1)×E_6上爱因斯坦度量.     

On a period map for SU(2) conformal vacua

We study some properties of a natural period map associated to SU(2) theta functions or, in other words, to SU(2) conformal vacua. In particular we will consider for such a map an analogue of the classical Torelli’s problem. The main result of the paper is a sufficient conditions for our period map to have the infinitesimal Torelli’s property.      

The Conditions for Some Linear Partial Differential Equations to Be Solvable in G

Using Bargmann's transformation and some basic results of theory of analytic functions with several complex variables, we have disscussed two classes of LPDOs in this paper. We prove that each operator of one class of them is surjective both from G to G and from L² to L², but not injective, and each operator of another class is injective from G to G but not surjective. And in the letter case, the necessary and suffcient conditions for the corresponding equations to be solvable in G are given.     

On spherical functions on the group SU(2) × SU(2) × SU(2)

We consider the group G:= SU(2) × ... × SU(2) (l factors), where SU(2) is the group of unitary 2 × 2 matrices with unit determinant. Let K ≃ SU(2) be the diagonal subgroup of G. We obtain the generating function of all K-spherical functions on G.     

Equivariant bordisms

The article investigates the category of manifolds acted upon by a Lie group G in such a way that the orbit of every point is equivariantly diffeomorphic to one of a fixed set S of homogeneous spaces of G. The bordism groups (S-equivariant bordism groups) are defined in a natural way. These groups are described in detail in the special cases of the quasicomplex action of the groups U(1) and SU(2) on quasicomplex manifolds.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 519–526, November, 1971.In conclusion, the author thanks S. P. Novikov and A. S. Mishchenko for their interest in the study.     

关于图是(g,f,n)-临界图的充分条件

设G是一个图. 设g和f是两个定义在V(G)上的整值函数使得对V(G)所有的顶点x有g(x)f(x). 图G被称为(g,f,n)-临界图,如果删去G的任意n个顶点后的子图都含有G的(g,f)-因子. 本文给出了图是(a,b,n)-临界图几个充分条件. 进一步指出这些条件是最佳的. 例如,如果对V(G)所有的顶点x和y都有g(x)＜f(x), n+g(x)dG(x)和g(x)/(dG(x)-n)f(y)/dG(y),则G是(g,f,n)-临界图.