Structure of Hyperbolic Unitary Groups II： Classification of E-Normal Subgroups

Abstract. This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group U2n（R, A） which are normalized by the elementary subgroup EU2n（R, A）, under the condition that R is a quasi-finite ring with involution, i.e., a direct limit of module finite rings with involution, and n ≥ 3. 2010 Mathematics Subject Classification： 20G35, 20H25

Constant terms of spherical Eisenstein series on quasi-split unitary groups

In this note, we prove a formula which expresses the constant term of the spherical Eisenstein series on a quasi-split unitary group as a linear combination of spherical Eisenstein series on smaller unitary groups.

STABILIZATION OF UNITARY GROUPSOVER POLYNOMIAL RINGS

The author studies the stabilization for the unitary groups over polynomial rings and obtainsfor them some results analogous to the results of linear groups and symplectic groups.It isespecially proved that K1 U(A) = K1 U(R) where A = R[X1,…, Xm], R is a ring of algebraicintegers in a quadratic field Q().

Matrix Elements of One— and Two—Body Operators in the Unitary Group Approach （Ⅱ）—Application

Simple analytical expressions for one- and two-body matrix elements in the unitary group approach to the configuration interaction problems of many-electron systems are obtained based on the previous results for general irreps.

Degenerate States in Nonlinear Sigma Model with SU(2) Symmetry

Entanglement in quantum theory is a peculiar concept to scientists. With this concept we are forced to re-consider the cluster property which means that one event is irrelevant to another event when they are fully far away. In the recent works we showed that the quasi-degenerate states induce the violation of cluster property in antiferromagnets when the continuous symmetry breaks spontaneously. We expect that the violation of cluster property will be observed in other materials too, because the spontaneous symmetry breaking is found in many systems such as the high temperature superconductors and the superfluidity. In order to examine the cluster property for these materials, we studied a quantum nonlinear sigma model with U(1) symmetry in the previous work. There we showed that the model does have quasi-degenerate states. In this paper we study the quantum nonlinear sigma model with SU(2) symmetry. In our approach we first define the quantum system on the lattice and then adopt the representation where the kinetic term is diagonalized. Since we have no definition on the conjugate variable to the angle variable, we use the angular momentum operators instead for the kinetic term. In this representation we introduce the states with the fixed quantum numbers and carry out numerical calculations using quantum Monte Carlo methods and other methods. Through analytical and numerical studies, we conclude that the energy of the quasi-degenerate state is proportional to the squared total angular momentum as well as to the inverse of the lattice size.

Basic 3-Transposittions of Unitary Group Un(2)

We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions.Let G=U_(n)(2).It is well known that G is a 3-transposition group with the set D,the conjugacy class consisting of its transvections,as the set of 3-transpositions.Let L be a set of basic transpositions in D.We give general descriptions of L and 1-(ν,κ,λ)designs D=(P,B),with P=D and B={L^(9)|g∈G}.The parameters k=|L|,λ and further properties of D are determined.We also,as examples,apply the method to the unitary simple groups U_(4)(2),U_(5)(2),U_(6)(2),U_(7)(2),U_(8)(2)and U_(9)(2).

Universal Jensen's Equations in Banach Modules over a C-Algebra and Its Unitary Group

In this paper,we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen's equations in Banach modules over a unital C~*-algebra.It is applied to show the stability of universal Jensen's equations in a Hilbert module over a unital C~*-algebra.Moreover,we prove the stability of linear operators in a Hilbert module over a unitat C~*-algebra.

On the group with the same orders of Sylow nor-malizers as the finite projective special unitary group

In this paper we prove that a finite group G is isomorphic to the finite projective special unitary group U n (q) if and only if they have the same order of Sylow r-normalizer for every prime r.

ON FACTORIZATION THEOREMS OF PLURIHARMONIC MAPS INTO THE UNITARY GROUP

The authors give some constructive factorization theorems for pluriharmonic maps from a Kaehler manifold into the unitary group U(N) and obtain some optimal upper bounds of minimal uniton numbers.

THE TRACE SPACE INVARIANT AND UNITARY GROUP OF C-ALGEBRA

Let A be a unital C-algebra, n ∈ N ∪ {∞}. It is proved that the isomorphism △n : is isometric for some suitable distances. Asan application, the author has the split exact sequence with iA contractive (and isometric if n = ∞) under certain condition of A.

Maps Preserving Commutators on the Standard Borel Subgroup of the Unitary Group over a Field

Let F be a field with charF ≠ 2 and ｜F｜ 〉 9, and let B2n（F） be the standard Borel subgroup of the unitary group U2n（F） over F. For n ≥ 3, we obtain a complete description of all bijective maps preserving commutators on B2n （F）.

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.

Approximate （m, n）-Cauchy-Jensen Additive Mappings in C^＊-algebras

Concerning the stability problem of functional equations, we introduce a general （m, n）- Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general （m, n）-Cauchy Jensen additive mappings in C^＊-algebras, which generalize the result＇s obtained for Cauchy-Jensen type additive mappings.

Surjective Lp-isometries on Grassmann spaces

Letϕ:Pc(M1)→Pc(M2)be a surjective Lp-isometry between Grassmann spaces of projections with the trace value c in semifinite factors M1 and M2.Based on the characterization of surjective Lp-isometries of unitary groups in finite factors,we show thatϕor I−ϕcan be extended to a∗-isomorphism or a∗-antiisomorphism.In particular,ϕis given by a∗-(anti-)isomorphism unless M1 and M2 are finite and c=12.

HORMANDER MULTIPLIER THEOREM ON SU(2)

A boundedness criterion is set up for some convolution operators on a compact Lie group.By this criterion a Hormander multiplier theorem is proved in the Hardy spaces on SU(2).

New method of applying conformal group to quantum fields

Most of previous work on applying the conformal group to quantum fields has emphasized its invariant aspects, whereas in this paper we find that the conformal group can give us running quantum fields, with some constants, vertex and Green functions running, compatible with the scaling properties of renormalization group method （RGM）. We start with the renormalization group equation （RGE）, in which the differential operator happens to be a generator of the conformal group, named dilatation operator. In addition we link the operator/spatial representation and unitary/spinor representation of the conformal group by inquiring a conformal-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. By this kind of application, we find out that quite a few interaction vertices are separately invaxiant under certain transformations （generators） of the conformal group. The significance of these transformations and vertices is explained. Using a particular generator of the conformal group, we suggest a new equation analogous to RGE which may lead a system to evolve from asymptotic regime to nonperturbative regime, in contrast to the effect of the conventional RGE from nonperturbative regime to asymptotic regime.

The Initial Boundary Value Problem for a Class ofGeneralized Nonlinear Schrodinger Equations

We discuss the initial boundary value problem of a class of nonlinear Schr6dinger equations with potential functions. By the theory of the group of unitary operators and the method ofthe prior estimate, we prove the global existence of the classical solutions of the nonlinear Schrodingerequations with potential functions.