Application of Angular Momentum Theory to Constructing Basis Functions of Irreducible Representations of Icosahedral Group

A new method based on angular momentum theory was proposed to construct the basis functions of the irreducible representations(IRs) of point groups. The transformation coefficients, i. e., coefficients S, are the components of the eigenvectors of some Hermitian matrices, and can be made as real numbers for all pure rotation point groups. The general formula for coefficient S was deduced, and applied to constructing the basis functions of single-valued irreducible representations of icosahedral group from the spherical harmonics with angular momentum j≤7.

Kaplansky Density and Kadison Transitivity Theorems for Irreducible Representations of Real C^-Algebras

We prove analogs of the Kaplansky Density Theorem and the Kadison Transitivity Theorem for irreducible representations of a real C＊-algebra on a real Hilbert space. Specifically, if a C＊-algebra is acting irreducibly on a real Hilbert space, then the Hilbert space has either a real, complex, or quaternionic structure with respect to which the density and transitivity theorems hold.

D Dimensions and N ×N Matrix Representations of Fermions

The connection between the number of dimensions and the size of the representation matrices in the Dirac equation has been discussed thoroughly and the restriction N2 = 2D was derived. In this summary, the result is brought again, this time with emphasis on the importance of irreducibility of the representations. As a counter example, the case of the neutrino is discussed where the above restriction does not hold, indicating that the Dirac equation, in this case, is reducible.

Irreducible * Representations of Real C~*-Algebras

In this paper,we show that a topologically irreducible * representation of a real C~*- algebra is also algebraically irreducible.Moreover,the properties of pure real states on a real C~*- algebra and their left kernels are discussed.

Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur＇s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if L is a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbert bundle H = （Hu）u∈G^0, then each Hu is finite dimensional. It is also shown that if L is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle （G^0, （Hu）,μ）, then dimHu = 1 （u ∈ G^0）. Furthermore it is proved that every square integrable representation of a locally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation. Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the left regular representation is square integrable.

Construct Weak Hopf Algebras by Using Borcherds Matrix

We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying jm = j for some integer m. We denote this algebra by wUqT（A）. This algebra is a weak Hopf algebra if and only if m = 2,3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra Uq （A） of a generalized Kac-Moody algebra A.

Uniqueness of twisted linear periods and twisted Shalika periods

Let k be a local field of characteristic zero.Letπbe an irreducible admissible smooth representation of GL2 n(k).We prove that for all but countably many charactersχ’s of GLn(k)×GLn(k),the space ofχ-equivariant(continuous in the archimedean case)linear functionals onπis at most one dimensional.Using this,we prove the uniqueness of twisted Shalika models.