Newton's formula for

This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra of the Lie algebra : one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton's formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.

     

is a CS-algebra

In this paper, it is shown that the group algebra is right CS if and only if . Moreover, when , then is also CS as a module over its center.

     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

Weyl's construction and tensor power decomposition for

Let be the 7-dimensional irreducible representations of . We decompose the tensor power into irreducible representations of and obtain all irreducible representations of in the decomposition. This generalizes Weyl's work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group .

     

Low-dimensional linear representations of

We classify all complex representations of the automorphism group of the free group of dimension Among those representations is a new representation of dimension which does not vanish on the group of inner automorphisms.

     

Irreducible representations of the Cuntz algebra

In this paper, we establish formulas for the configuration of a special class of irreducible representations of the Cuntz algebra , . These irreducible representations arise as subrepresentations of naturally occurring representations of acting in and arise from consideration of multiresolution wavelet filters.

     

systems of hyperbolic conservation laws

In this paper, we study the evolution of the distance of solutions for systems of hyperbolic conservation laws. For the approximate solutions constructed by Glimm's scheme with the aid of the wave tracing method, we introduce a nonlinear functional which is equivalent to the distance between solutions, nonincreasing in time, and expressed explicitly in terms of the wave patterns of the solutions. This functional reveals the nonlinear mechanism of wave interactions and coupling which affect the topology.

     

Copies of in topological Riesz spaces

The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of or .

     

On reflection of stationary sets in

Let be an inaccessible cardinal, and let and is regular and . It is consistent that the set is stationary and that every stationary subset of reflects at almost every .

     

On pointed Hopf algebras of dimension

In this note we describe nonsemisimple Hopf algebras of dimension with coradical isomorphic to , abelian of order , over an algebraically closed field of characteristic zero. If is cyclic or , then we also determine the number of isomorphism classes of such Hopf algebras.

     

Normal subgroups of are not finitely generated

As a generalization of Wedderburn's classic theorem, it is shown that the multiplicative group of a noncommutative finite dimensional division algebra cannot be finitely generated. Also, the following conjecture is investigated: An infinite non-central normal subgroup of cannot be finitely generated.

     

The nilpotence height of for odd primes

K. G. Monks has recently shown that the element has nilpotence height in the mod Steenrod algebra. Here the method and result are generalized to show that for an odd prime the element has nilpotence height in the mod Steenrod algebra.

     

Invertible completions of upper triangular operator matrices

In this note we prove that if

is a upper triangular operator matrix acting on the Banach space , then is invertible for some if and only if and satisfy the following conditions:

(i)

is left invertible;

(ii)

is right invertible;

(iii)

.

Furthermore we show that , where is the union of certain of the holes in which happen to be subsets of .

     

Level one representations of

We construct a level one representation of the quantum affine algebra by vertex operators from bosonic fields.

     

Regularization of semigroups that are strongly continuous for

Let be a Banach space and a strongly continuous semigroup with . We show that the generator of generates a regularized semigroup. Our construction of a regularizing operator uses an existence result of J. Esterle.

     

The structure of some virtually free pro- groups

We prove two conjectures on pro- groups made by Herfort, Ribes and Zalesskii. The first says that a finitely generated pro- group which has an open free pro- subgroup of index is a free pro- product , where the are free pro- of finite rank and the are cyclic of order . The second says that if is a free pro- group of finite rank and is a finite -group of automorphisms of , then is a free factor of . The proofs use cohomology, and in particular a ``Brown theorem' for profinite groups.

     

Subsystems of the Schauder system that are quasibases for

We show that if is a subsystem of the Faber-Schauder system, and if is complete in , then is a quasibasis for each space , . Although it follows from the work of Ul'yanov that each element of can be represented by a Schauder series that converges unconditionally to the function, in the metric of the space, it proves to be the case that none of the aforementioned systems is an unconditional quasibasis for any of the -spaces herein considered.

     

On the range and the kernel of the operator

Let denote the algebra of (bounded linear) operators on the separable complex Hilbert space , and let denote a norm ideal in . For , let the derivation be defined by , and let be defined by . The main result of this paper is to show that if , are contractions, then for every operator such that , then for all .

     

Normalizers of the congruence subgroups of the Hecke group

Let cos and let be the Hecke group associated to . In this article, we show that for a prime ideal in , the congruence subgroups of are self-normalized in .

     

On the unitary dual of

In this paper we begin a systematic study of the unitarity question for genuine representations of the group . The main result is that we find a class of unitary representations (mostly isolated) analogous to the special unipotent representations defined by D. Barbasch and D. Vogan. In particular the full unitary dual of should be obtainable from this set by complementary series.