Module Structures on U(ε^(-))for the Euclidean Algebra

We classify modules which are free of rank 1 when restricted to the negative part for the Euclidean Lie algebra.Sufficient and necessary conditions for the simplicity of these modules are given.From these modules we construct modules which are infinitely generated when restricted to the maximal toral subalgebra.

On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices

A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented.The key idea is to establish a relationship between a matrix and any of its full row rank submatrices.Based on the new result,the authors propose an algorithm for factorizing matrices and have implemented it on the computer algebra system Maple.Two examples are given to illustrate the effectiveness of the algorithm,and experimental data shows that the algorithm is efficient.

New Solution for the Earth’s Free Wobble and Its Geophysical Implications

In this paper, the theory of the free wobble of the triaxial Earth is developed and new conclusions are drawn: the Euler period should be actually expressed by the first kind of complete elliptic integral; the trace of the free polar motion is elliptic and the orientations of its semi-minor and major axes are approximately parallel to the Earth's principal axes A and B, respectively. In addition, the present theory shows that there is a mechanism of frequency-amplitude modulation in the Chandler wobble, which might be a candidate for explaining the correlation between the amplitude and period of the Chandler wobble.

Orders of Classical Groups over Finite Rings

It is observed that a classical group over a finite ring R with identity can be reduced to that over finite fields after the procedures of taking “modulo the radical”, “direct sum” and “tensor products”. Basing on that fact, we calculate the orders of classical groups over R and the number of k dimensional free submodules of an n dimensional free module over R .

The Lie Conformal Algebra of a Block Type Lie Algebra

Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with C[δ]- basis { Lα（w） ｜ α ∈ Z} and λ-brackets [Lα（w）λLβ（w）] = （αδ ＋ （α ＋β）A）Lα＋β（w）. Finally, we give a classification of free intermediate series B-modules.

Representations of Frobenius-type Triangular Matrix Algebras

The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.

Grbner-Shirshov Bases of Irreducible Modules of the Quantum Group of Type G_2

First, the authors give a GrSbner-Shirshov basis of the finite-dimensional irre- ducible module Vq（λ） of the Drinfeld-Jimbo quantum group Uq（G2） by using the double free module method and the known GrSbner-Shirshov basis of Uq（G2）. Then, by specializing a suitable version of Uq （G2） at q = 1, they get a GrSbner-Shirshov basis of the universal enveloping algebra U（G2） of the simple Lie algebra of type G2 and the finite-dimensional irreducible U（G2）-module V（λ）.

On Regular Power-Substitution

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.