A Ring with Left Identities anb Right Inverses Is a Skew Field

It is well known that a semigroup in which every element has a left identity and right inverses need not be a group. In this note we prove that if the semigroup is the multiplicative semigroup of a ring, then it is a group. In fact, we have the following theorem.     

On the Group of ρ-endotrivial κG-modules

In this paper, we define a group T_ρ(G) of p-endotrivial κG-modules and a generalized Dade group D_ρ(G) for a finite group G. We prove that T_ρ(G)? T_ρ(H)whenever the subgroup H contains a normalizer of a Sylow p-subgroup of G, in this case, K(G)? K(H). We also prove that the group D_ρ(G) can be embedded into T_ρ(G) as a subgroup.     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

Perturbation of Yamabe equation on Iwasawa N groups in presence of symmetry

Let G be a simple Lie group of real rank one and N be in the Iwasawa decomposition of G. Under the assumption of some symmetries, we obtain an existent result for the nonlinear equation △NU ＋ （1 ＋ ∈K（x, z））u2＊-1 = 0 on N, which generalizes the result of Malchiodi and Uguzzoni to the Kohn＇s subelliptic context on N in presence of symmetry.     

A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

本刊英文版Vol.26(2010),No.3论文摘要(英文)

<正>A Decomposition Theorem for G-GroupsⅠ.LIZASOAIN Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups(G-groups).In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations.Moreover,we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

RESEARCH REPORT ON THE GOLUB—WILKINSON’S PROBLEM

We conducted the research on the calculation of the Jordan normal form of a matrixfrom 1993 to 1998.In this period,we obtained some theoretical results,also developedand implemented a series of algorithms for the theoretical results.Now this research iscompleted to some degree.The Golub-Wilkinson’s ProblemComputing the Jordan decomposition of a matrix,introduced by G.H.Golub and J.     

自伴矩阵代数上约当半三可乘映射

In this paper,we show that every injective Jordan semi-triple multiplicative map on the Hermitian matrices must be surjective,and hence is a Jordan ring isomorphism.     

Plancherel Formula and Local Solvability for the Pre-Homogeneous Left Invariant PDO on the Nilpotent Lie Group H_nR~k

The nilpotent Lie group H_nR~k is a very important Lie group except the Heisenberg grollp.The Product of this group is where x,y,x＇y＇∈R~n,t,t＇∈R,z,z＇∈R~k.In this paper we give the precise form of the Plancherel formnula and study the local solvability of a speciaI kind of no- nhomogeIleOUS left invariant PDO in the dilatiorl sen8e——pre-homogeneous PDO. We prove that the WeyI pSelldodifferential operators have the propertie:     

The Decomposition and uniqueness of n-Lie Superalgebras

In this paper, the definitions and some properties of n-Lie superalgebras are presented. Our main aim is to study the decomposition and uniqueness of finite dimensional n-Lie superalgebras with trivial center. Aecoding to the decomposition of n-Lie superalgebras, we obtain the decomposition of inner derivation superalgebras and derivation superalgebras respectively. Furthermore, we discuss some properties about the centroid of n-Lie superalgebras, so we can see its application in the decomposition of n-Lie superalgebras.     

Multiplicative Jordan Decomposition in Group Rings of 3-Groups,II

In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ?[G] satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ?[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 3³ = 27.     

Decomposition of projective spaces defined by unit-regular jordan pairs

An analogue of the Bialynicki-Birula decomposition of a smooth algebraic variety under an action of the multiplicative group is shown to hold for spaces obtained from Jordan pairs by projective equivalence. The methods are Jordan-theoretic rather than algebraic-geometric and involve unit-regularity and rank functions for Jordan pairs.     

Groups with Certain Normality Conditions

We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.     

Jordan代数上的可乘Jordan导子

设A是Jordan代数,如果映射d:A→A满足任给a,b∈A,都有d(aob)=d(a)o b+aod(b),则称d为可乘Jordan导子.如果A含有一个非平凡幂等p,且A对于p的Peirce分解A=A_1⊕A_(1/2)⊕A_0满足:(1)设ai∈Ai(i=1,0),如果任给t_(1/2)∈A_(1/2),都有a_i○t_(1/2)=0,则a_i=0,则A上的可乘Jordan导子d.如果满足d(p)=0,则d是可加的.由此得到结合代数和三角代数满足一定条件时,其上的任意可乘Jordan导子是可加的.     

The Fields of Real and Complex Numbers with a Small Multiplicative Group

We consider the model theory of the real and complex fieldswith a multiplicative group having the Mann property. Amongthese groups are the finitely generated multiplicative groupsin these fields. As a by-product we obtain some results on groupswith the Mann property in rings of Witt vectors and in fieldsof positive characteristic.k 2000 Mathematics Subject Classification03C10, 03C35, 03C60, 03C64, 03C98, 13K05.     

Gelfand-Kirillov dimension in Jordan Algebras

In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.

     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

Wedderburnzerlegung für Jordan-Paare

In the first section we summarize some properties of Jordan pairs. Then we state some results about some groups defined by Jordan pairs.In the next section we construct a Lie algebra to a Jordan pair. This construction is a generalization of the wellknown Koecher-Tits-construction. We calculate the radical of this Lie algebra in terms of the given Jordan pair.In the last section we prove a Wedderburn decomposition theorem for Jordan pairs in the characteristic zero case. Some special cases in arbitrary characteristic have been shown by R. Carlson (see [5]). Also we show that any two such decompositions are conjugate under a certain group of automorphism. Analogous theorems will be shown for Jordan Triples.     

Additivity of multiplicative maps on triangular rings

In this paper we shall give a unified technique in the discussion of the additivity of n-multiplicative automorphisms, n-multiplicative derivations, n-elementary surjective maps, and Jordan multiplicative surjective maps on triangular rings.