On Locally Finite Skew Group Algebras

Mathematical Notes - In this note, we study the problem on the existence of noncyclic free subgroups of the skew group algebra of a locally finite group over a field.     

体上两个矩阵乘积的群逆

给出了体上两个矩阵乘积的群逆的存在性的一个等价条件及若干充分条件。     

Simplicity of Skew Group Rings of Abelian Groups

Necessary and sufficient conditions for simplicity of a general skew group ring A ?_σ G are not known. In this article, we show that a skew group ring A ?_σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ?_σ G is simple.     

Group inverse for the block matrices with an invertible subblock

Let (A is square) be a square block matrix with an invertible subblock over a skew field K. In this paper, we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for M under some conditions.     

Dubrovin赋值斜群环

本文中对一个斜群环为Dubrovin赋值环给出了一系列等价刻画,并且刻画了一个Dubrovin赋值斜群环的所有素理想.     

A Note on Free Groups in the Ring of Fractions of Skew Polynomial Rings

Let L be a function field over the rationals and let D denote the skew field of fractions of L[t;σ], the skew polynomial ring in t, over L, with automorphism σ. We prove that the multiplicative group D ^× of D contains a free noncyclic subgroup.     

Dubrovin Valuation Skew Group Rings

Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given. Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised. Received June 8, 1999, Revised June 4, 2001, Accepted July 20, 2001     

体上线性映射的子空间的维数及其应用

本文给出体上左向量空间的线性映射的某些子空间的维数恒等式,并讨论了它在体上矩阵秩的理论上的应用,其中一个有趣的应用是,由体上矩阵秩的恒等式来刻划体上某些矩阵的特征性质。 以下设Ω是一个体。对Ω上左向量空间V映入Ω上左向量空间V′的线性映射σ:V V~σ,记σ的核空间为:     

Leavitt Path Algebras as Partial Skew Group Rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory of the Cuntz–Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.     

On Cocycles with Values in the Group SU(2)

 In this paper we introduce the notion of degree for C ¹-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a C ¹-cocycle over an irrational rotation by α has nonzero degree, then the skew product

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

体上特征矩阵的简化形式与法式存在定理

谢邦杰在文献[2]—[5]中,研究了体上特征矩阵的简化形式与法式。本文在文献[1]的基础上,采用不同于[2]的方法,定义体上特征矩阵的另一种简化形式。对于我们所定义的简化形式,得到体上特征矩阵具有绝对唯一的简化形式,即法式存在的充分必要条件是矩阵满足中心化条件,从而使更广的一类特征矩阵存在法式。此外,我们还将[5]中体上矩阵的Caylcy-Hamilton定理再一次推广。     

Skew partial fields,multilinear representations of matroids,and a matrix tree theorem

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte?s definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte?s representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the matrix tree theorem for this class.     

Group actions on algebras and the graded Lie structure of Hochschild cohomology

Hochschild cohomology governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms. We examine the Gerstenhaber bracket with a view toward deformations and developing bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras.     

Skew Linear Unipotent Groups

A unipotent linear group over a skew field of characteristic2 is nilpotent. 2000 Mathematics Subject Classification 20H25(primary), 20F18 (secondary).     

关于体上分块矩阵的群逆

本文利用分块矩阵方法．研究了体上两个矩阵乘积的群逆的存在性及表示形式，给出了体上两个矩阵乘积群逆存在的充分必要条件和表示形式．并且在一定条件下．给出了体上分块矩阵的群逆存在性及表示形式．