Classification of Lie algebras of specific type in complexified Clifford algebras

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.     

关于某些椭圆曲线的Tate—Shafarevich群的注记

本文首先考虑某个四元代数，通过对此四元代数的算术的研究，得到某类丢番图方程的解数与某些虚二次域类数之间的关系，最后，应用Ｔｕｎｎｅｌｌ的定理，得到一簇椭圆曲线的Ｔａｔｅ－Ｓｈａｆａｒｅｖｉｃｈ群的阶的上界。     

Capelli elements of the group algebra

Inspired by the Capelli-type identities for group determinants researched by Tôru Umeda, we give Capelli identities for irreducible representations of any finite group, and Capelli elements of the group algebra associated with these identities. These elements construct a basis of the centre of the group algebra.     

可换四元数空间中某些双曲型方程的Riemann-Hilbert边值问题

考察了在可交换四元数空间(基元为1,i,j,k满足条件i~2=j~2=-1,ij=ji= k)中的某些双曲型方程的Riemann-Hilbert边值问题,分别在不同的情况下获得了问题的可解条件和通解.     

关于广义四元数代数上矩阵的迹的一个等式

众所周知,相似矩阵的迹相等对于非交换代数和环上的矩阵不一定成立,有趣的问题是给定一个条件使得相似矩阵的迹相等对于非交换代数或非交换环上的矩阵成立。本文对于特征不是2的任意域F上定义的广义四元数代数上的两个矩阵A和B,给出如果A和B相似并且它们的主对角线上的元素在F中,那么它们的迹相等。     

Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.     

Wakimoto modules, opers and the center at the critical level

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In this paper, we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc, giving another proof of the theorem of B. Feigin and the author.     

Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.     

The Partition Algebra as a Centralizer Algebra of the Alternating Group

ABSTRACT

In this article, we focus on the result of V.F.R. Jones which says that the partition algebra is the algebra of all transformations commuting with the action of the symmetric group on tensor products of its permutation representation. In particular, we restrict the action of the symmetric group to the action of the alternating group. In this context, we compute a basis for the centralizer algebra and show when the centralizer is isomorphic to the partition algebra.     

A spectral mapping theorem for representations of one-parameter groups

In this paper we present some generalization (at the same time a new and a short proof in the Banach algebra context) of the Weak Spectral Mapping Theorem (WSMT) for non-quasianalytic representations of one-parameter groups.

     

Real matrix representations of complex split quaternions with applications

In this paper, we introduce 8×8 real matrix representations of complex split quaternions. Then, the relations between real matrix representations of split and complex split quaternions are stated. Moreover, we investigate some linear split and complex split quaternionic equations with split Fibonacci and complex split Fibonacci quaternion coefficients. Finally, we also give some numerical examples as applications of real matrix representation of complex split quaternions.     

一对四元数矩阵方程的共轭延拓解

定义广义四元数共轭延拓矩阵的概念,利用矩阵分块和四元数矩阵的实表示方法,分别给出四元数矩阵方程AX=C和XB=D存在列共轭延拓解和行共轭延拓解的必要充分条件及解的表达式.     

Strong associativity of a group algebra

The concept of strongly associative algebra was introduced in Dokuchaev and Exel [Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans Amer Math Soc, 2005] roughly 2 years ago. In Section 3 of Dokuchaev and Exel (to appear), the authors gave an example of a non-strongly associative algebra. Following that example, the strong associativity of the group algebra of the cyclic group of order four was claimed. In this paper, we will prove that this group algebra is not strongly associative.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

A complex semigroup approach to group algebras of infinite dimensional Lie groups

A host algebra of a topological group G is a C ^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C ^*-algebra C ^*(S,α), and we describe a setting which permits us to conclude that this C ^*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future. To K.H. Hofmann on the occasion of his 75th birthday     

On the Divided Power Algebra and the Symplectic Group in Characteristic 2

Let V be an even dimensional vector space over a field K of characteristic 2 equipped with a nondegenerate alternating bilinear form f. The divided power algebra DV is considered as a complex with differential defined from f. We examine the cohomology modules as representations of the corresponding symplectic group.     

广义四元数的一次代数方程

本文运用广义四元数代数的矩阵表示讨论了两类广义四元数的一次代数方程的解问题，并得到了这两类代数方程有唯一解、无穷多解，无解的判别条件。     

A-扩张Lie Rinehart代数

The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary, commutative,associative algebra A.It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group,analogous to the well known relationship of Lie algebras and Lie groups.     

Stably Calabi-Yau algebras and skew group algebras

Let G be a finite group and A be a finite-dimensional selfinjective algebra over an algebraically closed field. Suppose A is a left G-module algebra. Some suficient conditions for the skew group algebra AG to be stably Calabi-Yau are provided, and some new examples of stably Calabi-Yau algebras are given as well.