Conjugation of elements in central simple algebras

We study the conjugacy classes of elements of a finite-dimensional central simple associative algebra with respect to scalar extensions. We apply the results to investigate the products of two (skew-)symmetric elements in central simple algebras with involution.     

EXACT SEQUENCES OF WITT GROUPS

ABSTRACT

An exact sequence of Witt groups, motivated by exact sequences obtained by Lewis and by Parimala, Sridharan and Suresh, is constructed. The behavior of the maps involved in these sequences with respect to isotropy is completely determined in the case of division algebras. In particular, the kernels of the maps involved in the previous sequences are explicitly given, leading to a new proof of their exactness. Similar exact sequences of equivariant Witt groups are constructed. As an application, relations between the cardinality of certain Witt groups are obtained.     

The Lie Algebra of a Hyperbolic Higher Degree Alternating Form

We compute the Lie algebra of a higher degree hyperbolic alternating form, which generalizes the symplectic algebra to higher degrees. We use this to show that hyperbolic forms descend under scalar extension. An immediate consequence is that a version of the weak Hasse–Minkowski Theorem is valid for alternating forms of higher degree.     

Induced Quadratic Modules

Positivity in *-algebras can be defined either algebraically, by quadratic modules, or analytically, by *-representations. By the induction procedure for *-representations, we can lift the analytical notion of positivity from a *-subalgebra to the entire *-algebra. The aim in this article is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the *-algebra is induced from its intersection with the *-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases.     

Symmetric Elements,Hermitian Forms,and Cyclic Involutions

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.     

Weyl and Lidski(i) Inequalities for General Hyperbolic Polynomials

The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidski(i) inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.     

On Generation of Sporadic Simple Groups by Three Involutions Two of Which Commute

We prove the following result: Let G be one of the 26 sporadic simple groups. The group G cannot be generated by three involutions two of which commute if and only if G is isomorphic to M ₁₁, M ₂₂, M ₂₃, or M ^cL .     

Involutions in triangular groups

We describe involutions, i.e. elements of order 2, in the groups T _n (K) – of upper triangular matrices of dimension n (n?∈??), and T _∞(K) – of upper triangular infinite matrices, where K is a field of characteristic different from 2. Using the obtained result, we give a formula for the number of all involutions in T _n (K) in the case when K is a finite field.     

Quadratic transform of complete ideals in regular local rings

In an upcoming article we study harmonic analysis on the quantum E(2) group within an algebraic framework: we explicitly construct Fourier transforms between quantum E(2) and its Pontryagin dual, involving q-Bessel functions as kernel, prove Plancherel &; inversion formulas etc. In the present paper we propose an algebraic setting in which to perform harmonic analysis on non-compact, non-discrete quantum groups and in particular on quantum E(2). We are mainly concerned with the construction of positive and faithful invariant functionals on an algebraic level, KMS properties, etc.     

Formally Real Involutions on Central Simple Algebras

An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r ^# r where r ? R is nonzero. Suppose that R is a central simple algebra (i.e., R = M _n (D) for some integer n and central division algebra D) and # is an involution on R of the form r ^# = a ^?1 r? a, where ? is some transpose involution on R and a is an invertible matrix such that a? = ±a. In Section 1 we characterize formal reality of # in terms of a and ?| _D . In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on D = (K/F, Φ) that extend to a formally real involution on the split algebra D ? _F K ? M _n (K). Every such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x ? tr(x ^# x) is not positive semidefinite.     

一种新的求解双曲型守恒律的三阶中心差分格式

本文提出了一种求解双曲型守恒律新的三阶中心差分格式，主要是引入了一种推广的三阶重构，并证明了这种重构在网格边界无振荡．所提的格式保持了中心差分格式简单的优点，不需用Riemann解算器，避免了进行特征解耦．数值试验结果表明本文格式是高精度、高分辨率的。     

约化Banach 代数的谱不变性和双曲群

主要研究了可数离散群的性质 SRD, 并证明了具有性质 SRD的群上的速降函数全体组成的空间是约化Banach代数 谱不变的稠密子代数. 最后作为例子给出了双曲群具有性质 SRD.     

Free Groups in Central Simple Algebras without Tits' Theorem

Let R be a noncommutative central simple algebra, the center k of which is not absolutely algebraic, and consider units a,b of R such that {a,a ^b } freely generate a free group. It is shown that such b can be chosen from suitable Zariski dense open subsets of R, while the a can be chosen from a set of cardinality |k| (which need not be open).     

Involutions for upper triangular matrix algebras

In this paper we describe completely the involutions of the first kind of the algebra UT_n(F) of n×n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT_n(F) when n is even and a single class in the odd case.Furthermore we consider the algebra UT₂(F) of the 2×2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution *, we describe the *-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras.Then we consider the *-polynomial identities for the algebra UT₃(F) over a field of characteristic zero. We describe a finite generating set of the ideal of *-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the *-identities for the algebra UT_n(F) of the n×n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.     

3维双曲空间中曲面的双曲Gauss映照和法Gauss映照

本文导出了3维双曲空间中曲面的双曲Gauss映照和法Gauss映照的关系,发现了一般的曲面由双曲Gauss映照和平均曲率函数唯一确定,并证明了双曲Gauss映照所满足的二阶线性椭圆方程,给出了两种形式的关于双曲Gauss映照的三阶非线性偏微分方程(组)的一个解.     

具有三角分解李代数的积分元和中心扩张

本文将应用广义限制李代数的概念来研究具有三角分解李代数的积分元和中心扩张的关系.对于给定的广义限制普遍包络代数,我们确定了它的积分元并且提供了一种计算dim H2(L,F)的方法.     

Frobenius Pairs with Perfect Involutions

An involution i of a group G is said to be perfect in G if any two non-commuting involutions in i^G are conjugated by an involution in the same class. We generalize theorems of Jordan and M. Hall concerning sharply doubly transitive groups, and the Shunkov theorem on periodic groups with a finite isolated subgroup of even order. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 751–762, November–December, 2005.     

Extensions of the Tensor Algebra and Their Applications

This article presents a natural extension of the tensor algebra. In addition to “left multiplications” by vectors, we can consider “derivations” by covectors as basic operators on this extended algebra. These two types of operators satisfy an analogue of the canonical commutation relations. This algebra and these operators have the following applications: (i) applications to invariant theory related to tensor products and (ii) applications to immanants. The latter includes a new method to study the quantum immanants in the universal enveloping algebras of the general linear Lie algebras and their Capelli type identities (the higher Capelli identities).     

不动点集为（■P（2r_i＋1））■（■S~nj）的对合的流形

本文讨论了不动点集Ｆ为任意多个奇维实数（复数、或者四元数）射影空间与球面不交并的闭流形Ｍ上对合Ｔ的流形（Ｍ，Ｔ），给出了这样的流形不协边的充要条件．