Nondegenerate Semiramified Valued and Graded Division Algebras

The main goal of this article is to give examples of division p-algebras that are not tensor product of cyclic algebras (Corollary 2.19) and to prove that nondegenerate tame semiramified division algebras of prime power degree over a Henselian valued field are indecomposable (Theorem 3.5). For this, we give new results concerning nicely semiramified division algebras over Henselian valued fields, and we develop a new study for nondegenerate valued and graded division algebras.     

Classification of Some Graded not Necessarily Associative Division Algebras I

We study not necessarily associative (NNA) division algebras over the reals. We classify in this paper series those that admit a grading over a finite group G, and have a basis {v _g |g ∈ G} as a real vector space, and the product of these basis elements respects the grading and includes a scalar structure constant with values only in {1, ? 1}. We classify here those graded by an abelian group G of order |G| ≤8 with G non–isomorphic to ?/8?. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non–associative division algebras.     

Subfields of Nondegenerate Tame Semiramified Division Algebras

We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of the center (Theorems 2.5, 2.12, and Proposition 2.16).     

Graded Lie Algebras of Maximal Class

We study graded Lie algebras of maximal class over a field of positive characteristic . A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct pairwise non-isomorphic such algebras, and soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.

     

New Hardy Spaces Associated with Herz Spaces and Beurling Algebras on Homogeneous Groups

The author introduces the Hardy spaces associated with the Herz spaces and the Beurling algebras on homogeneous groups and establishes their atomic decomposition characterizations. As the applications of this decomposition, the duals of these Hardy spaces and the boundedness of the central δ-Calderón-Zygmund operators on these Hardy spaces are studied. Received January 13, 2000, Accepted July 26, 2000     

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences, and their decompositions, are investigated.     

Characters and Equivalence Classes of Central Simple Group Graded Algebras

We introduce an equivalence between central simple strongly G-graded algebras. Such classes are associated in a natural way to absolutely irreducible characters of semisimple G-graded algebras. We study invariants of this equivalence relation, and also the structure of certain representatives of the equivalence classes.     

Clifford Algebras and Euclid’s Parametrization of Pythagorean Triples

We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R₂₁, whose minimal version may be conceptualized as a 4-dimensional real algebra of “kwaternions.” We observe that this makes Euclid’s parametrization the earliest appearance of the concept of spinors. We present an analogue of the “magic correspondence” for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.     

Simple Twisted Group Algebras of Dimension p4 and their Semi-Centers

For a simple twisted group algebra over a group G, if G^∣ is Hall subgroup of G, then the semi-center is simple. Simple twisted group algebras correspond to groups of central type. We classify all groups of central type of order p⁴ where p is prime and use this to show that for odd primes p there exists a unique group G of order p⁴, such that there exists simple twisted group algebra over G with a commutative semi-center. Moreover, if 1 < |G| <64, then the semi-center of simple twisted group algebras over G is noncommutative and this bounds are strict.     

Capable Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field

In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given.     

The Autotopism Group of A Semifield of Order p 4, p is an Odd Prime

The aim of this article is to investigate the autotopism group of a semifield of order p ⁴, p is an odd prime, admitting a four-group of automorphisms E? Z ₂ × Z ₂ acting freely on A.     

二元广义外代数的Hochschild同调群

设Aq=k／(x2,xy+qyx,y2)是含有两个变量的广义外代数,基于Buch- weitz等人构造的极小投射双模解,广义外代数的各阶Hochschild同调群的维数被清晰地计算．     

有一个稳定子群是F-群的素数阶自同构的有限π-群

本文引入了 ｃ－自同构的概念,在［１］和 ［２］的基础上得到了 Ｔｈｏｍｐｓｏｎ定理的三个推广．     

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).     

Finite Groups of Bounded Rank with an Almost Regular Automorphism of Prime Order

We prove that if a finite group G of rank r admits an automorphism of prime order having exactly m fixed points, then G has a -invariant subgroup of (r,m)-bounded index which is nilpotent of r-bounded class (Theorem 1). Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened. The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3). The general case reduces modulo available results to the case of finite p-groups. For reduction to Lie rings powerful p-groups are also used. For them a useful fact is proved which allows us to glue together nilpotency classes of factors of certain normal series (Theorem 2).