Involutory decomposition of a group and twisted subsets with few involutions

A subset K of some group C is called twisted if 1 ∈ K and x, y ∈ K implies that xy ^?1 x belongs to K. We use the concept of twisted subset to investigate and generalize the concept of involutory decomposition of a group. A group is said to admit involutory decomposition if it contains some involution such that the group is the product of the centralizer of the involution and the set of elements inverted by the involution. We study the twisted subsets with at most one involution. We prove that if a twisted subset has no involutions at all then it generates a subgroup of odd order.     

Minimal non-group twisted subsets containing involutions

A subset K of a group G is said to be twisted if 1 ∈ K and xy⁻¹x ∈ K for any x, y ∈ K. We explore finite twisted subsets with involutions which are themselves not subgroups but every proper twisted subset of which is. Groups that are generated by such twisted subsets are classified. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 459–482, July–August, 2007.     

Minimal non-group-like twisted subsets with involutions

A subset K of some group G is called twisted if 1 ∈ K and xy ^?1 x ∈ K for all x, y ∈ K. We study the finite twisted subsets with an involution which are not subgroups but whose every proper twisted subset is a subgroup. We also consider the groups generated by twisted subsets.     

Twisted calculus

A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis, the categories of twisted modules and integrable twisted differential modules are equivalent. As particular cases, one recovers classical results from the theory of finite difference equations or q-difference equations.     

The Collineation Groups of Generalized Twisted Field Planes

We investigate autotopisms and isotopisms of generalized twisted field planes. A revision of Alberts results is developed using a mixture of group-theoretical and algebraic methods. In particular, transitivity conditions of the autotopism group on the nonvertex points of one or more sides of the autotopism triangle are determined. Also, correlations and polarities in generalized twisted field planes are examined in detail. These results allow us to obtain several improvements of recent results in semifield planes.     

关于扭曲Lie结构范畴中的对偶函子

首先介绍扭曲Lie余代数．然后讨论扭曲Lie代数和扭曲Lie余代数范畴．主要给出扭曲Lie结构范畴之间函于的一些性质。     

扭Heisenberg-Virasoro代数上的Poisson结构

Poisson代数是指同时具有代数结构和李代数结构的一类代数,其乘法与李代数乘法满足Leibniz法则.扭Heisenberg-Virasoro代数是一类重要的无限维李代数,是次数不超过1的微分算子李代数W(0)的普遍中心扩张,与曲线的模空间有密切联系.本文主要研究扭Heisenberg-Virasoro代数上的Poisson结构,首先确定了李代数W(0)上的Poisson结构,进而给出了扭Heisenberg-Virasoro代数上的Poisson结构.     

Fibred knots and twisted Alexander invariants

We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.

     

D_l和E_6型Weyl群的扭子群的扩群

本文决定了 Dl 和 E6 型 Weyl群扭子群的所有扩群 ,这为确定相应 Chevalley群扭子群的所有扩群奠定了基础 .     

Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over Twisted Heisenberg-Virasoro Algebra

We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenber-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module （and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series）.     

Nilpotency of the derived subgroup of a finite tangled group

We study finite groups whose every subset containing the identity and closed with respect to the operation x ○ y = xy ^?1 x is a subgroup. We prove that the derived subgroup of such a group is nilpotent, which implies that the derived length of such a group is at most three.     

New construction of special Lagrangian submanifolds in T^＊R^n equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T~*R~n and a kind of austere submanifold from conormal bundle of minimal surface of R~3 are constructed.     

New construction of special Lagrangian submanifolds in <Emphasis Type="Italic">T</Emphasis>*R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> equipped with the standard metric

A class of twisted special Lagrangian submanifolds in T＊R^n and a kind of austere submanifold from conormal bundle of minimal surface of R^3 are constructed.     

Diverse <Emphasis Type="Italic">N</Emphasis>=(4,4) Twisted Multiplets in the <Emphasis Type="Italic">N</Emphasis>=(2,2) Superspace

We describe four different types of N=(4, 4) twisted supermultiplets in the two-dimensional N=(2, 2) superspace ℝ^(1,1|2,2). All these multiplets are represented by a pair of chiral and twisted chiral superfields and differ in the transformation properties under an extra hidden N=(2, 2) supersymmetry. The sigma-model N=(2, 2) superfield Lagrangians for each type of the N=(4, 4) twisted supermultiplet are real functions subjected to some differential constraints implied by the hidden supersymmetry. We prove that the general sigma-model action including all types of N=(4, 4) twisted multiplets and invariant under the N=(4, 4) supersymmetry reduces to a sum of sigma-model actions for separate types. An interaction between the multiplets of different sorts is possible only through the appropriate mass terms and only for those multiplets that belong to the same “self-dual” pair. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 66–86, October, 2005.     

Finite tangled groups

We study the so-called finite tangled groups. These are the groups whose every subset containing 1 and closed under the operation x ○ y = xy ^?1 x is a subgroup. The general problem of studying such groups reduces to the case of tangled groups of odd order. We classify all finite nilpotent tangled groups.     

Twisted Alexander norms give lower bounds on the Thurston norm

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

     

D_4型广义扭仿射李代数及其模

本文是[3]的继续,将讨论D4型的广义扭仿射李代数及其表示理论;证明作用在其不可约模上的一类算子的局部幂零性.     

Nakayama Twisted Centers and Dual Bases of Frobenius Cellular Algebras

For a Frobenius cellular algebra, we prove that, if the left (right) dual basis of a cellular basis is again cellular, then the algebra is symmetric. Moreover, some ideals of the center are constructed by using the so-called Nakayama twisted center.     

An explicit example of a noncrossed product division algebra

The paper presents an explicit example of a noncrossed product division algebra of index and exponent 8 over the field ?(s)(t). It is an iterated twisted function field in two variables D(x, σ)(y, τ ) over a quaternion division algebra D which is defined over the number field ?(√3,√?7). The automorphisms σ and τ are computed by solving relative norm equations in extensions of number fields. The example is explicit in the sense that its structure constants are known. Moreover, it is pointed out that the same arguments also yield another example, this time over the field ?((s))((t)), given by an iterated twisted Laurent series ring D((x, σ))((y, τ )) over the same quaternion division algebra D. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Edge-pancyclicity and Hamiltonian connectivity of twisted cubes

The twisted cube TQn is a variant of the hypercube Qn. It has been shown by Chang, Wang and Hsu [Topological properties of twisted cube. Information Science, 113, 147-167 （1999）] that TQn contains a cycle of every length from 4 to 2^n. In this paper, we improve this result by showing that every edge of TQn lies on a cycle of every length from 4 to 2^n inclusive. We also show that the twisted cube are Hamiltonian connected.