Azumaya Categories

We define the notions of Azumaya category and Brauer group in category theory enriched over some very general base category V. We prove the equivalence of various definitions, in particular in terms of separable categories or progenerating bimodules. When V is the category of modules over a commutative ring R with unit, we recapture the classical notions of Azumaya algebra and Brauer group and provide an alternative, purely categorical treatment of those theories. But our theory applies as well to the cases of topological, metric or Banach modules, to the sheaves of such structures or graded such structures, and many other examples.     

H-separable Hopf Galois Extensions and Azumaya Algebra

1 IntroductionLet A/R be a ring extension with the common identity 1. A/R is said to be separable if theA-bimodule homomorphism of A @R A onto A defined by a @ 5-a6 splits. A separableextension over a non-commutative ring generalizes that over a commutative ring which wasdiscussed in [1]. Hirata introduced anOther kind of separable extensions called H-separabeones (see [2]). A/R is said to be H-separable if A @R A is isomorphic as an A-bimoduleto a direct sumrnand of A". riom {2, Theor…     

The equivariant Brauer group of a group

We consider the Brauer group of a group (finite or infinite) over a commutative ring with identity. A split exact sequence

is obtained. This generalizes the Fröhlich-Wall exact sequence from the case of a field to the case of a commutative ring, and generalizes the Picco-Platzeck exact sequence from the finite case of to the infinite case of . Here is the Brauer-Taylor group of Azumaya algebras (not necessarily with unit). The method developed in this paper might provide a key to computing the equivariant Brauer group of an infinite quantum group.

     

The Brauer group of Yetter-Drinfel'd module algebras

Let be a Hopf algebra with bijective antipode. In a previous paper, we introduced -Azumaya Yetter-Drinfel'd module algebras, and the Brauer group classifying them. We continue our study of , and we generalize some properties that were previously known for the Brauer-Long group. We also investigate separability properties for -Azumaya algebras, and this leads to the notion of strongly separable -Azumaya algebra, and to a new subgroup of the Brauer group .

     

The Invariant Subrings of an Azumaya Galois Extension

Let B be an Azumaya Galois extension or a DeMeyer-Kanzaki Galois extension with Galois group G. Equivalent conditions are given for a separable subextension of a Galois extension in the skew group ring B * G being an invariant subring of a subgroup of the Galois group G.AMS Subject Classification (2000): 16S35, 16W20.     

A Note on K-Theory of Azumaya Algebras

For an Azumaya algebra A which is free over its centre R, we prove that K-theory of A is isomorphic to K-theory of R up to its rank torsions. We conclude that K _i (A, ?/m) = K _i (R, ?/m) for any m relatively prime to the rank and i ≥ 0. This covers, for example, K-theory of division algebras, K-theory of Azumaya algebras over semilocal rings, and K-theory of graded central simple algebras indexed by a totally ordered abelian group.     

Graded quaternion symbol equivalence of function fields

We present criteria for a pair of maps to constitute a quaternion-symbol equivalence (or a Hilbert-symbol equivalence if we deal with global function fields) expressed in terms of vanishing of the Clifford invariant. In principle, we prove that a local condition of a quaternion-symbol equivalence can be transcribed from the Brauer group to the Brauer-Wall group.     

On Compact Graphs

Let G be a finite simple graph with adjacency matrix A, and let P（A） be the convex closure of the set of all permutation matrices commuting with A. G is said to be compact if every doubly stochastic matrix which commutes with A is in P（A）. In this paper, we characterize 3-regular compact graphs and prove that if G is a connected regular compact graph, G - v is also compact, and give a family of almost regular compact connected graphs.     

Quantum Yang-Baxter module algebras

LetH be a quantum group over a commutative ringR. We introduce the concept of quantum Yang-BaxterH-module algebra, generalizing the notion ofH-dimodule algebra in the case whereH is commutative, cocommutative and faithfully projective. After discussing some examples, we introduceH-Azumaya algebras. The set of quivalence classes ofH-Azumaya algebras can be made into a group, called the Brauer group of the quantum groupH. This group is a generalization of the Brauer-Long group.This author wishes to thank the Department of Mathematics, UIA, for its hospitality and financial support during the time when most of this paper was written.     

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Γ=Cay(G,S)is said to be normal if G is normal in Aut Γ.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Γ of finite nonabelian simple groups G,where the vertex stabilizer Av is soluble for A=Aut Γ and v ∈ VΓ.We prove that either Γ is normal or G=A5,A10,A54,A274,A549 or A1099.Further,11-valent symmetric nonnormal Cayley graphs of A5,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.     

Subgroups of the Brauer Group of a Cocommutative Coalgebra

We introduce Shur and projective Schur subgroup of the Brauer group of a cocommutative coalgebra by means of twisted cogroup coalgebras and we study their properties. In particular we show that these subgroups are always torsion (in contrast with the whole Brauer group). Moreover, when C is coreflexive and irreducible both subgroups coincide with the coradical ones. We illustrate the theory with several examples.     

On the Genus of the Nilpotent Graphs of Finite Groups

The nilpotent graph of a group G is a simple graph whose vertex set is G?nil(G), where nil(G) = {y ∈ G | ? x, y ? is nilpotent ? x ∈ G}, and two distinct vertices x and y are adjacent if ? x, y ? is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.     

On Locally Primitive Graphs of Order 18p

The authors investigate locally primitive bipartite regular connected graphs of order $18p$. It is shown that such a graph is either arc-transitive or isomorphic to one of the Gray graph and the Tutte $12$-cage.     

半二面体群的小度数Cayley图

群G的一个Cayley图X=Cay(G,S)称为正规的,如果右乘变换群R(G)在Aut X中正规.研究了4m阶半二面体群G=〈a,b a2m=b2=1,ab=am-1〉的3度和4度Cayley图的正规性,其中m=2r且r>2,并得到了几类非正规的Cayley图.     

On Cubic Graphs Admitting an Edge-Transitive Solvable Group

Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip₃ or the complete graph K ₄, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of an arc-transitive solvable group on a connected cubic simple graph is at most 3-arc-transitive. As an application, a new infinite family of semisymmetric cubic graphs, arising as regular elementary abelian covering projections of K _3,3, is constructed.     

Cayley图的泛因子性质

一个κ-正则图若满足对任意正整数s,1≤s≤κ,均存在一个s-因子或一个2[s/2]因子,则称其有泛因子或偶泛因子性质.本文证明了每个奇度Cayley图是泛因子的,每个偶度Carley图是偶泛因子的.同时证明了二面体群上的每个Cayley图均是泛因子的.