Brauer Characters of Finite Monoids

We introduce the concept of a Brauer character of a representation of a finite monoid M in characteristic p>0. When p does not divide the order of any subgroup of M, we develop a theory of p-monoid quivers. We apply our results to the full transformation semigroup.     

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ?-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ?-situation.     

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.

     

Commuting Homotopy Limits and Smash Products

In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization that involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Bökstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.     

On distance two graphs of upper bound graphs

For a poset P=(X,≤), the upper bound graph (UB-graph) of P is the graph U=(X,E_U), where uv∈E_U if and only if u≠v and there exists m∈X such that u,v≤m. For a graph G, the distance two graph DS₂(G) is the graph with vertex set V(DS₂(G))=V(G) and u,v∈V(DS₂(G)) are adjacent if and only if d_G(u,v)=2. In this paper, we deal with distance two graphs of upper bound graphs. We obtain a characterization of distance two graphs of split upper bound graphs.     

A modified Brauer algebra as centralizer algebra of the unitary group

The centralizer algebra of the action of on the real tensor powers of its natural module, , is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for and with the decomposition of into irreducible submodules is considered.

     

Planarity of Cayley graphs of graph products of groups

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.     

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 .

     

On the Brauer group of a two-dimensional local field

The two-dimensional local field K = F _q((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have x ^p ? x = ut ^?p =: h, then the condition that (y, f | h]_K = 0 for any y ε K is equivalent to the condition f ε Im(Nm(L*)).     

The Brauer group of an affine double plane associated to a hyperelliptic curve

We study the Brauer group of an affine double plane π:X→𝔸² defined by an equation of the form z² = f in two separate cases. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface ?¹×C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y²?p(x) is the equation of an affine hyperelliptic curve. For π as well as the unramified part of π, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology.     

Orthogonal bases of Brauer symmetry classes of tensors for the dihedral group

In this article necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the dihedral group.     

End-regularity of zero-divisor graphs of group rings

A graph Γ is said to be End-regular if its endomorphism monoid End(Γ) is regular. D. Lu and T. Wu [25 Lu, D., Wu, T. (2008). On endomorphism-regularity of zero-divisor graphs. Discrete Math. 308:4811–4815.[Crossref], [Web of Science ®] , [Google Scholar]] posed an open problem: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? and they solved the problem for R a commutative ring with at least one nontrivial idempotent. In this paper, we solve this problem for zero-divisor graphs of group rings.     

A bound on the treewidth of planar even-hole-free graphs

The class of planar graphs has unbounded treewidth, since the k×k grid, ∀k∈N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an even-hole-free planar graph, then it does not contain a 9×9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 49.     

About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

Note on upper bound graphs and forbidden subposets

In the upper bound graph of a poset P, the vertex set is V(P) and xy is an edge if there exists an m∈V(P) with x,y≤_Pm. We show some characterizations on split upper bound graphs, threshold upper bound graphs and difference upper bound graphs in terms of m-subposets and canonical posets.     

The Brauer group of a noncomplete real algebraic surface

The Brauer group of a noncomplete real algebraic surface is calculated. The calculations make use of equivariant cohomology. The resulting formula is similar to the formula for a complete surface, but the proof is substantially different. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 355–359, March, 2000.