Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

The Braided product of Ockham algebras

In this paper we introduce an algebraic concept of the product of Ockham algebras called the Braided product. We show that ifL _i MS(i=1, 2, ,n) then the Braided product ofL _i(i=1, 2, ,n) exists if and only ifL ₁, ,L _n have isomorphic skeletons.     

Sharply transitive linear algebraic groups

We study Zariski-closed linear groupsG GL _n (k) over fieldsk of characteristic 0 which act sharply transitively on the non-zero vectors ofk ⁿ . For square-freen, orn15, or ifk has cohomological dimension 1 we obtain a complete classification (i.e. a reduction to questions about associative division algebras). The main tools are representation theory of Lie algebras over algebraically closed and non-closed fields, and results about simple associative algebras in order to control the interplay between linear Lie algebras and the associative algebras generated by them. The relation to nearfields and left-symmetric division algebras is also discussed.     

Disproof of a conjecture in graph reconstruction theory

In his thesis [3] B. D. Thatte conjectured that ifG=G ₁,G ₂,...G _n is a sequence of finitely many simple connected graphs (isomorphic graphs may occur in the sequence) with the same number of vertices and edges then their shuffled edge deck uniquely determines the graph sequence (up to a permutation). In this paper we prove that there are such sequences of graphs with the same shuffled edge deck.This research was partially supported by Hungarian National Foundation of Scientific Research Grant no. 1812     

Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by N_{n, 7} and N_{n, 16} are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that N_{n, 7} and N_{n, 16} serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a u-product G₁ ○ G₂ of two u-groups G₁ and G₂, and prove that G₁ ○ G₂ is also a u-group and that every u-group, which contains G₁ and G₂ as subgroups and is generated by these, is a homomorphic image of G₁ ○ G₂. It is stated that if G is a u-group then the coordinate group of an affine space Gⁿ is equal to G ○ F_n, where F_n is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 601–621, September–October, 2005. Supported by RFBR grant No. 05-01-00292, by FP “Universities of Russia” grant No. 04.01.053, and by RF Ministry of Education grant No. E00-1.0-12.     

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.     

On finite groups with a certain number of centralizers

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)?A₅, then #Cent(G)=22 or 32. Moroever, we prove that A₅ is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A₅ in terms of the number of centralizers     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.     

Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD _w .D ₁ is the only invariant control set, and the subsetW(S)={w:D _w =D ₁} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold. Research partially supported by CNPq grant n^o 50.13.73/91-8     

An algorithm for constructing a variety of arbitrary finite dimension

The dimension of a variety V of algebras is the greatest length of a basis (i.e., of an independent generating set) for an SC-theory SC(V), consisting of strong Mal'tsev conditions satisfied in V. The variety V is assumed infinite-dimensional if the lengths of the bases in SC(V) are not bounded. A simple algorithm is found for constructing a variety of any finite dimension r≥1. Using the sieve of Eratosthenes, r distinct primes p₁, p₂,…, p_r are written and their product n=p₁p₂…p_r computed. The variety G_n of algebras (A, f) with one n-ary operation satisfying the identity f(x₁, x₂,…,x_n)=f(x₂,…,x_n, x₁) has, then, dimension r. Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 167–180, March–April, 1998.     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

Krull Dimension and Deviation in Certain Parafree Groups

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) ? n, where μ(G) is the minimum possible number of generators of G.

Let G be fg; then Hom(G, SL ₂?) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer.     

n-Dimensional algebras over a field with a cyclic extension of degree n

We give a geometric method of classifying algebras A _n,K , n-dimensional over a field K, with a cyclic extension of degree n. Algebras A _n,K without zero divisors satisfying some conditions are classified. In particular, we determine all n-dimensional division algebras over a finite field F _q when n is prime and q is large enough.This research was supported in part by a grant from the M U R S T (40 % funds).     

Riemannian almost-product structures with maximal mobility

The aim of the paper is to give a classification up to isomorphism of (local or simply-connected global) Riemannian almost-product structures (i.e. O(n ₁) × ... × O(n _t)-structures) whose automorphism group has maximal dimension (the so-called -structures). The describing objects are Lie algebras with some additional structure, and the methods are mainly Lie algebraic. The results obtained are applied to determine all -structures on simply-connected spaces of constant curvature.     

A simple bijectio n betwee n sta ndard 3× n tableaux a nd irreducible webs for $\mathfrak{sl}_{3}$

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and \mathfrak sl₂\mathfrak {sl}_{2}. Building on a result of Kuperberg, Khovanov–Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3×n and irreducible webs for \mathfraksl₃\mathfrak{sl}_{3} whose boundary vertices are all sources.     

Greatest common subgraphs with specified properties

A graphG without isolated vertices is a greatest common subgraph of a setG of graphs, all having the same size, ifG is a graph of maximum size that is isomorphic to a subgraph of every graph inG. A number of results concerning greatest common subgraphs are presented. For several graphical propertiesP, we discuss the problem of determining, for a given graphG with propertyP, the existence of two non-isomorphic graphsG ₁ andG ₂ of equal size, also with propertyP, such thatG is the unique greatest common subgraph ofG ₁ andG ₂. In particular, this problem is solved whenP is the property of being 2-connected and whenP is the property of having chromatic numbern.     

Counting models in universal Horn classes

Definen _K (λ) to be either ω, or the number of non-isomorphic models inK having cardinality α, whichever cardinal is larger. This paper contains a proof that for a congruence modular variety ⋎ of algebras of countable similarity type, there are only six possible functionsn _⋎. It is also proved that ifn _K (λ)≠2^λ for some λ, andK is a universal Horn class of models for a countable language, thenK must satisfy two conditions, one of which is quite restrictive and requires that the members ofK are all in a certain sense Abelian. Presented by B. Jonsson.     

On J. H. C. Whitehead's aspherical question I

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] _f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l ₁,…,l _k}⊆L such thatL is the normal closure inG of {l ₁,…,l _k}. We prove that a [G, 2] _f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e ₁ ² ,…,e _k ² } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] _f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.