Amalgamation in varieties of distributive double p-algebras

Distributive double p-algebras and regular double p-algebras are shown to have the amalgamation property (AP). Finitely generated varieties of distributive double p-algebras with the AP are characterized, and also varieties generated by algebras with a bounded finite number of maximal or minimal prime ideals.Dedicated to Bjarni Jónsson on his 70th birthdayPresented by G. McNulty.The support of the NSERC is gratefully acknowledged by both authors.     

Equimorphy in varieties of distributive double p-algebras

Any finitely generated regular variety V of distributive double p-algebras is finitely determined, meaning that for some finite cardinal n(V), any subclass S V of algebras with isomorphic endomorphism monoids has fewer than n(V) pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double p-algebras must be almost regular.     

On regular incidence quasi-polytopes

It is shown that under certain conditions the regularization of a pair of regular incidence polytopes is not itself an incidence polytope. Thus there exist regular incidence quasi-polytopes which are not incidence polytopes.     

On a classifying property of regular representations

We show that, for each connected compact Lie group G, the Hilbert G-space L ₂(G) and the Banach G-space C(G;?) classify the G-spaces.     

A conjecture on bipartite graphical regular representations

In this paper we are concerned with the classification of the finite groups admitting a bipartite DRR and a bipartite GRR.First, we find a natural obstruction that prevents a finite group from admitting a bipartite GRR. Then we give a complete classification of the finite groups satisfying this natural obstruction and hence not admitting a bipartite GRR. Based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite GRR.Next, we prove the existence of bipartite DRRs for most of the finite groups not admitting a bipartite GRR found in this paper. Actually, we prove a much stronger result: we give an asymptotic enumeration of the bipartite DRRs over these groups. Again, based on these results and on some extensive computer computations, we state a conjecture aiming to give a complete classification of the finite groups admitting a bipartite DRR.     

Graphical regular representations of free products of groups

If X is a Cayley graph of a group G possessing a normal subgroup N, then there is a quotient graph of X which is a Cayley graph of $\text{G}\text{N}$. With the aid of this result, it is shown that the free product of at least two and at most countably many groups, each of which is at most countably generated, admits a graphical regular representation.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

Decomposition of highest-weight representations of a semisimple Lie algebra into representations of regular subalgebras

A formula is derived for the decomposition of the highest-weight representation of a semisimple Lie algebra into irreducible representations of its regular subalgebra. In particular the case of finite-dimensional representations is investigated.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 703–710, December, 1970.     

On graphical regular representations of direct products of groups

A given group G may or may not have the property that there exists a graph X such that the automorphism group of X is regular, as a permutation group, and isomorphic to G. Mark E. Watkins has shown that the direct product of two finite groups has this property if each factor has this property and both factors are different from the cyclic group of order 2. Later, Wilfried Imrich generalized this result to infinite groups. In this paper, a new proof of this result for finite groups is given. The proof rests heavily on the result which states that if X is a graphical regular representation of the group G, then X is not self-complementary.     

Algebras having bases that consist solely of strongly regular elements

“Locally invertible” algebras, those algebras which have a basis consisting solely of strongly regular elements, are introduced as a generalization of “invertible algebras,” that is, algebras which have a basis consisting solely of units. While this new family properly contains the family of (necessarily unital) invertible algebras, its definition does not assume the existence of a multiplicative identity. Because of this, we consider both unital and non-unital examples of locally invertible algebras. In particular, we show that under a mild condition on the basis of a not necessarily unital R-algebra A, the R-algebras $M_n(A)$ of finite matrix rings over the R-algebra A. Furthermore, many infinite matrix algebras are also locally invertible, but not all. Also it is shown that all semiperfect D-algebras over a division ring D are locally invertible.     

On the sign of regular algebraic polarizable automorphic representations

We remove a parity condition from the construction of automorphic Galois representations carried out in the Paris Book Project. We subsequently generalize this construction to the case of ‘mixed-parity’ (but still regular essentially self-dual) automorphic representations over totally real fields, finding associated geometric projective representations. Finally, we optimize some of our previous results on finding geometric lifts, through central torus quotients, of geometric Galois representations, and apply them to the previous mixed-parity setting.     

Regular incidence quasi-polytopes and regular maps

We define incidence quasi-polytopes and give a procedure for constructing regular incidence quasi-polytopes. We use this procedure to construct a finite map of type {a,b} for all even a and b, and infinitely many such maps when a or b is divisible by 4 and both are greater than or equal to 4.