L_{\infty _\omega }free algebras

In this paper the study of which varieties, in a countable similarity type, have non-free (or equivalently ₁-free) algebras is completed. It was previously known that if a variety satisfies a property known as the construction principle then there are such algebras. If a variety does not satisfy the construction principle then either every -free algebra is free or for every infinite cardinalk, there is a k⁺-free algebra of cardinality k⁺ which is not free. Under the set theoretic assumption V=L, for any varietyV in a countable similarity type, either the class of free algebras is definable in or it is not definable in any .In Memory of Evelyn NelsonPresented by Ralph McKenzie.Research partially supported by NSERC of Canada Grant #A8948.Research partially supported by NSERC of Canada. The research for this paper was begun while the second author was visiting Simon Fraser University.     

Reduction of structure for torsors over semilocal rings

Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H ¹(R, S) → H ¹(R, G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H ¹(X, S) → H ¹(X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras and essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper. V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant. Z. Reichstein was partially supported by NSERC Discovery and Accelerator Supplement grants.     

Completeness theory for finite partial algebras

LetA be a finite universe of cardinality ¦A¦ >- 3. We characterize the non quasi-diagonai strongly reflexive relations that determine maximal partial clones onA. Combining this with some known results, we deduce a general completeness criterion for finite partial algebras.Presented by J. Berman.Research partially supported by NSERC Canada.     

Thin-tall Boolean algebras and their automorphism groups

Our main result is a general construction to show that for each countable groupG there are 2¹ pairwise non-isomorphic thin-tall superatomic Boolean algebras, all of them representingG as a group of non-trivial automorphisms.Presented by J. Mycielski.Research supported by the NSERC of Canada.     

Full groups of Cantor minimal systems

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K ⁰-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC ^*-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics. Research supported in part by operating grants from NSERC (Canada). Research supported in part by the Norwegian Research Council for Science and Humanities.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

κ-Gorenstein Modules

Let A and F be artin algebras and ∧UГa paper, we first introduce the notion of k-Gorenstein faithfully balanced selforthogonal bimodule. In this modules with respect to ∧UГ and then characterize it in terms of the U-resolution dimension of some special injective modules and the property of the functors Ext^i （Ext^i （-, U）, U） preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of ∧UГwith finite left or right injective dimension.     

Group algebras with symmetric units satisfying a group identity

We study group algebras FG for which the symmetric units under the natural involution: g^*=g⁻¹ satisfy a group identity. For infinite fields F of characteristic ≠2, a classification of torsion groups G whose symmetric units U⁺(FG) satisfy a group identity was given in [3] by Giambruno-Sehgal-Valenti. We extend this work to non torsion groups. Research supported by NSERC of Canada and MIUR of Italy.     

On the discretization of differential and Volterra integral equations with variable delay

In this paper we analyze the attainable order ofm-stage implicit (collocation-based) Runge-Kutta methods for differential equations and Volterra integral equations of the second kind with variable delay of the formqt (0<q<1). It will be shown that, in contrast to equations without delay, or equations with constant delay, collocation at the Gauss (-Legendre) points will no longer yield the optimal (local) orderO(h ^2m ). This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Research Grant OGP0009406).     

The orders of nonsingular derivations of Lie algebras of characteristic two

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. A study of the set of positive integers which occur as orders of nonsingular derivations of finite-dimensional nonnilpotent Lie algebras of characteristic p > 0 was initiated by Shalev and continued by the present author. In this paper we continue this study in the case of characteristic two. Among other results, we prove that any divisor n of 2^k − 1 with n ⁴ > (2^k − n)³ belongs to . Our methods consist of elementary arguments with polynomials over finite fields and a little character theory of finite groups. This work was partially supported by Ministero dell’Istruzione e dell’Università, Italy, through PRIN “Graded Lie algebras and pro-p-groups of finite width”.     

Stability Properties of Neighbourly Random Polytopes

We introduce a quantitative parameter measuring m-neighbourliness of symmetric convex polytopes in ℝ ^k . We discuss this parameter for random polytopes generated by subgaussian vectors and show its stability properties. Research of P. Mankiewicz was partially supported by KBN Grant no. 1 P03A 015 27. N. Tomczak-Jaegermann holds the Canada Research Chair in Geometric Analysis.     

n-representation infinite algebras

From the viewpoint of higher dimensional Auslander–Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n  -regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext¹${\mathrm{Ext}}^1$-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras.     

Some universal graphs

It is shown that various classes of graphs have universal elements. In particular, for eachn the class of graphs omitting all paths of lengthn and the class of graphs omitting all circuits of length at leastn possess universal elements in all infinite powers. Research partially supported by Hungarian Science Research Fund No. 1805. Research partially supported by NSERC of Canada Grant #A8948.     

Small Forbidden Configurations IV: The 3 Rowed Case

The present paper continues the work begun by Anstee, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let F be a k×l (0,1)-matrix (the forbidden configuration). Small refers to the size of k and in this paper k = 3. Assume A is an m×n simple matrix which has no submatrix which is a row and column permutation of F. We define forb(m,F) as the best possible upper bound on n, for such a matrix A, which depends on m and F. We complete the classification for all 3-rowed (0,1)-matrices of forb (m,F) as either Θ(m), Θ(m²) or Θ(m³) (with constants depending on F). * Research is supported in part by NSERC. † Research was done while the second author visited the University of British Columbia supported by NSERC of first author. Research was partially supported by Hungarian National Research Fund (OTKA) numbers T034702 and T037846.     

Cohen–Macaulay Isolated Singularities with a Dualizing Module

We generalize results of Foxby concerning a commutative Nötherian ring to a certain noncommutative Nötherian algebra Λ over a commutative Gorenstein complete local ring. We assume that Λ is a Cohen–Macaulay isolated singularity having a dualizing module. Then the same method as in the commutative cases works and we obtain a category equivalence between two subcategories of mod Λ, one of which includes all finitely generated modules of finite Gorenstein dimension. We give examples of such algebras which are not Gorenstien; orders related to almost Bass orders and some k-Gorenstein algebras for an integer k.Presented by I. Reiten ^★The author is supported by Grant-in-Aid for Scientific Researches B(1) No. 14340007 in Japan.     

Iyama’s finiteness theorem via strongly quasi-hereditary algebras

Let Λ be an artin algebra and X a finitely generated Λ-module. Iyama has shown that there exists a module Y such that the endomorphism ring Γ of X⊕Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Γ is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length n must have global dimension at most 2n−2. We want to show that Iyama’s better bound is related to the fact that the ring Γ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1.     

Differentiability of convex functions and the convex point of continuity property in Banach spaces

We show that ifX is a separable Banach space, then every continuous, convex, Gateaux differentiable function onX is Fréchet differentiable on a dense set if and only ifX* has theweak*-Convex Point of Continuity Property (C*PCP). Research completed while a visitor at the University of Alberta. Research supported in part by an H. R. MacMillan Fellowship from the University of British Columbia. Research partially supported by NSERC (Canada).     

Strong Morita equivalence of higher-dimensional noncommutative tori. II

We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K ₀-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, held by George A. Elliott.     

A Family of Complete Caps in PG (<Emphasis Type="Italic">n</Emphasis>, 2)

We give a combinatorial construction of a one-parameter and a two-parameter family of complete caps in finite projective spaces over GF(2). As an application of our construction we find, for each α ε[1.89,2], a sequence of complete caps in PG(n,2) whose sizes grow roughly as α ⁿ . We also discuss the relevance of our caps to the problem of finding the least dependent caps of a given cardinality in a given dimension. *Research partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). An erratum to this article is available at .