The Loewy length of a tensor product of modules of a dihedral two-group

While the finite-dimensional modules of the dihedral 2-groups over fields of characteristic 2 were classified over 30 years ago, very little is known about the tensor products of such modules. In this article, we compute the Loewy length of the tensor product of two modules of a dihedral 2-group in characteristic 2. As an immediate consequence, we determine when such a tensor product has a projective direct summand.     

The Ring of Modules with Endo-permutation Source

In this paper we consider certain subalgebras of the Green algebra (representation algebra) of a finite group G. One such algebra is spanned by the isomorphism classes of all indecomposable modules whose source is an endo-permutation module. This algebra can be embedded into a finite direct product of Laurent polynomial rings in finitely many variables over a field. Another such algebra is spanned by the isomorphism classes of all irreducibly generated modules. When G is p-solvable then this algebra is finite-dimensional and split semisimple.R. Boltje was supported by the NSF, DMS-0200592 and 0128969. B. Külshammer was supported by the DAAD.     

Limits of small functors

For a small category enriched over a suitable monoidal category , the free completion of under colimits is the presheaf category . If is large, its free completion under colimits is the -category of small presheaves on , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on .     

Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander [Coherent functors. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). Springer, Berlin, pp. 189–231, 1966] and strict polynomial functors in the sense of Friedlander and Suslin (Invent. Math. 127(2), 209–270, 1997). Applications to functor cohomology are discussed. V. Franjou is partially supported by the LMJL—Laboratoire de Mathématiques Jean Leray, CNRS: Université de Nantes, école Centrale de Nantes, and acknowledges the hospitality and support of CRM Barcelona where the final corrections on this paper were implemented.     

An equivariant tensor product on Mackey functors

For all subgroups H of a cyclic p-group G we define norm functors that build a G-Mackey functor from an H-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.     

Is topologically simple simple?

In this note we determine when topological simplicity of a subsemigroup implies its simplicity. A measure theoretic approach is used. The main result is: In a locally compact completely simple semigroup whose maximal subgroups are compact, every subsemigroup which is either locally compact or of non-empty interior is completely simple. As a corollary, we have: In a compact semigroup, any topologically simple subsemigroup which is either locally compact or of non-empty interior is simple.     

Finite-product-preserving functors,Kan extensions,and strongly-finitary 2-monads

We study those 2-monads on the 2-categoryCat of categories which, as endofunctors, are the left Kan extensions of their restrictions to the sub-2-category of finite discrete categories, describing their algebras syntactically. Showing that endofunctors of this kind are closed under composition involves a lemma on left Kan extensions along a coproduct-preserving functor in the context of cartesian closed categories, which is closely related to an earlier result of Borceux and Day.The first author gratefully acknowledges the support of the Australian Research Council.     

Locally strong endomorphisms of paths

We determine the number of locally strong endomorphisms of directed and undirected paths—direction here is in the sense of a bipartite graph from one partition set to the other. This is done by the investigation of congruence classes, leading to the concept of a complete folding, which is used to characterize locally strong endomorphisms of paths. A congruence belongs to a locally strong endomorphism if and only if the number l of congruence classes divides the length of the original path and the points of the path are folded completely into the l classes, starting from 0 to l and then back to 0, then again back to l and so on. It turns out that for paths locally strong endomorphisms form a monoid if and only if the length of the path is prime or equal to 4 in the undirected case and in the directed case also if the length is 8. Finally some algebraic properties of these monoids are described.     

The Lagrangian Gauss image of a compact surface in Minkowski 3-Space

A generic compact surfaceQ in Minkowski 3-Space is naturally stratified by the loci where the orthogonal line bundle is tangent to the next lower stratum,SP D ⁰ Q M³. To each component inD ⁰ we associate a light-like hypersurface and in turn a Lagrangian loop in the cotangent bundle of the circle. We then establish an inequality relating the Euler characteristic of the indefinite component ofQ with the total Gauß-Maslov index of the associated Lagrangian loops.     

A Tate cohomology sequence for generalized Burnside rings

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G, then the mark morphism φ:D₊→D⁺ is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D⁺. As a consequence, we obtain an exact sequence of Mackey functors

     

Lifting retracted diagrams with respect to projectable functors

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean 〈∨ ,0 〉-semilattices with 〈∨ ,0 〉-embeddings, can be lifted, with respect to the Con_c functor, by a diagram of lattices, then so can every diagram, indexed by a lattice, of finite distributive 〈∨ ,0 〉-semilattices with 〈∨ ,0 〉-embeddings. If the premise of this statement held, this would solve in turn the (still open) problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lattice. We also outline potential applications of our method to other functors, such as the functor on von Neumann regular rings. Received August 12, 2004; accepted in final form June 6, 2005.     

Affine Jacquet functors and Harish-Chandra categories

We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman. The latter provides an extension of the works of Rocha-Caridi and Wallach [A. Rocha-Caridi, N.R. Wallach, Projective modules over infinite dimensional graded Lie algebras, Math. Z. 180 (1982) 151-177] and Deodhar, Gabber and Kac [V. Deodhar, O. Gabber, V. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. Math. 45 (1982) 92-116] on block decompositions of BGG categories for Kac-Moody algebras. We also derive a compatibility relation between the affine Jacquet functor and the Kazhdan-Lusztig tensor product and apply it to prove that the affine Harish-Chandra category is stable under fusion tensoring with the Kazhdan-Lusztig category. This compatibility will be further applied in studying translation functors for the affine Harish-Chandra category, based on the fusion tensor product.     

Natural base-extending operators of foliations into foliations on the Weil functors

For any Weil algebraA we establish a bijection between the set of all natural base-extending operators of foliations into foliations on the Weil functor ofA-velocities and the set of all pairs (I, J) of ideals inA withI J.     

On the continuity of separately continuous bihomomorphisms

Separately continuous bihomomorphisms on a product of convergence or topological groups occur with great frequency. Of course, in general, these need not be jointly continuous. In this paper, we exhibit some results of Banach-Steinhaus type and use these to derive joint continuity from separate continuity. The setting of convergence groups offers two advantages. First, the continuous convergence structure is a powerful tool in many duality arguments. Second, local compactness and first countability, the usual requirements for joint continuity, are available in much greater abundance for convergence groups.     

The Green rings of pointed tensor categories of finite type

In this paper, we compute the Clebsch–Gordan formulae and the Green rings of connected pointed tensor categories of finite type.