Cofiniteness conditions, projective covers and the logarithmic tensor product theory

We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C₁-cofinite in the sense of Li. 2. There exists a positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra A_N(V) is finite dimensional. This result shows that the category of grading-restricted generalized V-modules is a finite abelian category over C. Using the existence of projective covers, we prove that if such a vertex operator algebra V also satisfies Condition 3 that irreducible V-modules are R-graded and C₁-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under operations ?_P(z) for z∈C^×. We also prove that other conditions for applying the logarithmic tensor product theory developed by Lepowsky, Zhang and the author hold. Consequently, for such V, this category has a natural structure of braided tensor category. In particular, when V is of positive energy and C₂-cofinite, Conditions 1-3 are satisfied and thus all the conclusions hold.     

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract

The category Hopf ? of Hopf monoids in a symmetric monoidal category ?, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ? preserve directed colimits one has the following results: (1) If, in ?, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ? and comonadic over the category of monoids in C. (2) If, in ?, extremal monomorphisms are stable under tensor squaring, then Hopf ? is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ?.     

Formal Hopf algebra theory I: Hopf modules for pseudomonoids

In this article we develop some of the basic constructions of the theory of Hopf algebras in the context of autonomous pseudomonoids in monoidal bicategories. We concentrate on the notion of Hopf modules. We study the existence and the internalisation of this notion, called the Hopf module construction. Our main result is the equivalence between the existence of a left dualization for A (i.e., A is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case a Hopf module construction for A always exists. We recover from the general theory developed here results on coquasi-Hopf algebras.     

On the strong almost convergence of double sequences

In this paper, necessary and sufficient conditions that the real doubly infinite matrixA sums every strongly almost convergent double sequence, leaving the limit invariant, have been determined.     

On geometrically transitive Hopf algebroids

This paper contributes to the characterization of a certain class of commutative Hopf algebroids. It is shown that a commutative flat Hopf algebroid with a non zero base ring and a nonempty character groupoid is geometrically transitive if and only if any base change morphism is a weak equivalence (in particular, if any extension of the base ring is Landweber exact), if and only if any trivial bundle is a principal bi-bundle, and if and only if any two objects are fpqc locally isomorphic. As a consequence, any two isotropy Hopf algebras of a geometrically transitive Hopf algebroid (as above) are weakly equivalent. Furthermore, the character groupoid is transitive and any two isotropy Hopf algebras are conjugated. Several other characterizations of these Hopf algebroids in relation to transitive groupoids are also given.     

Additivity of the crossing number of graphs with connectivity 2

We prove that the crossing number of graphs with connectivity 2 has in certain cases an additive property analogous to that of crossing number of graphs with connectivity ≤1.     

Finite products are biproducts in a compact closed category

If a compact closed category has finite products or finite coproducts then it in fact has finite biproducts, and so is semi-additive.     

Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.     

A theory of recursive dimension for ordered sets

The classical theorem of R. P. Dilworth asserts that a partially ordered set of width n can be partitioned into n chains. Dilworth's theorem plays a central role in the dimension theory of partially ordered sets since chain partitions can be used to provide embeddings of partially ordered sets in the Cartesian product of chains. In particular, the dimension of a partially-ordered set never exceeds its width. In this paper, we consider analogous problems in the setting of recursive combinatorics where it is required that the partially ordered set and any associated partition or embedding be described by recursive functions. We establish several theorems providing upper bounds on the recursive dimension of a partially ordered set in terms of its width. The proofs are highly combinatorial in nature and involve a detailed analysis of a 2-person game in which one person builds a partially ordered set one point at a time and the other builds the partition or embedding.This paper was prepared while the authors were supported, in part, by NSF grant ISP-80-11451. In addition, the second author received support under NSF grant MCS-80-01778 and the third author received support under NSF grant MCS-82-02172.     

Twisted dendriform algebras and the pre-Lie Magnus expansion

In this paper an application of the recently introduced pre-Lie Magnus expansion to Jackson’s q-integral and q-exponentials is presented. Twisted dendriform algebras, which are the natural algebraic framework for Jackson’s q-analogs, are introduced for that purpose. It is shown how the pre-Lie Magnus expansion is used to solve linear q-differential equations. We also briefly outline the theory of linear equations in twisted dendriform algebras.     

The total graph and regular graph of a commutative ring

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z(R) is not an ideal. In this paper we show that if T(Γ(R)) is a connected graph, then . Also, we prove that if R is a finite ring, then T(Γ(R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.     

Stochastic geometry,random zero point field and quantum correction to the metric

The random zero point field induces the probabilistic aspect in the geometry of background spacetime. The corrections to the metric tensor for Riemannian or pseudo-Euclidean spaces are calculated by averaging over the ensemble of random A (x). This provides a cut-off procedure which yields a finite energy density for the vacuum state.     

The Hilbert functions which force the Weak Lefschetz Property

The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [A. Wiebe, The Lefschetz property for componentwise linear ideals and Gotzmann ideals, Comm. Algebra 32 (12) (2004) 4601-4611]) whose Gotzmann ideals have the WLP.This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti numbers) have the WLP as well. However, we will answer in the negative, even in the case of level algebras, the most natural question that one might ask after reading the previous sentence: If A is an artinian algebra enjoying the WLP, do all artinian algebras with the same Hilbert function as A and Betti numbers lower than those of A have the WLP as well?Also, as a consequence of our result, we have another (simpler) proof of the fact that all codimension 2 algebras enjoy the WLP (this fact was first proven in [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for Artinian K-algebras, J. Algebra 262 (2003) 99-126], where it was shown that even the Strong Lefschetz Property holds).     

The 2-category theory of quasi-categories

In this paper we re-develop the foundations of the category theory of quasi-categories (also called ∞-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2-category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples.     

Planar graphs and poset dimension

We view the incidence relation of a graph G=(V. E) as an order relation on its vertices and edges, i.e. a<_G b if and only of a is a vertex and b is an edge incident on a. This leads to the definition of the order-dimension of G as the minimum number of total orders on V E whose intersection is <_G. Our main result is the characterization of planar graphs as the graphs whose order-dimension does not exceed three. Strong versions of several known properties of planar graphs are implied by this characterization. These properties include: each planar graph has arboricity at most three and each planar graph has a plane embedding whose edges are straight line segments. A nice feature of this embedding is that the coordinates of the vertices have a purely combinatorial meaning.