A note on the equivalence of m-periodic derived categories

Let A and B be finite-dimensional algebras over a field k of finite global dimension. Using some results of Gorsky in “Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras”, we prove that if A and B are derived equivalent, then the corresponding m-periodic derived categories are triangulated equivalent.     

Circuit preserving edge maps II

In Section 1 of this article we prove the following. Let f: G → G′ be a circuit surjection, i.e., a mapping of the edge set of G onto the edge set of G′ which maps circuits of G onto circuits of G′, where G, G′ are graphs without loops or multiple edges and G′ has no isolated vertices. We show that if G is assumed finite and 3-connected, then f is induced by a vertex isomorphism. If G is assumed 3-connected but not necessarily finite and G′ is assumed to not be a circuit, then f is induced by a vertex isomorphism. Examples of circuit surjections f: G → G′ where G′ is a circuit and G is an infinite graph of arbitrarily large connectivity are given. In general if we assume G two-connected and G′ not a circuit then any circuit surjection f: G → G′ may be written as the composite of three maps, f(G) = q(h(k(G))), where k is a 1-1 onto edge map which preserves circuits in both directions (the “2-isomorphism” of Whitney (Amer. J. Math. 55 (1933), 245–254) when G is finite), h is an onto edge map obtained by replacing “suspended chains” of k(G) with single edges, and G is a circuit injection (a 1-1 circuit surjection). Let f: G → M be a 1-1 onto mapping of the edges of G onto the cells of M which takes circuits of G onto circuits of M where G is a graph with no isolated vertices, M a matroid. If there exists a circuit C of M which is not the image of a circuit in G, we call f nontrivial, otherwise trivial. In Section 2 we show the following. Let G be a graph of even order. Then the statement “no trivial map f: G → M exists, where M is a binary matroid,” is equivalent to “G is Hamiltonian.” If G is a graph of odd order, then the statement “no nontrivial map f: G → M exists, where M is a binary matroid” is equivalent to “G is almost Hamiltonian,” where we define a graph G of order n to be almost Hamiltonian if every subset of vertices of order n − 1 is contained in some circuit of G.     

Groupoid normalizers of tensor products

We consider an inclusion B⊆M of finite von Neumann algebras satisfying B^′∩M⊆B. A partial isometry v∈M is called a groupoid normalizer if vBv^∗,v^∗Bv⊆B. Given two such inclusions Bi⊆Mi, i=1,2, we find approximations to the groupoid normalizers of in , from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis , i=1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v∈M satisfying vBv^∗⊆B and v^∗v,vv^∗∈B.     

The higher rank rigidity theorem for manifolds with no focal points

We say that a Riemannian manifold M has rank M ≥ k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns–Spatzier, later generalized by Eberlein–Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank k ≥ 2 (the “higher rank” assumption) whose isometry group Γ satisfies the condition that the Γ-recurrent vectors are dense in SM is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein–Heber to prove a generalization of this theorem where the manifold M is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann–Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.     

Permanent formulae from the Veronesean

The two formulae for the permanent of a d × d matrix given by Ryser (1963) and Glynn (2010) fit into a similar pattern that allows generalization because both are related to polarization identities for symmetric tensors, and to the classical theorem of P. Serret in algebraic geometry. The difference between any two formulae of this type corresponds to a set of dependent points on the “Veronese variety” (or “Veronesean”) v _d ([d ? 1]), where v _d ([n]) is the image of the Veronese map v _d acting on [n], the n-dimensional projective space over a suitable field. To understand this we construct dependent sets on the Veronesean and show how to construct small independent sets of size nd + 2 on v _d ([n]). For d = 2 such sets of 2n + 2 points in [n] have been called “associated” and we observe that they correspond to self-dual codes of length 2n + 2.     

Decompositions of modules lacking zero sums

A module over a semiring lacks zero sums (LZS) if it has the property that v +w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, this property always holds for modules over an idempotent semiring and related semirings, so arises for example in tropical mathematics.A direct sum decomposition theory is developed for direct summands (and complements) of LZS modules: The direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated “strongly projective” module is a finite direct sum of summands of R (assuming the mild assumption that 1 is a finite sum of orthogonal primitive idempotents of R). This leads to an analog of the socle of “upper bound” modules. Some of the results are presented more generally for weak complements and semi-complements. We conclude by examining the obstruction to the “upper bound” property in this context.     

The Fundamental Pro-groupoid of an Affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don’t remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π ₁(spec(R)), i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π ₁(spec(R)) in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π ₁ for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the étale fundamental group of a scheme preserves finite products but not all products.     

Cyclic Hilbert Spaces and Connes’ Embedding Problem

Let M be a II ₁-factor with trace τ, the finite dimensional subspaces of L ²(M, τ) are not just common Hilbert spaces, but they have an additional structure. We introduce the notion of a cyclic linear space by taking these additional properties as axioms. In Sect. 3 we formulate the following problem: “does every cyclic Hilbert space embed into L ²(M, τ), for some M?”. An affirmative answer would imply the existence of an algorithm to check Connes’ embedding Conjecture. In Sect. 4 we make a first step towards the answer of the previous question.     

Completeness of the Space of Separable Measures in the Kantorovich-Rubinshtein Metric

We consider the space M(X) of separable measures on the Borel σ-algebra ?(X) of a metric space X. The space M(X) is furnished with the Kantorovich-Rubinshtein metric known also as the “Hutchinson distance” (see [1]). We prove that M(X) is complete if and only if X is complete. We consider applications of this theorem in the theory of selfsimilar fractals.     

Surface basic sets with wildly embedded supporting surfaces

The situation where a “nice” diffeomorphism f of a 3-manifold has a wildly embedded invariant surfaceM for which the restriction g = f| _M : M → M is “nice” is considered.     

Characterization and representation problems for intersection betweennesses

For a set system M=(M_v)_v∈V indexed by the elements of a finite set V, the intersection betweenness B(M) induced by M consists of all triples (u,v,w)∈V³ with M_u∩M_w⊆M_v. Similarly, the strict intersection betweenness B_s(M) induced by M consists of all triples (u,v,w)∈B(M) such that u, v, and w are pairwise distinct. The notion of a strict intersection betweenness was introduced by Burigana [L. Burigana, Tree representations of betweenness relations defined by intersection and inclusion, Math. Soc. Sci. 185 (2009) 5-36]. We provide axiomatic characterizations of intersection betweennesses and strict intersection betweennesses. Our results yield a simple and efficient algorithm that constructs a representing set system for a given (strict) intersection betweenness. We study graphs whose strict shortest path betweenness is a strict intersection betweenness. Finally, we explain how the algorithmic problem related to Burigana’s notion of a partial tree representation can be solved efficiently using well-known algorithms.     

Fixed Points of Compact Quantum Groups Actions on Cuntz Algebras

Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using Kirchberg classification results. Under certain conditions, we prove that the fixed point algebra is purely infinite and simple. We further identify it as a C ^*-algebra, compute its K-theory and prove a “stability property”: the fixed points only depend on the CQG via its fusion rules. We apply the theory to SU _q (N) and illustrate by explicit computations for SU _q (2) and SU _q (3). This construction provides examples of free actions of CQG (or “principal noncommutative bundles”).     

Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measurev on [?1, 1] are studied whenv is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in [?1, 1]. We prove some weighted norm inequalities for the partial sum operatorsS _n, their maximal operatorS ^*, and the commutator [M _b, S_n], whereM _b denotes the operator of pointwise multiplication byb ∈BMO. We also prove some norm inequalites forS _n whenv is a sum of a Laguerre weitht onR ⁺ and a positive mass on 0.     

Lines imply spaces in density Ramsey theory

Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of Fⁿ there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see [9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in [3, 6] we obtain various consequences: a “group-theoretic” version of Roth's Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥F∥ = 2.     

Note on a conjecture of Kojima and Saigal

We prove there exist a finite set of (real) matrices of order n with positive determinant that, collectively, “see” all such matrices.     

Coactions on Spaces of Morphisms

We study certain comodule structures on spaces of linear morphisms between H-comodules, where H is a Hopf algebra over the field k. We apply the results to show that H has non-zero integrals if and only if there exists a non-zero finite dimensional injective right H-comodule. Using this approach, we prove an extension of a result of Sullivan, by showing that if H is involutory and has non-zero integrals, and there exists an injective indecomposable right comodule whose dimension is not a multiple of char(k), then H is cosemisimple. Also we prove without using character theory that if H is cosemisimple and M is an absolutely irreducible right H-comodule, then char(k) does not divide dim(M).     

*-Generalized polynomial identities of finite dimensional central simple algebras

LetD be a finite dimensional central simple algebra with involution * of the first kind. We prove (in a fixed number of variables) the ideal of *-GPI’s ofD is generated by a finite collection of elements of the form [v _ij ,v _pq ] andv _ij ?v _ij ^* where thev _ij ’s are first degree generalized polynomials.     

The isogeny conjecture for t-motives associated to direct sums of Drinfeld modules

Let K be a finitely generated field of transcendence degree 1 over a finite field. Let M be a t-motive over K of characteristic p₀, which is semisimple up to isogeny. The isogeny conjecture for M says that there are only finitely many isomorphism classes of t-motives M^′ over K, for which there exists a separable isogeny M^′→M of degree not divisible by p₀. For the t-motive associated to a Drinfeld module this was proved by Taguchi. In this article we prove it for the t-motive associated to any direct sum of Drinfeld modules of characteristic p₀≠0.     

Additive representation of symmetric inverse M-matrices and potentials

In this article we characterize the closed cones respectively generated by the symmetric inverse M-matrices and by the inverses of symmetric row diagonally dominant M-matrices. We show the latter has a finite number of extremal rays, while the former has infinitely many extremal rays. As a consequence we prove that every potential is the sum of ultrametric matrices.