Abelian Structure Research Articles

Compatible weak factorization systems and model structures

In this paper, the concept of compatible weak factorization systems in general categories is introduced as a counterpart of compatible complete cotorsion pairs in abelian categories. We describe a method to construct model structures on general categories via two compatible weak factorization systems satisfying certain conditions, and hence, generalize a very useful result by Gillespie for abelian model structures. As particular examples, we show that weak factorization systems associated to some classical model structures (for example, the Kan-Quillen model structure on sSet) satisfy these conditions.

Cotorsion pairs and model structures on Morita rings

We study cotorsion pairs and abelian model structures on Morita rings Λ=(ANBAMABB) which are Artin algebras. Given cotorsion pairs (U,X) and (V,Y) in A-Mod and B-Mod, respectively, one can construct four cotorsion pairs in Λ-Mod:((XY)⊥,(XY)),(Δ(U,V),Δ(U,V)⊥),((UV),(UV)⊥),(∇⊥(X,Y),∇(X,Y)). These cotorsion pairs have relations:Δ(U,V)⊥⊆(XY),∇⊥(X,Y)⊆(UV). An important feature is that they are not equal, in general. In fact, there even exists a Morita algebra Λ, such that the four cotorsion pairs are pairwise different. The problem of identifications, i.e., when these inclusions are the same, are studied; the heredity and completeness of these cotorsion pairs are investigated; and finally, various model structures on Λ-Mod are obtained, by explicitly giving the corresponding Hovey triples and Quillen's homotopy categories. In particular, cofibrantly generated Hovey triples, and the Gillespie-Hovey triples induced by compatible generalized projective (respectively, injective) cotorsion pairs, are explicitly constructed. All these Hovey triples obtained are pairwise different and “new” in some sense. Some results are even new when M=0 or N=0.

Coderived and contraderived categories of locally presentable abelian DG-categories

The concept of an abelian DG-category, introduced by the first-named author in Positselski (Exact DG-categories and fully faithful triangulated inclusion functors. arXiv:2110.08237 [math.CT]), unites the notions of abelian categories and (curved) DG-modules in a common framework. In this paper we consider coderived and contraderived categories in the sense of Becker. Generalizing some constructions and results from the preceding papers by Becker (Adv Math 254:187–232, 2014. arXiv:1205.4473 [math.CT]) and by the present authors (Positselski and Št’ovíček in J Pure Appl Algebra 226(#4):106883, 2022. arXiv:2101.10797 [math.CT]), we define the contraderived category of a locally presentable abelian DG-category B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{B}$$\\end{document} with enough projective objects and the coderived category of a Grothendieck abelian DG-category A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{A}$$\\end{document}. We construct the related abelian model category structures and show that the resulting exotic derived categories are well-generated. Then we specialize to the case of a locally coherent Grothendieck abelian DG-category A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{A}$$\\end{document}, and prove that its coderived category is compactly generated by the absolute derived category of finitely presentable objects of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{A}$$\\end{document}, thus generalizing a result from the second-named author’s preprint (Št’ovíček in On purity and applications to coderived and singularity categories. arXiv:1412.1615 [math.CT]). In particular, the homotopy category of graded-injective left DG-modules over a DG-ring with a left coherent underlying graded ring is compactly generated by the absolute derived category of DG-modules with finitely presentable underlying graded modules. We also describe compact generators of the coderived categories of quasi-coherent matrix factorizations over coherent schemes.

On Hawaiian homology groups

In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group is isomorphic to the abelianization of the first Hawaiian group for path-connected and locally path-connected topological spaces. Since Hawaiian homology has concrete elements and abelian structure, its calculations are more routine. Thus we use Hawaiian homology groups to compare Hawaiian groups, and then we obtain some information about Hawaiian groups of some wild topological spaces.

K\"{a}hler-Norden structures on Hom-Lie groups and Hom-Lie algebras

The aim of this paper is to describe two geometric notions, holomorphic Norden structures and K\"{a}hler-Norden structures on Hom-Lie groups, and study their relationships in the left invariant setting. We study K\"{a}hler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex structure) is abelian.

Abelian model structures on comma categories

UDC 512.64 Let A and B be bicomplete Abelian categories, which both have enough projectives and injectives and let T : A → B be a right exact functor. Under some mild conditions, we show that hereditary Abelian model structures on A and B can be amalgamated into a global hereditary Abelian model structure on the comma category ( T ↓ B ) . As an application of this result, we give an explicit description of a subcategory that consists of all trivial objects of the Gorenstein flat model structure on the category of modules over a triangular matrix ring.

On splittings and integration of almost periodic functions with and without geometry

AbstractRecently, the authors introduced the notion of weighted semigroups which apply to sun-dual semigroups and especially to the translation semigroup on the space of left continuous functions with values in dual spaces. In this article, we will show that it is sufficient that we either assume geometry on the Banach, or an abelian structure on the minimal subgroup to prove almost periodicity. This yields a different approach to the almost periodicity of semigroups and integrals, by extending Basit generalized Kadets result to general groups, to obtain almost periodicity.

Generalized tilting theory in functor categories

AbstractThis paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories $\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$ with $\mathcal{C}$ an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory $\mathcal{T}$ of $\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C})$ is constructed. Some applications of these two results include the equivalence of Grothendieck groups $K_0(\mathcal{C})$ and $K_0(\mathcal{T})$ , the existences of a new abelian model structure on the category of complexes $\mathop{\textrm{C}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$ , and a t-structure on the derived category $\mathop{\textrm{D}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C}))$ .

On the algebra of elliptic curves

AbstractIt is argued that a nonsingular elliptic curve admits a natural or fundamental abelian heap structure uniquely determined by the curve itself. It is shown that the set of complex analytic or rational functions from a nonsingular elliptic curve to itself is a truss arising from endomorphisms of this heap.

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ , the unrolled restricted quantum group of $$\mathfrak {sl}_3$$ at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of $$W^0_{\!A_2}(2)$$ -modules. Here, $$W^0_{\!A_2}(2)$$ is an orbifold of the octuplet vertex algebra $$W_{\!A_2}(2)$$ of Semikhatov, the latter being the natural $$\mathfrak {sl}_3$$ -analogue of the well known triplet algebra. Moreover, $$W^0_{\!A_2}(2)$$ is the parafermionic coset of the affine vertex algebra $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ . We formulate an explicit conjecture relating the representation theory of $$W^0_{\!A_2}(2)$$ and $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ and work out the resulting structures of the corresponding $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ -modules. In particular, we obtain conjectural Loewy diagrams for the latter’s projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde’s formula. Finally, we give analogous results for $$W_{\!A_2}(2)$$ .

Lifting recollements of abelian categories and model structures

We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwanaga-Gorenstein rings.

On pp-elimination and stability in a continuous setting

We generalize pp-elimination for modules, or more generally, abelian structures, to a continuous logic setting where the abelian structure is equipped with a homomorphism to a compact Hausdorff group. We conclude that the continuous logic theory of such a structure is stable.

The homotopy category of acyclic complexes of pure-projective modules

Abstract Let R be any ring with identity. We show that the homotopy category of all acyclic chain complexes of pure-projective R-modules is a compactly generated triangulated category. We do this by constructing abelian model structures that put this homotopy category into a recollement with two other compactly generated triangulated categories: The usual derived category of R and the pure derived category of R.

Unified products for Jordan algebras. Applications

Given a Jordan algebra A and a vector space V, we describe and classify all Jordan algebras containing A as a subalgebra of codimension dimk(V) in terms of a non-abelian cohomological type object JA(V,A). Any such algebra is isomorphic to a newly introduced object called unified productA♮V. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology Hnab2(V,A) associated to two Jordan algebras A and V which classifies all extensions of V by A is also constructed. Several applications and examples are given: we prove that Hnab2(k,kn) is identified with the set of all matrices D∈Mn(k) satisfying 2D3−3D2+D=0, where we consider the abelian Jordan algebra structure on k and kn.

Odd‐dimensional counterparts of abelian complex and hypercomplex structures

AbstractWe introduce the notion of abelian almost contact structures on an odd‐dimensional real Lie algebra . We investigate correspondences with even‐dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when carries a compatible inner product. The classification of 5‐dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3‐contact structures on real Lie algebras of dimension , obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3‐contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.

A note on twisted crossed products and spectral triples

Starting with a spectral triple on a unital C⁎-algebra A with an action of a discrete group G, if the action is uniformly bounded (in a Lipschitz sense) a spectral triple on the reduced crossed product C⁎-algebra A⋊rG is constructed in [23]. The main instrument is the Kasparov external product. We note that this construction still works for twisted crossed products when the twisted action is uniformly bounded in the appropriate sense. Under suitable assumptions we discuss some basic properties of the resulting triples: summability and regularity. Noncommutative coverings with finite abelian structure group are among the most basic, still interesting, examples of twisted crossed products; we describe their main features.

K‐flat complexes and derived categories

Let R $R$ be a ring with identity. Inspired by recent work in Emmanouil, Preprint, 2021, we show that the derived category of R $R$ is equivalent to the chain homotopy category of all K-flat complexes with pure-injective components. This is implicitly related to a recollement we exhibit. It expresses D p u r ( R ) $\mathcal {D}_{pur}(R)$ , the pure derived category of R $R$ , as an attachment of the usual derived category D ( R ) $\mathcal {D}(R)$ with Emmanouil's quotient category D K − f l a t ( R ) : = K ( R ) / K − F l a t $\mathcal {D}_{\textnormal {K-flat}}(R):=K(R)/K\textnormal {-Flat}$ , which we call here the K-flat derived category. It follows that this Verdier quotient is a compactly generated triangulated category. We obtain our results by using methods of cotorsion pairs to construct (cofibrantly generated) monoidal abelian model structures on the exact category of chain complexes along with the degreewise pure exact structure. In fact, most of our model structures are obtained as corollaries of a general method which associates an abelian model structure to any class of so-called C $\mathcal {C}$ -acyclic complexes, where C $\mathcal {C}$ is any given class of chain complexes. Finally, we also give a new characterization of K-flat complexes in terms of the pure derived category of R $R$ .

An exact quantum hidden subgroup algorithm and applications to solvable groups

We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $\Z_{m^k}^n$. The algorithm uses the quantum Fourier transform modulo $m$ and does not require factorization of $m$. For smooth $m$, i.e., when the prime factors of $m$ are of size $(\log m)^{O(1)}$, the quantum Fourier transform can be exactly computed using the method discovered independently by Cleve and Coppersmith, while for general $m$, the algorithm of Mosca and Zalka is available. Even for $m=3$ and $k=1$ our result appears to be new. We also present applications to compute the structure of abelian and solvable groups whose order has the same (but possibly unknown) prime factors as $m$. The applications for solvable groups also rely on an exact version of a technique proposed by Watrous for computing the uniform superposition of elements of subgroups.

Decay of Correlations in Finite Abelian Lattice Gauge Theories

In this paper, we study lattice gauge theory on mathbb {Z}^4 with finite Abelian structure group. When the inverse coupling strength is sufficiently large, we use ideas from disagreement percolation to give an upper bound on the decay of correlations of local functions. We then use this upper bound to compute the leading-order term for both the expected value of the spin at a given plaquette as well as for the two-point correlation function. Moreover, we give an upper bound on the dependency of the size of the box on which the model is defined. The results in this paper extend and refine results by Chatterjee and Borgs.

Pluriclosed and Strominger Kähler–like metrics compatible with abelian complex structures

We show that the existence of a left-invariant pluriclosed Hermitian metric on a unimodular Lie group with a left-invariant abelian complex structure forces the group to be 2-step nilpotent. Moreover, we prove that the pluriclosed flow starting from a left-invariant Hermitian metric on a 2-step nilpotent Lie group preserves the Strominger Kähler–like condition.