Category Of Algebras Research Articles

Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras

In this paper, the categories of braided categorical associative algebras and braided crossed modules of associative algebras are studied. These structures are also correlated with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras.

Realisability problem in arrow categories

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category $\mathcal{C}$ and for arbitrary groups $H\le G_1\times G_2$, is there an object $\phi \colon A_1 \rightarrow A_2$ in $\operatorname{Arr}(\mathcal{C})$ such that $\operatorname{Aut}_{\operatorname{Arr}(\mathcal{C})}(\phi) = H$, $\operatorname{Aut}_{\mathcal{C}}(A_1) = G_1$ and $\operatorname{Aut}_{\mathcal{C}}(A_2) = G_2$? We are interested in solving this problem when $\mathcal C =\mathcal{H}oTop_*$, the homotopy category of pointed topological spaces. To that purpose, we first settle that question in the positive when $\mathcal C = \mathcal{G}raphs$. Then, we construct an almost fully faithful functor from $\mathcal{G}raphs$ to $\operatorname{CDGA}$, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when $\mathcal C = \operatorname{CDGA}$ and, as long as we work with finite groups, when $\mathcal C =\mathcal{H}oTop_*$. Some results on representability of concrete categories are also obtained.

The category of implicative algebras and realizability

AbstractIn this paper, we continue with the algebraic study of Krivine’s realizability, completing and generalizing some of the authors’ previous constructions by introducing two categories with objects the abstract Krivine structures and the implicative algebras, respectively. These categories are related by an adjunction whose existence clarifies many aspects of the theory previously established. We also revisit, reinterpret, and generalize in categorical terms, some of the results of our previous work such as: the bullet construction, the equivalence of Krivine’s, Streicher’s, and bullet triposes and also the fact that these triposes can be obtained – up to equivalence – from implicative algebras or implicative ordered combinatory algebras.

Graded tilting for gauged Landau–Ginzburg models and geometric applications

In this paper we develop a graded tilting theory for gauged Landau-Ginzburg models of regular sections in vector bundles over projective varieties. Our main theoretical result describes - under certain conditions - the bounded derived category of the zero locus $Z(s)$ of such a section $s$ as a graded singularity category of a non-commutative quotient algebra $\Lambda/\langle s\rangle$: $D^b(\mathrm{coh} Z(s))\simeq D^{\mathrm{gr}}_{\mathrm{sg}}(\Lambda/\langle s\rangle)$. Our geometric applications all come from homogeneous gauged linear sigma models. In this case $\Lambda$ is a non-commutative resolution of the invariant ring which defines the $\mathbb{C}^*$-equivariant affine GIT quotient of the model. We obtain purely algebraic descriptions of the derived categories of the following families of varieties: - Complete intersections. - Isotropic symplectic and orthogonal Grassmannians. - Beauville-Donagi IHS 4-folds.

On Morita weak equivalences of simplicial algebraic theories and operads

Generalizing a classical result of Dwyer and Kan for simplicial categories, we characterize the morphisms of multi-sorted simplicial algebraic theories and simplicial coloured operads which induce a Quillen equivalence between the corresponding categories of algebras.

CHOICE-FREE STONE DUALITY

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

A Hopf algebra on subgraphs of a graph

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category.

Energy bounds condition for intertwining operators of types 𝐵, 𝐶, and 𝐺₂ unitary affine vertex operator algebras

The energy bounds condition for intertwining operators of unitary rational vertex operator algebras (VOAs) was studied, first by A. Wassermann for type A A affine VOAs, and later by T. Loke for c > 1 c>1 Virasoro VOAs, and by V. Toledano-Laredo for type D D affine VOAs. In this paper, we extend their results to affine VOAs of types B B , C C , and G 2 G_2 . As a consequence, the modular tensor categories of these unitary vertex operator algebras are unitary.

Prelinear Hilbert algebras

In this paper we give an explicit description of the left adjoint of the forgetful functor from the algebraic category of Gödel algebras (i.e., prelinear Heyting algebras) to the algebraic category of bounded prelinear Hilbert algebras. We apply this result in order to study possible descriptions of the coproduct of two finite algebras in the algebraic category of prelinear Hilbert algebras.

Relations for Grothendieck groups and representation-finiteness

For an exact category E, we study the Butler's condition “AR=Ex”: the relation of the Grothendieck group of E is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that E has finitely many indecomposables. This can be applied to functorially finite torsion(free) classes and contravariantly finite resolving subcategories of the module category of an artin algebra, and the category of Cohen-Macaulay modules over an order which is Gorenstein or has finite global dimension. Also we showed that under some weaker assumption, AR=Ex implies that the category of syzygies in E has finitely many indecomposables.

Compatible actions of Lie algebras

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object L from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in XModL(LieR).

The Japanese Pioneering Contribution to the Investigation of Modular Structures

AbstractMonoarchetypal modular crystal structures are built by regularly juxtaposing modules obtained from the same archetype. At the interface the coordination environment may altered, leading to a chemical modulation that accompanies the structural variability. When this does not occur the result is a series of polytypes, whose building modules are not limited to layers, although the latter represent the most common examples. The symmetry theory of polytypes was developed by the Order–Disorder (OD) school, which however considered only layers as building modules, and only those polytypes in which pairs of layers are geometrically equivalent. Well before the OD school, Ito in Japan recognized the need of a wider algebraic category, that he called “twinned space groups”, in which he considered only digonal operations. His approach was later expanded by Sadanaga, who recognized that Ito's “twinned space groups” are space groupoids. In this article the contribution of the Japanese school to the field is traced back, and it is shown that the fundamental concepts later developed in a more systematic way, were already present, in a nutshell, in the works of Ito and Sadanaga; finally, a slightly revised classification of monoarchetypal modular structures is proposed which better reflects the original contribution by the Japanese school.

A Geometric Model for the Module Category of a Gentle Algebra

AbstractIn this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra.

Unitarity of the Modular Tensor Categories Associated to Unitary Vertex Operator Algebras, II

This is the second half of a two-part series studying tensor categories of unitary vertex operator algebras from a unitary point of view.

Cluster structures in Schubert varieties in the Grassmannian

AbstractIn this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is encoded using (target labelings of) Postnikov's plabic graphs. This result generalizes a theorem of Scott [Proc. Lond. Math. Soc. (3) 92 (2006) 345–380] for the Grassmannian, and proves a folklore conjecture for Schubert varieties that has been believed by experts since Scott's result [Proc. Lond. Math. Soc. (3) 92 (2006) 345–380], though the statement was not formally written down until Muller–Speyer explicitly conjectured it [Proc. Lond. Math. Soc. (3) 115 (2017) 1014–1071]. To prove this conjecture we use a result of Leclerc [Adv. Math. 300 (2016) 190–228] who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety can be identified with cluster algebras. Our proof also uses a construction of Karpman [J. Combin. Theory Ser. A 142 (2016) 113–146] to build plabic graphs associated to reduced expressions. We additionally generalize our result to the setting of skew‐Schubert varieties; the latter result uses generalized plabic graphs, that is, plabic graphs whose boundary vertices need not be labeled in cyclic order.

On prelinear Hilbert algebras with successor

In this paper we give an explicit description of the left adjoint of the forgetful functor from the algebraic category of Gödel algebras with successor to the algebraic category of bounded prelinear Hilbert algebras with successor. We apply this result in order to study possible descriptions of the coproduct of two finite algebras in the algebraic category of prelinear Hilbert algebras with successor.

A topological origin of quantum symmetric pairs

It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are {mathbb {Z}}_2-orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Examples of these are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra.

Pseudo effect algebras are algebras over bounded posets

We prove that there is a monadic adjunction between the category of bounded posets and the category of pseudo effect algebras.

An extension of de Vries duality to normal spaces and locally compact Hausdorff spaces

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras. In a recent article, we have extended de Vries duality to completely regular spaces by generalizing de Vries algebras to de Vries extensions. To illustrate the utility of this point of view, we show how to use this new duality to obtain algebraic counterparts of normal and locally compact Hausdorff spaces in the form of de Vries extensions that are subject to additional axioms which encode the desired topological property. This, in particular, yields a different perspective on de Vries duality. As a further application, we show that a duality for locally compact Hausdorff spaces due to Dimov can be obtained from our approach.

Kernels and cokernels in the category of augmented involutive stereotype algebras

We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: (1) this category has kernels and cokernels, (2) the cokernel is preserved under the passage to the group stereotype algebras, and (3) the notion of cokernel allows to prove that the continuous envelope [Formula: see text] of the group algebra of a compact buildup of an abelian locally compact group is an involutive Hopf algebra in the category of stereotype spaces [Formula: see text]. The last result plays an important role in the generalization of the Pontryagin duality for arbitrary Moore groups.