Article
Mathematics
Ivan Losev
Summary: This paper examines derived equivalences for symplectic reflection algebras, establishing a version of the derived localization theorem between categories of modules over these algebras and categories of coherent sheaves over quantizations of Q-factorial terminalizations of the symplectic quotient singularities. The construction of a Procesi sheaf on the terminalization and demonstration that the quantizations of the terminalization are simple sheaves of algebras are key components of the study. Additionally, some applications to the generalized Bernstein inequality and perversity of wall crossing functors are outlined.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics
Nick Galatos, Adam Prenosil
Summary: We introduce (t)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially ordered (lattice-ordered) set. Bimonoids form an appropriate framework for the study of a general notion of complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses in monoids. We prove that each commutative (8-)bimonoid embeds into a complete complemented commutative .@-bimonoid in a doubly dense way reminiscent of the Dedekind-MacNeille completion. This construction of the algebra of fractions in fact yields a categorical equivalence between varieties of integral and of involutive residuated structures which subsumes as special cases the known equivalences between Abelian l-groups and their negative cones, and between Sugihara monoids and their negative cones.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics, Applied
Jean B. Nganou
Summary: This paper proves that the profinite completion construction is a covariant functor from the category of (universal) algebras of a given type into the category of the corresponding Stone algebras. A Grothendieck problem for finitely presented MV-algebras is also formulated and solved. Finally, we characterize finitely presented MV-algebras for which profinite completions and MacNeille completions coincide.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics
S. Ludkowski
Summary: In this study, infinite dimensional Cayley-Dickson algebras and their metrizability are closely examined. The completions of generalized Cayley-Dickson algebras with respect to metrics are analyzed, and the continuity of their homomorphisms is investigated.
LINEAR & MULTILINEAR ALGEBRA
(2021)
Article
Mathematics
T. Ait Aissa, M. W. Mansouri
Summary: In this work, we studied the case where the left symmetric product associated with a symplectic Lie algebra is Novikov, and proved that any SNLA is completely reducible and two-step solvable. The classification of four-dimensional and nilpotent six-dimensional Lie algebras with an SNLA structure was also presented, along with methods for constructing various examples. Finally, a geometric study of the affine connection linked to an SNLA was conducted.
COMMUNICATIONS IN ALGEBRA
(2022)
Article
Mathematics
Johannes Christensen, Sergey Neshveyev
Summary: Motivated by the problem of characterizing KMS states on the reduced C*-algebras of etale groupoids, the authors investigate the influence of the reduced norm on the group algebras of isotropy groups, and show that the norm is still the reduced one for some classes of graded groupoids.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics, Applied
Aleksandar Milivojevic
Summary: This paper answers questions regarding the realizability of symplectic algebras by symplectic manifolds in dimensions divisible by four and all even dimensions. It also addresses the possibility of algebraic conditions on the rational homotopy minimal model of a closed smooth manifold implying the existence of a symplectic structure on the manifold in even dimensions six and higher.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
J. Harding, F. M. Lauridsen
Summary: This study explores the relationships between centrally supplemented Heyting algebras, their centrally supplemented extensions, and hyper-MacNeille completions. It introduces an algebraic approach to describe hyper-MacNeille completions and discusses varieties of Heyting algebras that are closed under hyper-MacNeille completions.
Article
Mathematics
Mohamed Boucetta, Hamza El Ouali, Hicham Lebzioui
Summary: In this paper, the authors study flat symplectic Lie groups and prove that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to the symplectic form. They also show that a flat symplectic Lie group must be nilpotent with a degenerate center, and that the connection of a flat symplectic Lie group is always complete. The authors further prove that the flat symplectic double extension process can be used to characterize all flat symplectic Lie algebras.
COMMUNICATIONS IN ALGEBRA
(2023)
Article
Mathematics, Applied
Piergiulio Tempesta, Giorgio Tondo
Summary: A tensorial approach based on the geometry of Haantjes tensors is proposed for the theory of classical Hamiltonian integrable systems. The concept of symplectic-Haantjes manifolds is introduced as a natural setting for formulating integrability. It is shown that the existence of suitable Haantjes algebras with vanishing torsion is a necessary and sufficient condition for integrability in the Liouville-Arnold sense, leading to the discovery of new integrable models from Haantjes geometry. An application of this approach to the Post-Winternitz system and a stationary flow of the KdV hierarchy is also presented.
ANNALI DI MATEMATICA PURA ED APPLICATA
(2022)
Article
Mathematics
Valdis Laan, Jianjun Feng, Xia Zhang
Summary: In this article, we investigate ordered universal algebras and present a method of constructing a join-completion for them using D-ideals. We demonstrate that this construction possesses a universal property, which leads to a reflector between a certain category of ordered algebras and the category of sup-algebras. Our findings extend previous knowledge on various ordered structures.
ALGEBRA UNIVERSALIS
(2023)
Article
Computer Science, Theory & Methods
Zhongxi Zhang
Summary: In this paper, the authors consider a full subcategory S of the category CLQ of stratified Q-closure spaces and construct a reflection from the category CLQ onto S, called the S-completion. They introduce the concept of Z-convergence Q-closure space by considering fuzzy subset systems Z on the category CLQ, and show that the category S is a category of Z-convergence spaces for some fuzzy subset system Z. They also establish connections between fuzzy subset systems on CLQ and those on the category ORDQ of Q-ordered sets, providing a general approach to generating various completions of Q-ordered sets from the S-completion.
FUZZY SETS AND SYSTEMS
(2023)
Article
Mathematics
Szilvia Homolya, Jeno Szigeti, Leon van Wyk, Michal Ziembowski
Summary: The paper focuses on generating Lie K-algebras of matrix algebras under certain conditions, studying the relation between Lie centralizers and Lie-nilpotent associative subalgebras, and proposing conjectures towards proving higher nilpotency indices of Lie-nilpotent associative subalgebras.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics
Ken Brown, Milen Yakimov
Summary: This article discusses two approaches to studying irreducible representations of orders (via traces and Poisson orders) and introduces a natural compatibility relation between the two approaches, leading to the notion of Poisson trace orders.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
H. Abchir, Ilham Ait Brik, Mohamed Boucetta
Summary: This paper explores k-Para-Kahler Lie algebras, a generalization of para-Kahler Lie algebras, introducing new structures that extend the notion of left symmetric algebra and determining specific dimensions of Lie algebras.
COMMUNICATIONS IN ALGEBRA
(2021)
