Article
Mathematics
Katsuyuki Naoi
Summary: The generalized quantum affine Schur-Weyl duality functor establishes an equivalence between two different finite-dimensional module categories, which is of significant importance in the field of algebra.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Andrea Appel, Tomasz Przezdziecki
Summary: This paper investigates the relationship between quantum symmetric pair subalgebra and orientifold KLR algebra, and constructs a functor eF that maps their corresponding finite-dimensional modules to each other. It provides a boundary model of the generalized Schur-Weyl duality.
ADVANCES IN MATHEMATICS
(2023)
Article
Mathematics
Yun Gao, Naihuan Jing, Limeng Xia, Honglian Zhang
Summary: We introduce the concept of quantum N-toroidal algebras as a generalization of quantum toroidal algebras and extended quantized GIM algebras of N-fold affinization. We demonstrate that the quantum N-toroidal algebras are quotients of the extended quantized GIM algebras of N-fold affinization, which extends a well-known result of Berman and Moody for Lie algebras.
COMMUNICATIONS IN MATHEMATICS AND STATISTICS
(2023)
Article
Mathematics
Nicolas Libedinsky, Leonardo Patimo, David Plaza
Summary: For any affine Weyl group, the pre-canonical bases are introduced as a set of bases that interpolate between the standard basis and the canonical basis in the spherical Hecke algebra. The expansion of these bases is generally simple and it is conjectured to be positive in type A.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Jordan Disch
Summary: We construct infinite-dimensional analogues of finite-dimensional simple modules for the nonstandard q-deformed enveloping algebra U-q' (so(n)) and the classical universal enveloping algebra U(so(n)). Rational matrix coefficients are provided for these infinite-dimensional modules and used to embed the respective algebras into skew group algebras of shift operators. The Casimir elements of U-q'(so(n)) are given, and a commutative subalgebra generated by these elements and a corresponding subalgebra of U(so(n)) are considered. The embeddings of these subalgebras into skew group algebras yield invariant algebras under certain group actions, and it is shown that they are Harish-Chandra subalgebras of U-q'(so(n)) and U(so(n)).
JOURNAL OF ALGEBRA
(2023)
Article
Physics, Multidisciplinary
Vicente Said Morales-Salgado
Summary: This study investigates deformations of polynomial Heisenberg algebras and establishes a connection with extended affine Weyl groups of type A(m)((1)), contributing to a better understanding of quantum systems and their algebraic structures.
Article
Physics, Mathematical
Thomas Creutzig, Duiliu-Emanuel Diaconescu, Mingyang Ma
Summary: A family of vertex algebras with universal Verma modules equal to the cohomology of affine Laumon spaces is discovered, based on an explicit expression for the generating function of Poincare polynomials of these spaces. A variant of quantum Hamiltonian reduction, called iterated W-algebras, realizes vertex algebras, and the main conjecture is that the vertex algebras associated with affine Laumon spaces are subalgebras of iterated W-algebras.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Physics, Mathematical
Hans Plesner Jakobsen
Summary: In the framework of quantized holomorphic functions defined on non-commutative spaces, understanding the representations induced by these functions and determining the corresponding algebras of differential operators is crucial. This study investigates three different pairings and shows the fundamental role of the quantum Weyl algebra in expressing these representations.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Physics, Mathematical
Ryo Fujita, Se-jin Oh
Summary: This paper introduces the concept of Q-datum and develops a unified theory describing twisted Auslander-Reiten quivers and twisted adapted classes, along with applications in finite-dimensional representation theory. It also provides an alternative description for block decomposition results and presents a unified formula for the denominators of normalized R-matrices between Kirillov-Reshetikhin modules, as well as computes invariants Lambda (V, W) and Lambda (infinity) (V, W) for each pair of simple modules V and W introduced previously.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics
Ying Ma, Toshiaki Shoji, Zhiping Zhou
Summary: This paper considers the construction of canonical basis and canonical signed basis for Cartan data of symmetric type, and the natural bijection between them, and provides a construction method in the case where the order of sigma is odd.
JOURNAL OF ALGEBRA
(2023)
Article
Physics, Multidisciplinary
Slaven Kozic
Summary: This study extends the Etingof-Kazhdan construction to h-adic quantum vertex algebras associated with trigonometric R-matrices in types B, C, and D, and demonstrates that restricted modules for quantum affine algebras in these types possess the structure of phi-coordinated modules for the aforementioned h-adic quantum vertex algebras.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Article
Mathematics
Sanu Bera, Snehashis Mukherjee
Summary: This article explicitly computes the polynomial identity (PI) degree of the multiparameter quantized Weyl algebras at roots of unity.
PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES
(2023)
Article
Physics, Mathematical
Justine Fasquel
Summary: This article proves the rationality of the exceptional W-algebras W(k) (g, f) associated with the simple Lie algebra g = sp(4) and a subregular nilpotent element f = f(subreg), providing a new particular case of the Kac-Wakimoto conjecture. Moreover, the simple W-k (g, f)-modules are described and their characters are computed. The nontrivial action of the component group on the set of simple W-k (g, f)-modules is also analyzed.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Ryo Fujita
Summary: The paper presents a simple unified formula for expressing the denominators of normalized R-matrices between fundamental modules over quantum loop algebras of type ADE, which can be interpreted in terms of representations of Dynkin quivers and proven using geometry of graded quiver varieties. Additionally, a geometric interpretation of Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor is obtained when it arises from a family of fundamental modules. The study also explores cases where graded quiver varieties are isomorphic to unions of graded nilpotent orbits of type A.
SELECTA MATHEMATICA-NEW SERIES
(2022)
Article
Mathematics, Applied
Guiyu Yang
Summary: This paper proves that the triangular matrix algebra Lambda is an affine quasi-hereditary algebra if and only if H is an affine quasi-hereditary algebra, and describes some important properties of Lambda using those of H.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics
Jason Gaddis, Kenneth L. Price
COMMUNICATIONS IN ALGEBRA
(2017)
Article
Mathematics
Jason Gaddis, Ellen Kirkman, W. Frank Moore
JOURNAL OF ALGEBRA
(2017)
Article
Mathematics
Jason Gaddis, Robert Won, Daniel Yee
ALGEBRAS AND REPRESENTATION THEORY
(2019)
Article
Mathematics, Applied
Jason Gaddis, Ellen Kirkman, W. Frank Moore, Robert Won
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2019)
Article
Mathematics
Jason Gaddis, Daniel Yee
COMMUNICATIONS IN ALGEBRA
(2019)
Article
Mathematics
Jason Gaddis, Robert Won
JOURNAL OF ALGEBRA
(2019)
Article
Mathematics
Jason Gaddis, Xingting Wang
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
(2020)
Article
Mathematics
Jason Gaddis, Ho Phuong
COMMUNICATIONS IN ALGEBRA
(2020)
Article
Mathematics
Zachary Cline, Jason Gaddis
JOURNAL OF ALGEBRA
(2020)
Article
Mathematics, Applied
Jason Gaddis, Daniel Rogalski
Summary: The study discusses graded twisted Calabi-Yau algebras of dimension 3 and their possible types under the additional assumption of polynomial growth, with a near complete answer provided for the case where Q has at most 3 vertices.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2021)
Article
Mathematics
Jacob Barahona Kamsvaag, Jason Gaddis
Summary: This paper studies the Auslander map in the setting of nonconnected graded Calabi-Yau algebras, and presents the main result.
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES
(2022)
Article
Mathematics
Jason Gaddis, Robert Won
Summary: In this study, we investigate the actions of pointed Hopf algebras in the Z-graded setting. Our main result provides a classification of inner-faithful actions of generalized Taft algebras on quantum generalized Weyl algebras that respect the Z-grading. We also demonstrate that the invariant rings of Taft actions on quantum generalized Weyl algebras are typically commutative rings.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Jason Gaddis, Daniele Rosso, Robert Won
Summary: We study a class of Z-graded algebras that generalize the construction of rank one generalized Weyl algebras (GWAs) introduced by Bell and Rogalski. We establish certain ring-theoretic properties of these algebras and investigate their connection to GWAs. We classify the simple weight modules in the case of infinite orbits and partially classify them in the case of orbits of finite order.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics
Luigi Ferraro, Jason Gaddis, Robert Won
JOURNAL OF ALGEBRA
(2020)
Article
Mathematics, Applied
Manoj K. Keshari, Sampat Sharma
Summary: Assuming R is an affine algebra of dimension d > 4 over a perfect field k of char = 2 and I is an ideal of R. (1) M Sd+1(R) is uniquely divisible prime to char k if R is reduced and k is infinite with c.d.(k) < 1. (2) Umd+1(R, I)/Ed+1(R, I) has a nice group structure if c.d.2(k) < 2. (3) Umd(R, I)/Ed(R, I) has a nice group structure if k is algebraically closed of char k = 2, 3 and either (i) k = Fp or (ii) R is normal.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Mathieu Anel, Georg Biedermann, Eric Finster, Andre Joyal
Summary: In this article, the work of Toen-Vezzosi and Lurie on Grothendieck topologies is revisited using the new tools of acyclic classes and congruences. An extended Grothendieck topology on any 8-topos is introduced and it is proven that the poset of extended Grothendieck topologies is isomorphic to that of topological localizations, hypercomplete localizations, Lawvere-Tierney topologies, and covering topologies. The notions of cotopological morphism, hypercompletion, hyperdescent, hypercoverings, hypersheaves, and forcing are also discussed.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Daniel Dugger, Christy Hazel, Clover May
Summary: This article provides a complete description of the derived category of perfect complexes of modules over the constant Mackey ring Z/$ for the cyclic group C2. While it is simple for $ odd, it relies on a new splitting theorem when $ = 2. The splitting theorem also allows for computing the associated Picard group and Balmer spectrum for compact objects in the derived category. Additionally, it gives a complete classification of finite modules over the C2-equivariant Eilenberg-MacLane spectrum HZ/2 and provides new proofs for some facts about RO(C2)-graded Bredon cohomology.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Mikhailo Dokuchaev, Itailma Rocha
Summary: In this study, we construct an abelian group C(Theta/R) formed by the isomorphism classes of partial generalized crossed products related to a unital partial representation Theta of a group G into the Picard semigroup PicS(R) of a non-necessarily commutative unital ring R. We identify an appropriate second partial cohomology group of G with a naturally defined subgroup C0(Theta/R) of C(Theta/R). Using these results, we generalize the works by Kanzaki and Miyashita by giving an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of rings and a unital partial representation of an arbitrary group into the monoid of R-subbimodules.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Markus Thuresson
Summary: Hereditary algebras are quasi-hereditary and exhibit certain regularity properties with respect to adapted partial orders. This article investigates the Ext-algebra of standard modules over path algebras of linear quivers and provides necessary and sufficient conditions for regular exact Borel subalgebras. The findings have implications for the understanding of linear quivers with arbitrary orientations.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Cordian Riener, Robin Schabert
Summary: This article focuses on the geometry of a class of hyperbolic polynomial families determined by linear conditions on the coefficients. These polynomials have all their roots on the real line. The set of hyperbolic polynomials is stratified according to the multiplicities of the real zeros, and this stratification also applies to the hyperbolic slices. The study shows that the local extreme points of hyperbolic slices correspond to hyperbolic polynomials with at most k distinct roots, and that the convex hull of such a family is generally a polyhedron. The article also explores the implications of these results for symmetric real varieties and symmetric semi-algebraic sets, particularly in terms of sparse representations and sampling.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Edward L. Green, Sibylle Schroll
Summary: This paper studies the ideal C in the path algebra KQ, proving that KQ/C is always finite dimensional with finite global dimension, and it is Morita equivalent to an incidence algebra.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Alexei Entin, Noam Pirani
Summary: This paper proves the existence of a Galois extension with ramification only at infinity for symmetric and alternating groups over finite fields of odd characteristic.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Yves Baudelaire Fomatati
Summary: This paper improves the algorithm for matrix factorization of polynomials, obtaining better results by refining the construction of one of the main ingredients of the algorithm.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Ippei Nagamachi, Teppei Takamatsu
Summary: In this paper, we study the invariants and related phenomena of regular varieties and rings over imperfect fields. We give a criterion for geometric normality of such rings, study the Picard schemes of curves, and define new invariants relating to δ-invariants, genus changes, conductors, and Jacobian numbers. As an application, we refine Tate's genus change theorem and show that the Jacobian number of a curve is 2p/(p - 1) times the genus change.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Zongzhu Lin, Li Qiao
Summary: This article studies the Rota-Baxter algebra structure on the field A = k((t)), with P being the projection map. The representation theory and regular-singular decompositions of finite dimensional A-vector spaces are examined. The main result shows that the category of finite dimensional representations is semisimple, consisting of three isomorphism classes of one-dimensional irreducible representations. Additionally, the article uses the result to compute the generalized class number. (c) 2023 Elsevier B.V. All rights reserved.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Stephen Lack, Giacomo Tendas
Summary: In this paper, we characterize accessible V-categories with limits of a specified class by introducing the notion of companion C for a class of weights & psi;. We then characterize these categories as accessibly embedded and C-virtually reflective in a presheaf V-category, as well as the V-categories of C-models of sketches. Our theorem extends to the case of any weakly sound class & psi; and provides a new perspective on weakly locally presentable categories.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Pradeep K. Rai
Summary: In 1956, Green provided a bound on the order of the Schur multiplier of p-groups. This bound, which depends on the order of the group, is the best possible. Over time, the bound has been improved by incorporating additional factors such as the minimal number of generators and the order of the derived subgroup. We further enhance these bounds by considering the group's nilpotency class, with special emphasis on the cases of class 2 and maximal class.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Benjamin Dequene
Summary: Gentle algebras are a class of finite-dimensional algebras introduced by I. Assem and A. Skowronski in the 1980s. Modules over such algebras can be described using string and band combinatorics in the associated gentle quiver, as studied by M.C.R. Butler and C.M. Ringel. Nilpotent endomorphisms of quiver representations induce linear transformations over vector spaces at each vertex. Among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. This paper focuses on subcategories generated by the indecomposable representations of a gentle quiver, including a fixed vertex in their support, and characterizes the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
Article
Mathematics, Applied
Mark Lawson, Aidan Sims, Alina Vdovina
Summary: We construct a family of groups that are higher dimensional generalizations of the Thompson groups using suitable higher rank graphs. Inspired by the K-theory of C*-algebras, we introduce group invariants and demonstrate that many of our groups are non-isomorphic to the Brin-Thompson groups nV, where n ≥ 2.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2024)
