Article
Mathematics
Jun Tong, Bing Duan, Yan-Feng Luo
Summary: In this paper, Hernandez and Leclerc proved that the Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional modules of an untwisted quantum affine algebra has a cluster algebra structure. They also described a cluster algebra algorithm to compute the q-characters of minimal affinizations of type G(2) and proved the conjectured geometric q-character formula.
COMMUNICATIONS IN ALGEBRA
(2023)
Article
Mathematics, Applied
Ryo Takenaka
Summary: This paper presents the construction and properties of the standard module and its related subspaces for the affine Lie algebra of type A(2) 2l. Character formulas are obtained and a conjecture regarding vacuum modules is settled.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2023)
Article
Mathematics
Rekha Biswal, Travis Scrimishaw
Summary: A combinatorial crystal structure is constructed on the Kirillov-Reshetikhin crystal B-7, B-s in type E-7((1)), with 7 being the unique node in the orbit of 0 in the affine Dynkin diagram. The combinatorial R-matrix R: B-7, B-s circle times B-7, B-s '-> B-7, B-s 'circle times B-7, B-s is then described.
COMMUNICATIONS IN ALGEBRA
(2022)
Article
Mathematics
Gurbir Dhillon, Sam Raskin
Summary: In this article, we prove a localization theorem for affine W-algebras by drawing on the ideas of Beilinson-Bernstein and Kashiwara-Tanisaki. Specifically, for any non-critical regular weight lambda, we establish a correspondence between lambda-monodromic Whittaker D-modules on the enhanced affine flag variety and a full subcategory of Category O for the W-algebra. To determine the essential image of our functor, we introduce a new realization of Category O for affine W-algebras using Iwahori-Whittaker modules for the corresponding Kac-Moody algebra. Additionally, we provide a new proof of Arakawa's character formulae for simple positive energy representations of the W-algebra.
ADVANCES IN MATHEMATICS
(2023)
Article
Mathematics
S. Eswara Rao, Sachin S. Sharma, Sudipta Mukherjee
Summary: This article classifies the irreducible integrable modules for the loop affine-Virasoro algebra, which is important for understanding the related algebraic structures.
COMMUNICATIONS IN ALGEBRA
(2021)
Article
Mathematics, Applied
Xiangui Zhao
Summary: This paper studies the growth and Gelfand-Kirillov dimension (GK-dimension) of generalized Weyl algebra (GWA) A = D(sigma, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are provided. Specifically, the paper establishes a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, demonstrating that GKdim(A) is either 3 or infinity in this case. The results generalize existing findings in the literature and can be used to determine the growth, GK-dimension, simplicity, and cancellation properties of certain GWAs.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Mathematics
Gabriel Frieden
Summary: In this paper, the author further investigates the geometric crystal structure and R-matrix, finding that R is an isomorphism of geometric crystals satisfying the Yang-Baxter relation, and also traces back to the birational action of the symmetric group found in literature.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Masato Okado, Ryo Takenaka
Summary: In this study, we use the method proposed by Butorac and Sadowski to construct bases of principal subspaces for twisted affine Lie algebras except A(2l)((2)). These bases are used to obtain the character formulas of highest weight modules and parafermionic spaces for the same affine Lie algebras, confirming the conjecture made by Hatayama et al. (2001).
ALGEBRAS AND REPRESENTATION THEORY
(2022)
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
Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park
Summary: The paragraph discusses concepts and constructions related to the quantum affine algebra of untwisted affine ADE type, the Hernandez-Leclerc category of finite-dimensional U-q'(g)-modules, and monoidal categorification of cluster algebra.
PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES
(2021)
Article
Mathematics, Applied
S. Eswara Rao
Summary: This paper studies the representations of loop Affine-Virasoro algebras, defining Verma modules and their irreducible quotients due to their canonical triangular decomposition. Necessary and sufficient conditions for finite dimensional weight spaces of irreducible highest weight modules are provided, along with a proof that irreducible integrable modules are either highest weight or lowest weight when the canonical central element acts non-trivially. Affine central operators are constructed for each integer, and they commute with the action of the Affine Lie algebra.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics
Naihuan Jing, Fei Kong, Haisheng Li, Shaobin Tan
Summary: This paper investigates (G, chi(phi))-equivariant phi-coordinated quasi modules for nonlocal vertex algebras, establishing conceptual results and constructing coordinated quasi modules for lattice vertex algebras using Lepowsky's work.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics
Thomas Creutzig, Naoki Genra, Shigenori Nakatsuka
Summary: This paper proves Feigin-Frenkel type dualities between subregular W-algebras of type A and principal W-superalgebras of type B, as well as between the principal W-superalgebras of types sl(1 | n) and osp(2 | 2n). The results include isomorphisms of Heisenberg cosets at dual levels, and a novel Kazama-Suzuki type coset construction. Additionally, it is shown that the simple principal W-superalgebra and its Heisenberg coset are rational and/or C-2-cofinite under certain conditions.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Haijun Tan
Summary: In this paper, the authors classified a class of simple modules over untwised affine Kac-Moody Lie algebras, where each weight vector of the positive parts of affine Kac-Moody Lie algebras acts locally finitely. They also classified all simple modules with a similar property for all affine-Virasoro algebras. They identified precisely three classes of simple modules on which each weight vector of the positive part of any affine-Virasoro algebra acts locally finitely: simple highest weight or Whittaker Virasoro algebra modules, simple highest weight or Whittaker affine Lie algebra modules, and simple highest weight or Whittaker affine-Virasoro algebra modules which are neither simple Virasoro algebra modules nor simple affine Lie algebra modules.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics, Applied
Katsuyuki Naoi, Travis Scrimshaw
Summary: In this study, it is proved that Kirillov-Reshetikhin modules associated with the node adjacent to the adjoint one have a crystal pseudobase in certain types, by applying a criterion introduced by Kang et al. Some statements concerning values of a bilinear form need to be proved in order to apply the criterion, which is achieved by using the global bases of extremal weight modules.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2021)
