Vertex representation of quantum N-toroidal algebra for type F4

2020 ◽  
Vol 48 (9) ◽  
pp. 3780-3799
Author(s):  
Cenlei Ying ◽  
Limeng Xia ◽  
Honglian Zhang
Keyword(s):  
Toroidal Algebra ◽  
Vertex Representation

scholarly journals Level -1/2 Realization of Quantum N-Toroidal Algebra in Type Cn

2022 ◽  
Vol 29 (01) ◽  
pp. 79-98
Author(s):  
Naihuan Jing ◽  
Qianbao Wang ◽  
Honglian Zhang
Keyword(s):  
Natural Generalization ◽  
Type Formula ◽  
Toroidal Algebra ◽  
Usual Quantum ◽  
Vertex Representation ◽  
Level 1

We construct a level -1/2 vertex representation of the quantum [Formula: see text]-toroidal algebra of type [Formula: see text], which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type [Formula: see text] as a by-product.

scholarly journals Vertex representations of quantum N-toroidal algebras for type C

2020 ◽  
pp. 2150185
Author(s):  
Naihuan Jing ◽  
Zhucheng Xu ◽  
Honglian Zhang
Keyword(s):  
Quantum Affine Algebras ◽  
Type Formula ◽  
Toroidal Algebra ◽  
Affine Algebras ◽  
Type C ◽  
Vertex Representation ◽  
Special Case

Quantum [Formula: see text]-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper, we construct a level-one vertex representation of the quantum [Formula: see text]-toroidal algebra for type [Formula: see text]. In particular, we also obtain a level-one module of the quantum toroidal algebra for type [Formula: see text] as a special case.

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

scholarly journals Intertwining operator and integrable hierarchies from topological strings

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jean-Emile Bourgine
Keyword(s):  
Topological Strings ◽  
Vertex Operator ◽  
Integrable Hierarchies ◽  
Kp Hierarchy ◽  
Tau Function ◽  
Time Deformation ◽  
Toroidal Algebra ◽  
Crystal Model ◽  
Melting Crystal ◽  
Definition Of

Abstract In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

scholarly journals Toward AGT for Parabolic Sheaves

2020 ◽  
Author(s):  
Andrei Neguţ
Keyword(s):  
Field Theory ◽  
Gauge Theory ◽  
Conformal Field Theory ◽  
Moduli Spaces ◽  
Type A ◽  
Conformal Field ◽  
Toroidal Algebra ◽  
Agt Correspondence ◽  
K Theory ◽  
The Ideal

Abstract We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted quantum toroidal algebra of type $A$ and show that these elements act by 0 on the $K$-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$ for arbitrary nilpotent, which would imply a $q$-deformed version of the Alday-Gaiotto-Tachikawa (AGT) correspondence between gauge theory with surface operators and conformal field theory.

scholarly journals (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

2018 ◽  
Vol 2018 (3) ◽  
Author(s):  
Hidetoshi Awata ◽  
Hiroaki Kanno ◽  
Andrei Mironov ◽  
Alexei Morozov ◽  
Kazuma Suetake ◽  
...  
Keyword(s):  
Partition Functions ◽  
Toroidal Algebra ◽  
Nekrasov Partition Functions

The Fermion Representation of Quantum Toroidal Algebra on 3D Young Diagrams

2014 ◽  
Vol 31 (7) ◽  
pp. 070502 ◽  
Author(s):  
Li-Qiang Cai ◽  
Li-Fang Wang ◽  
Ke Wu ◽  
Jie Yang
Keyword(s):  
Young Diagrams ◽  
Toroidal Algebra

scholarly journals Using 2-Opt based evolution strategy for travelling salesman problem

2016 ◽  
Vol 6 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Kenan Karagul ◽  
Erdal Aydemir ◽  
Sezai Tokat
Keyword(s):  
Search Algorithm ◽  
Travelling Salesman Problem ◽  
Harmony Search ◽  
Heuristic Method ◽  
Harmony Search Algorithm ◽  
Evolution Strategy ◽  
Local Search Algorithm ◽  
Travelling Salesman ◽  
Selection Operators ◽  
Vertex Representation

Harmony search algorithm that matches the (µ+1) evolution strategy, is a heuristic method simulated by the process of music improvisation. In this paper, a harmony search algorithm is directly used for the travelling salesman problem. Instead of conventional selection operators such as roulette wheel, the transformation of real number values of harmony search algorithm to order index of vertex representation and improvement of solutions are obtained by using the 2-Opt local search algorithm. Then, the obtained algorithm is tested on two different parameter groups of TSPLIB. The proposed method is compared with classical 2-Opt which randomly started at each step and best known solutions of test instances from TSPLIB. It is seen that the proposed algorithm offers valuable solutions.

scholarly journals A Computationally-Realizable Rigorous Canonical Numbering Algorithm for Chemical Graphs with its Open-Source Implementation in Rust

2021 ◽  
Author(s):  
Qin Wan
Keyword(s):  
Open Source ◽  
High Performance ◽  
The Novel ◽  
Graph Reduction ◽  
Rigorous Approach ◽  
Open Issue ◽  
Vertex Representation ◽  
Representation Method ◽  
Symmetry Perception ◽  
Short Time

Canonical numbering of the vertices from a graph has been a challenging open issue for decades not only in the domain of graph theory but also in the cheminformatic applications. This paper presents an efficient, fast and rigorous approach for canonical numbering and symmetry perception as the first workable solution with theoretical completeness. The methodology is composed of a set of algorithms including extendable representation of vertex, high-performance sorting and graph reduction, etc. The canonical numbering of vertices can be generated in a short time through the novel vertex representation method. Furthermore, a new concept of graph reduction decreases the amount of computation to determine constitutional symmetry of complex graphs into the range of hardware capability. An open-source version of algorithms overall is implemented in Rust thanks to the features of safety, performance and robust abstraction of this modern programming language. The results of experiments on more than 2 million molecules from ChEMBL database has been given at the end.

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