Quantum Torus Lie Algebra in the q-Deformed Kadomtsev–Petviashvili System

2019 ◽  
Vol 26 (04) ◽  
pp. 579-588
Author(s):  
Chuanzhong Li ◽  
Xinyue Li ◽  
Fushan Li
Keyword(s):  
Lie Algebra ◽  
Kp Hierarchy ◽  
Quantum Torus ◽  
Tau Function ◽  
Ghost Symmetry

Based on the W∞ symmetry of the q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy, which is a q-deformation of the KP hierarchy, we construct the quantum torus symmetry of the q-KP hierarchy, which further leads to the quantum torus constraint of its tau function. Moreover, we generalize the system to a multi-component q-KP hierarchy that also has the well-known ghost symmetry.

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 ON CONNECTION BETWEEN TOPOLOGICAL LANDAU-GINZBURG GRAVITY AND INTEGRABLE SYSTEMS

1995 ◽  
Vol 10 (29) ◽  
pp. 4161-4178 ◽  
Author(s):  
A. LOSEV ◽  
I. POLYUBIN
Keyword(s):  
Generating Function ◽  
Complex Plane ◽  
Integrable Systems ◽  
Target Space ◽  
Kp Hierarchy ◽  
Tau Function ◽  
Topological Theory ◽  
Topological Gravity ◽  
Dispersionless Limit

We study flows on the space of topological Landau-Ginzburg theories coupled to topological gravity. We argue that flows corresponding to gravitational descendants change the target space from a complex plane to a punctured complex plane and lead to the motion of punctures. It is shown that the evolution of the topological theory due to these flows is given by the dispersionless limit of KP hierarchy. We argue that the generating function of correlators in such theories is equal to the logarithm of the tau function of the generalized Kontsevich model.

Integrable Representations for the Extended Affine Lie Algebra Coordinated by Quantum Tori in Two Variables

2014 ◽  
Vol 21 (03) ◽  
pp. 535-540 ◽  
Author(s):  
Fei Kong ◽  
Zhiqiang Li ◽  
Shaobin Tan ◽  
Qing Wang
Keyword(s):  
Lie Algebra ◽  
Type A ◽  
Quantum Torus ◽  
The Core ◽  
Quantum Tori ◽  
Finite Dimensional ◽  
Affine Lie Algebra ◽  
Extended Affine Lie Algebra ◽  
Integrable Modules

In this paper we classify the irreducible integrable modules for the core of the extended affine Lie algebra of type Ad-1 coordinated by ℂq with finite-dimensional weight spaces and the center acting trivially, where ℂq is the quantum torus in two variables.

scholarly journals SDIFF(2) KP HIERARCHY

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 889-922 ◽  
Author(s):  
KANEHISA TAKASAKI ◽  
TAKASHI TAKEBE
Keyword(s):  
Central Extension ◽  
Hilbert Problem ◽  
Poisson Algebra ◽  
Theoretical Description ◽  
Minimal Models ◽  
Kp Hierarchy ◽  
Tau Function ◽  
General Solutions ◽  
Area Preserving ◽  
Riemann Hilbert Problem

An analogue of the KP hierarchy, the SDiff(2) KP hierarchy, related to the group of area-preserving diffeomorphisms on a cylinder is proposed. An improved Lax formalism of the KP hierarchy is shown to give a prototype of this new hierarchy. Two important potentials, S and τ, are introduced. The latter is a counterpart of the tau function of the ordinary KP hierarchy. A Riemann-Hilbert problem relative to the group of area-diffeomorphisms gives a twistor theoretical description (nonlinear gravition construction) of general solutions. A special family of solutions related to topological minimal models are identified in the framework of the Riemann-Hilbert problem. Further, infinitesimal symmetries of the hierarchy are constructed. At the level of the tau function, these symmetries obey anomalous commutation relations, hence leads to a central extension of the algebra of infinitesimal area-preserving diffeomorphisms (or of the associated Poisson algebra).

scholarly journals Ghost symmetry of the discrete KP hierarchy

2015 ◽  
Vol 180 (4) ◽  
pp. 815-832 ◽  
Author(s):  
Chuanzhong Li ◽  
Jipeng Cheng ◽  
Kelei Tian ◽  
Maohua Li ◽  
Jingsong He
Keyword(s):  
Kp Hierarchy ◽  
Discrete Kp Hierarchy ◽  
Discrete Kp ◽  
Ghost Symmetry

scholarly journals IRREDUCIBLE HARISH CHANDRA MODULES OVER THE DERIVATION ALGEBRAS OF RATIONAL QUANTUM TORI

2013 ◽  
Vol 55 (3) ◽  
pp. 677-693 ◽  
Author(s):  
GENQIANG LIU ◽  
KAIMING ZHAO
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Quantum Torus ◽  
Quantum Tori

AbstractLet d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra $\mathfrak{g}$=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight $\mathfrak{g}$-modules $\mathcal{V}$α (V,W) with three parameters: a vector α∈ℂd, an irreducible $\mathfrak{gl}$d-module V and a graded-irreducible $\mathfrak{gl}$N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded) $\mathfrak{g}$-module M is isomorphic to $\mathcal{V}$α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).

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