Quantum Torus Research Articles

Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

Derivations of the Positive Part of the Two-parameter Quantum Group of Type G2

In this paper, we compute the derivations of the positive part of the two-parameter quantum group of type G2 by embedding it into a quantum torus. We also show that the first Hochschild cohomology group of this algebra is a two-dimensional vector space over the complex field.

Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori

Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.

Quantum (non-commutative) toric geometry: Foundations

In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we show that their category is equivalent to a certain category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantum Toric Varieties.Unlike classical toric varieties, quantum toric varieties admit moduli and we define their moduli spaces, prove that these spaces are orbifolds and, in favorable cases, up to homotopy, they admit a complex structure.

Additional symmetries of the dispersionless extended noncommutative Gelfand–Dickey hierarchy

This paper aims at additional symmetries of the unextended and extended, commutative and noncommutative dispersionless Gelfand–Dickey (dGD) hierarchies. Being similar to the Lax formalism of the Gelfand–Dickey (GD) hierarchy, we construct the function [Formula: see text] and Orlov–Schulman function [Formula: see text] of the hierarchies. Meanwhile, the additional symmetry will be studied with the infinite flows of [Formula: see text] and [Formula: see text] function of the dGD hierarchy and one can find that only a part of additional flows can survive under the GD constraints with the corresponding string equation. Furthermore, we pay attention to the additional symmetries of the dispersionless extended Gelfand–Dickey (dEGD) hierarchy which has a quantum torus algebraic structure and show the flows in detail. The additional symmetry of dispersionless noncommutative Gelfand–Dickey (dNCGD) hierarchy and dispersionless extended noncommutative Gelfand–Dickey (dENCGD) hierarchy are studied.

Stated Skein Modules of Marked 3-Manifolds/Surfaces, a Survey

We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces) and describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representation theory of the skein algebra. In particular, we show that the skein algebra of non-closed marked surface at any root of 1 is a maximal order. We give a full description of the Azumaya locus of the skein algebra of the puncture torus and give partial results for closed surfaces.

Cuspidal modules for the derivation Lie algebra over a rational quantum torus

Let ℂ Q denote a rational quantum torus with d variables, and be the centre of ℂ Q . In this paper we give a explicit description of the structure of the cuspidal modules for the derivation Lie algebra over ℂ Q , with an extra associative -action.

Symmetries of theq-NKdV hierarchy

In this paper, the additional symmetries and the string equation of the [Formula: see text]-NKdV hierarchy are established basing on the Orlov–Shulman’s [Formula: see text] operator, and then some properties are also given. Next, we construct the quantum torus symmetry of the [Formula: see text]-NKdV hierarchy and further derive the quantum torus constraints on the tau function of the [Formula: see text]-NKdV hierarchy. Comparing to the [Formula: see text] infinite-dimensional Lie symmetry, this quantum torus symmetry has a nice algebraic structure with double indices. Finally, we study the ghost symmetry of the multicomponent [Formula: see text]-NKdV hierarchy.

The Fermionic-bosonic Representations of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie Superalgebras

Lie groups, Lie algebras and their representation theories are important parts of mathematical physics. They play a crucial character in symmetries. As a generalization of Lie algebra, Lie superalgebras are from the comprehending and description for supersymmetry of mathematical physics. Unlike the semisimple Lie algebras, understanding the representation theory of Lie superalgebras is a difficult problem. Lie superalgebras graded by root supersystems are Lie superalgebras of great significance. In recent years, the representations of types <I>B</I>(<i>m,n</i>)<i>, C</i>(<i>n</i>)<i>, D</i>(<i>m,n</i>)<i>, P</i>(<i>n</i>) and <I>Q</I>(<i>n</i>)-graded Lie superalgebras coordinatized by quantum tori have been studied. In this paper, we construct fermionic-bosonic representations for a class of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions. At first, we introduce the background of the research on the graded Lie superalgebras and present some basics on it. Then, a set of bases for <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras and the multiplication operations among them are given specifically to present the construction of the vector space. By using the tensor product of fermionic and bosonic module, the operators and their operation relations are derived. Finally, we obtain a brief and pretty representation theorem of <I>A</I>(<I>M</I>-1,<I>N</I>-1)-graded Lie superalgebras with nontrivial central extensions.

Zero Modes of Velocity Field and Topological Invariant in Quantum Torus

Zero Modes of Velocity Field and Topological Invariant in Quantum Torus

Irreducible modules over the divergence zero algebras and their [formula omitted]-analogues

In this paper, we study a class of Zd-graded modules, which are constructed using Shen–Larsson’s functor from sld-modules V, for the Lie algebras of divergence zero vector fields on tori and quantum tori. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.

A Quantized Riemann–Hilbert Problem in Donaldson–Thomas Theory

Abstract We introduce Riemann–Hilbert problems determined by the refined Donaldson–Thomas theory. They involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a quantum torus algebra. We study the simplest case in detail and use the Barnes double gamma function to construct a solution.

Vertex algebras and extended affine Lie algebras coordinated by rational quantum tori

Let slNˆ(Cq) be the core of extended affine Lie algebra of type AN−1 coordinated by the rational quantum 2-torus Cq. In this paper, we first prove that for any complex number ℓ, the category of restricted slNˆ(Cq)-modules of level ℓ is canonically isomorphic to the category of twisted modules for the vertex algebra VCgˆ(ℓ,0) arising from a conformal Lie algebra Cg, where Cgˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer ℓ, the integrable restricted slNˆ(Cq)-modules of level ℓ are exactly the twisted modules for the quotient vertex algebra LCgˆ(ℓ,0) of VCgˆ(ℓ,0). Finally, we classify irreducible graded twisted LCgˆ(ℓ,0)-modules.

Positive Representations of Split Real Simply-Laced Quantum Groups

We construct the positive principal series representations for \mathcal U_q(\mathfrak g_\mathbb R) where \mathfrak g is of simply-laced type, parametrized by \mathbb R_{\geq 0}^r where r is the rank of \mathfrak g . We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group \mathbf U_{\mathfrak q \tilde{\mathfrak q}}(\mathfrak g_\mathbb R) of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.

Noncommutative coverings of quantum tori

We investigate a framework for coverings of noncommutative spaces. Furthermore, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

Multi-cover skeins, quivers, and 3d mathcal{N} = 2 dualities

The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver descriptions of the geometry. Embedding into M-theory, a large class of dualities of 3d mathcal{N} = 2 theories associated to quivers is obtained. The multi-cover skein relation admits a compact formulation in terms of quantum torus algebras associated to the quiver and in this language the relations are similar to wall-crossing identities of Kontsevich and Soibelman.

The covariant Gromov–Hausdorff propinquity

We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems, which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that our resulting metric is zero between two Lipschitz dynamical systems if and only if there exists an equivariant full quantum isometry between. We also present sufficient conditions for Cauchy sequences to converge for our new metric, thus exhibiting certain complete classes of Lipschitz dynamical systems. We apply our work to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.

On Kauffman bracket skein modules of marked 3–manifolds and the Chebyshev–Frobenius homomorphism

In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary components. The addition of unmarked components allows us to develop a surgery theory which enables us to extend the Chebyshev homomorphism of Bonahon and Wong between skein algebras of unmarked surfaces to a "Chebyshev-Frobenius homomorphism" between skein modules of marked 3-manifolds. We show that the image of the Chebyshev-Frobenius homomorphism is either transparent or skew-transparent. In addition, we make use of the Muller algebra method to calculate the center of the skein algebra of a marked surface when the quantum parameter is not a root of unity.

Three-Partition Hodge Integrals and the Topological Vertex

A conjecture on the relation between the cubic Hodge integrals and the topological vertex in topological string theory is resolved. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of the generalized KdV (aka Gelfand-Dickey) hierarchies.

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

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.