Quantum Torus Research Articles

Summability of double Fourier series on quantum tori

Summability of double Fourier series on quantum tori

Sobolev projection on quantum torus, its complete boundedness and applications

In this paper, we establish the complete boundedness of Sobolev projection from L1(Tθd) into L1,∞(Tθd). In the special case θ=0, our results strengthen the classical results due to Berkson, Bourgain, Pelczynski and Wojciechowski.

Triangular-$$\theta $$ summability of double Fourier series on quantum tori

Triangular-$$\theta $$ summability of double Fourier series on quantum tori

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Naturality of $\mathrm{SL}_{3}$ quantum trace maps for surfaces

Fock–Goncharov’s moduli spaces \mathscr{X}_{\mathrm{PGL}_3,\mathfrak{S}} of framed \mathrm{PGL}_{3} -local systems on punctured surfaces \mathfrak{S} provide prominent examples of cluster \mathscr{X} -varieties and higher Teichmüller spaces. In a previous paper of the author (2022), building on the works of others, the so-called \mathrm{SL}_{3} quantum trace map is constructed for each triangulable punctured surface \mathfrak{S} and an ideal triangulation \Delta of \mathfrak{S} , as a homomorphism from the stated \mathrm{SL}_{3} -skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster \mathscr{X} -chart for \mathscr{X}_{\mathrm{PGL}_3,\mathfrak{S}} associated to \Delta . We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the \mathrm{SL}_{3} quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum \mathrm{SL}_{3} - \mathrm{PGL}_{3} duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.

Howe duality in the toroidal setting

In this paper, we construct and study various dual pairs acting on the oscillator modules of the symplectic toroidal Lie algebras coordinated by irrational quantum tori. This extends the classical Howe dual pairs to the toroidal setup.

Almost representations of algebras and quantization

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

Heat-smoothing for holomorphic subalgebras of free group von Neumann algebras

The heat semigroup on discrete hypercubes is well-known to be contractive over L p L_p -spaces for 1 > p > ∞ 1>p>\infty . A question of Mendel and Naor [Publ. Math. Inst. Hautes Études Sci. 119 (2014), pp. 1–95] concerns a stronger contraction property in the tail spaces, which is known as the heat-smoothing conjecture. Eskenazis and Ivanisvili [Israel J. Math. (2022), pp. 1–17] considered a Gaussian analog of this conjecture and resolved some special cases. In particular, they proved that heat-smoothing type conjecture holds for holomorphic functions in the Gaussian spaces with sharp constants. In this paper, we prove analogous sharp inequalities for holomorphic subalgebras of free group von Neumann algebras. Similar results also hold for q q -Gaussian algebras and quantum tori. In the case of free group von Neumann algebras, the weaker formulation of heat-smoothing is proved with optimal order.

Asymptotics of singular values for quantised derivatives on noncommutative tori

We obtain Weyl type asymptotics for pseudodifferential operators on quantum torus Tθd of the form TIθ−1. Here T stands for classical pseudodifferential operators on quantum torus of order 0, and Iθ−1 is the Riesz potential of order −1. As an application, we obtain Weyl type asymptotics for the quantised derivative ▪ of an operator x from the homogeneous Sobolev space W˙d1(Tθd). The asymptotic coefficient is equivalent to the norm of ▪ in the principal ideal Ld,∞, as well as the norm of x in W˙d1(Tθd). We precise and rectify earlier results in our previous paper (Comm. Math. Phys. 371 (2019), no. 3, 1231–1260) on quantum integration formula for ▪.

Quantization of the derivation Lie algebra over quantum torus

Recently, Lie bialgebra structures of the derivation Lie algebra over quantum torus were determined. In this paper, we use the general method of quantization by a Drinfel’d twist to quantize these algebras with their Lie bialgebra structures and present the explicit formulae.

Irreducible integrable modules for the full toroidal Lie algebras coordinated by rational quantum torus

Let Cq be the non-commutative Laurent polynomial ring associated with a (n+1)×(n+1) rational quantum matrix q. Let sld(Cq)⊕HC1(Cq) be the universal central extension of Lie subalgebra sld(Cq) of gld(Cq). We take the Lie algebra τ=gld(Cq)⊕HC1(Cq). Let Der(Cq) be the Lie algebra of all derivations of Cq and consider the Lie algebra τ˜=τ⋊Der(Cq), called the full toroidal Lie algebra coordinated by rational quantum tori. In this paper we get a classification of irreducible integrable modules with finite dimensional weight spaces for τ˜ with nonzero central action on the modules.

Quantum traces and embeddings of stated skein algebras into quantum tori

The stated skein algebra of a punctured bordered surface (or equivalently, a marked surface) is a generalization of the well-known Kauffman bracket skein algebra of unmarked surfaces and can be considered as an extension of the quantum special linear group \({\mathcal {O}}_{q^2}(SL_2)\) from a bigon to general surfaces. We show that the stated skein algebra of a punctured bordered surface with non-empty boundary can be embedded into quantum tori in two different ways. The first embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the shear coordinates of the enhanced Teichmüller space, and is a lift of Bonahon-Wong’s quantum trace map. The second embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the lambda length coordinates of the decorated Teichmüller space, and is an extension of Muller’s quantum trace map. We explain the relation between the two quantum trace maps. We also show that the quantum cluster algebra of Muller is equal to a reduced version of the stated skein algebra. As applications we show that the stated skein algebra is an orderly finitely generated Noetherian domain and calculate its Gelfand-Kirillov dimension.

Irreducible modules for the loop of derivations of rational quantum torus

Let [Formula: see text] be a rational quantum torus associated with the matrix [Formula: see text]. Let [Formula: see text] be the Lie algebra of derivations of [Formula: see text]. In this paper, we consider the Lie algebra [Formula: see text], where [Formula: see text] is a commutative associative unital algebra over [Formula: see text] and classify its irreducible modules with finite-dimensional weight spaces.

Topological Invariant of Velocity Field in Quantum Systems

AbstractA velocity field approach is proposed to characterize the topological invariants of quantum states based on the Poincar–Hopf theorem. It is found that the zero modes of the velocity field flow play the roles of effective topological charges to dominate the topological invariants against the parameter deformation. The validity of this approach is demonstrated by using the quantum sphere and torus models. These results are consistent with the mathematical results of the vector fields in the manifolds of the sphere and torus, Euler characteristic for sphere and for torus. The non‐Hermitian quantum torus model and differences in the topological invariants obtained using the velocity field and Chern number methods are discussed. The topological invariant characterized by the velocity field is homeomorphic in the Brillouin zone and the subbase manifold of the SU(2)‐bundle of the system, whereas the Chern number characterizes a homotopic invariant that is associated with the exceptional points in the Brillouin zone. These results enrich the topological invariants of quantum states and provide novel insights into the topological invariants of quantum states.

The Chebyshev–Frobenius homomorphism for stated skein modules of 3-manifolds

We study the stated skein modules of marked \(3-\)manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of \(3-\)manifolds. We show that there exists a Chebyshev–Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev–Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev–Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of \({\mathcal {O}}_{q^2}(SL(2))\) through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig’s Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev–Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.

Gelfand-Kirillov dimensions of simple modules over twisted group algebras $k \ast A$

For the $n$-dimensional multi-parameter quantum torus algebra $\Lambda_{\mathfrak q}$ over a field $k$ defined by a multiplicatively antisymmetric matrix $\mathfrak q = (q_{ij})$ we show that, in the case when the torsion-free rank of the subgroup of $k^\times$ generated by the $q_{ij}$ is large enough, there is a characteristic set of values (possibly with gaps) from $0$ to $n$ that can occur as the Gelfand-Kirillov dimensions of simple modules. The special case when $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = n - 1$ and $\Lambda_{\mathfrak q}$ is simple, studied in A. Gupta, $\mathrm{GK}$-dimensions of simple modules over $K[X^{\pm 1}, \sigma]$, Comm. Algebra, 41(7) (2013), 2593-2597, is considered without assuming the simplicity, and it is shown that a dichotomy still holds for the GK dimension of simple modules.

Zero modes of velocity field and topological invariant in quantum torus

We propose a velocity field approach to characterize topological invariants of quantum states. We introduce the indexes of the velocity field flow based on the zero modes of the velocity field and find that these zero modes play the role of the effective topological charges or defects linking to Euler characteristic by the Poincaré–Hopf theorem. The global property of the indexes is topological invariants against the parameter deformation. We demonstrate this approach by the quantum torus model and compare the topological invariant with that obtained by the Chern number. We find that the physical mechanism of the topological invariant based on the zero modes of the velocity field is different from that of the topological invariant by the Chern number. The topological invariant characterized by the velocity field describes a homeomorphic topological invariant associated with the zero modes on the submanifold of the base manifold of the SU(2)-fiber bundle for quantum torus, whereas the Chern number characterizes a homotopy invariant associated with the exceptional points in the Brillouin zone. We also propose a generalized winding number in terms of the velocity field for both Hermitian and non-Hermitian systems. This gives a connection between the zero mode and winding number in the velocity space. These results enrich the topological invariants of quantum states and provide a novel insight to understanding topological invariants of quantum states as well as expected to be further applied in more generic models.

Bloch waves and non-commutative tori of magnetic translations

We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction, the Zak’s magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant, which is the non-commutative torus of magnetic translations in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. The main focus of our review is the Kähler structure on the Hilbert space of Bloch waves and its inherent quantum toric geometry. We reveal that the metaplectic group Mp(2,R) of the automorphisms of magnetic translation algebras is represented by the quantum optics squeezing operators.

Sharp Estimates of Noncommutative Bochner–Riesz Means on Two-Dimensional Quantum Tori

In this paper, we establish the full $$L_p$$ boundedness of noncommutative Bochner–Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in Chen et al. (Commun Math Phys 322(3):755–805, 2013) in the sense of the $$L_p$$ convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner–Riesz means.

Scattering diagrams, theta functions, and refined tropical curve counts

Working over various graded Lie algebras and in arbitrary dimension, we express scattering diagrams and theta functions in terms of counts of tropical curves/disks, weighted by multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, our tropical curve counts are known to give certain descendant log Gromov-Witten invariants. Working over the quantum torus algebra yields theta functions for quantum cluster varieties, and our tropical description sets up for geometric interpretations of these. As an immediate application, we prove the quantum Frobenius conjecture of Fock and Goncharov. We also prove a refined version of the Carl-Pumperla-Siebert result on consistency of theta functions, and we prove the non-degeneracy of the trace-pairing for the Gross-Hacking-Keel Frobenius structure conjecture.