Quantum Torus Research Articles

Classification of irreducible Harish-Chandra modules over gap-p Virasoro algebras

We prove that any irreducible Harish-Chandra module for a class of Lie algebras, which we call gap-p Virasoro algebras, must be a highest weight module, a lowest weight module, or a module of intermediate series. These algebras are closely related to the Heisenberg-Virasoro algebra and the algebra of derivations over a quantum torus. They also contain subalgebras which are isomorphic to the Virasoro algebra Vir, but graded by pZ (unlike Vir by Z).

Mapping properties of operator-valued pseudo-differential operators

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. We prove the boundedness of regular symbols on Sobolev spaces H2α(Rd;L2(M)) and Besov spaces Bp,qα(Rd;Lp(M)) for α∈R and 1≤p,q≤∞, as well as the boundedness of forbidden symbols on H2α(Rd;L2(M)) and Bp,qα(Rd;Lp(M)) for α>0 and 1≤p,q≤∞. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces F1α,c(Rd,M) obtained in our previous paper, we establish the F1α,c-regularity of regular symbols for every α∈R, and the F1α,c-regularity of forbidden symbols for α>0. As applications, we obtain the same results on the usual and quantum tori.

Quantum linear Galois orders

We define a class of quantum linear Galois algebras which include the universal enveloping algebra the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand–Zetlin algebras of type A, the subalgebras of G-invariants of the quantum affine space, quantum torus for and of the quantum Weyl algebra for G = Sn. We show that all quantum linear Galois algebras satisfy the quantum Gelfand-Kirillov conjecture. Moreover, it is shown that the subalgebras of invariants of the quantum affine space and of quantum torus for the reflection groups and of the quantum Weyl algebra for symmetric groups are, in fact, Galois orders over an adequate commutative subalgebras and free as right (left) modules over these subalgebras. In the rank 1 cases the results hold for an arbitrary finite group of automorphisms when the field is

Modules of highest weight type for the algebra of derivations over a rational quantum torus

For the Lie algebra of derivations over a rational quantum torus, we construct a class of irreducible weight modules with all finite dimensional weight spaces, which are called the modules of highest weight type, generating the notion for the case of commutative torus.

Automorphisms, derivations, and central extensions of Lie algebras arising from quantum tori

For a Lie algebra related to a quantum torus, we compute its automorphisms, derivations, and universal central extension. This Lie algebra is isomorphic to a subalgebra of the Lie algebra of derivations over the quantum torus, and moreover contains a subalgebra isomorphic to the centerless higher rank Virasoro algebra.

Quantum Differentiability on Quantum Tori

We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case.

Quantum torus symmetries of multicomponent modified KP hierarchy and reductions

In this paper, we construct the multicomponent modified KP hierarchy and its additional symmetries. The additional symmetries constitute an interesting multi-folds quantum torus type Lie algebra. By a reduction, we also construct the constrained multicomponent modified KP hierarchy and its Virasoro type additional symmetries.

Finite dimensional quantum Teichmüller space from the quantum torus at root of unity

Representation theory of the quantum torus Hopf algebra, when the parameter q is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a ‘multiplicity module’ tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map T between the multiplicity modules, called the 6j-symbols. We study the left and right dual representations, and correspondingly, the left and right representations on the Hom spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map A on a multiplicity module, encoding the permutation of the roles of the irreducible representations in the identification of the multiplicity module as the space of intertwiners between representations. We show that T and A satisfy certain consistency relations, forming a Kashaev-type quantization of the Teichmüller spaces of bordered Riemann surfaces. All constructions and proofs in the present work use only plain representation theoretic language with the help of the notions of the left and the right dual and Hom representations, and therefore can be applied easily to other Hopf algebras for future works.

Cluster realization of $$\mathcal {U}_q(\mathfrak {g})$$ U q ( g ) and factorizations of the universal R-matrix

For each simple Lie algebra $$\mathfrak {g}$$ , we construct an algebra embedding of the quantum group $$\mathcal {U}_q(\mathfrak {g})$$ into certain quantum torus algebra $$\mathcal {D}_\mathfrak {g}$$ via the positive representations of split real quantum group. The quivers corresponding to $$\mathcal {D}_\mathfrak {g}$$ is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.

Metrized Quantum Vector Bundles over Quantum Tori Built from Riemannian Metrics and Rosenberg's Levi-Civita Connections

We build metrized quantum vector bundles, over a generically transcendental quantum torus, from Riemannian metrics, using Rosenberg's Levi-Civita connections for these metrics. We also prove that two metrized quantum vector bundles, corresponding to positive scalar multiples of a Riemannian metric, have distance zero between them with respect to the modular Gromov-Hausdorff propinquity.

Zero product determined Lie algebras

A Lie algebra L over a field $$\mathbb {F}$$ is said to be zero product determined (zpd) if every bilinear map with the property that $$f(x,y)=0$$ , whenever x and y commute, is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra , where V is a simple $$\mathfrak {sl}_2$$ -module, is zpd if and only if $$\dim V =2$$ or $$\dim V$$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $$\mathscr {L}_q$$ and $$\mathscr {L}^+_q$$ , the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most 3, while the class of non-zpd Lie algebras includes the (4-dimensional) aging Lie algebra and all Lie algebras of dimension more than 3 in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of zpd Lie algebra by using it in the study of commutativity preserving linear maps.

Conjugacy of Cartan subalgebras in EALAs with a non-fgc centreless core

We establish the conjugacy Cartan subalgebras for extended affine Lie algebras whose centreless core is of type A, i.e., matrices over a quantum torus Q whose trace lies in the commutator space Q. This settles the last outstanding part the conjugacy problem for Extended Affine Lie Algebras that remained open.

Quantum torus symmetries of the CKP and multi-component CKP hierarchies

In this paper, we construct a series of additional flows of the C type Kadomtsev-Petviashvili (CKP) hierarchy and the multi-component CKP hierarchy, and these flows constitute an N-fold direct product of the positive half of the quantum torus symmetry. Comparing to the W∞ infinite dimensional Lie symmetry, this quantum torus symmetry has a nice algebraic structure with double indices.

A strong Dixmier–Moeglin equivalence for quantum Schubert cells

Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier–Moeglin equivalence.We define quantities which measure how “close” an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the algebra satisfies the strong Dixmier–Moeglin equivalence. Using the example of the universal enveloping algebra of sl2(C), we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence.For a simple complex Lie algebra g, a non-root of unity q≠0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subalgebra Uq[w] of the quantised enveloping K-algebra Uq(g). These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence.

Quantum Torus Algebras and B(C)-Type Toda Systems

In this paper, we construct a new even constrained B(C) type Toda hierarchy and derive its B(C) type Block type additional symmetry. Also we generalize the B(C) type Toda hierarchy to the $N$-component B(C) type Toda hierarchy which is proved to have symmetries of a coupled $\bigotimes^NQT_+ $ algebra ( $N$-folds direct product of the positive half of the quantum torus algebra $QT$).

Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy basing on the $W_{\infty}$ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchy. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an $N$-folds direct product of the positive half of the quantum torus symmetries.

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W ⊗ W) is trivial.

Noncommutative solenoids and the Gromov-Hausdorff propinquity

We prove that noncommutative solenoids are limits, in the sense of the Gromov-Hausdorff propinquity, of quantum tori. From this observation, we prove that noncommutative solenoids can be approximated by finite dimensional quantum compact metric spaces, and that they form a continuous family of quantum compact metric spaces over the space of multipliers of the solenoid, properly metrized.

The Krull and global dimension of the tensor product of quantum tori

An [Formula: see text]-dimensional quantum torus is defined as the [Formula: see text]-algebra generated by variables [Formula: see text] together with their inverses satisfying the relations [Formula: see text], where [Formula: see text]. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of [Formula: see text] that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

Representations of the n-Dimensional Quantum Torus

The n-dimensional quantum torus 𝒪 q((F×)n) is defined as the associative F-algebra generated by x1,…, xn together with their inverses satisfying the relations xixj = qijxjxi, where q = (qij). We show that the modules that are finitely generated over certain commutative sub-algebras ℬ are ℬ-torsion-free and have finite length. We determine the Gelfand–Kirillov dimensions of simple modules in the case whenwhere K.dim stands for the Krull dimension. In this case, if M is a simple 𝒪 q((F×)n)-module, then 𝒢𝒦-dim(M) = 1 orwhere 𝒵(C) stands for the center of an algebra C. We also show that there always exists a simple F*A-module satisfying the above inequality.