Rational Quantum Research Articles

The irreducible modules for the derivations of the rational quantum torus

Let Cq be the quantum torus associated with the d×d matrix q=(qij), where qij are roots of unity with qii=1 and qij−1=qji for all 1≤i,j≤d. Let Der(Cq) be the Lie algebra of all the derivations of Cq. In this paper we define the Lie algebra Der(Cq)⋉Cq and classify its irreducible modules with finite dimensional weight spaces. These modules under certain conditions turn out to be of the form V⊗Cq, where V is a finite dimensional irreducible gld-module.

The Brauer loop scheme and orbital varieties

A. Joseph invented multidegrees in Joseph (1984) to study orbital varieties, which are the components of an orbital scheme, itself constructed by intersecting a nilpotent orbit with a Borel subalgebra. Their multidegrees are known as Joseph polynomials, and these polynomials give a basis of a (Springer) representation of the Weyl group. In the case of the nilpotent orbit {M2=0}, the orbital varieties can be indexed by noncrossing chord diagrams in the disk.In this paper we study the normal cone to the orbital scheme inside this nilpotent orbit {M2=0}. This gives a better-motivated construction of the Brauer loop scheme we introduced in Knutson and Zinn-Justin (2007), whose components are indexed by all chord diagrams (now possibly with crossings) in the disk.The multidegrees of its components, the Brauer loop varieties, were shown to reproduce the ground state of the Brauer loop model in statistical mechanics (Di Francesco and Zinn-Justin, 2006). Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik–Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the ei and fi generators of the Brauer algebra. The proof of the analogous statement in Knutson and Zinn-Justin (2007) was slightly roundabout; we verified the fi equation using the geometry of multidegrees, and used algebraic results of Di Francesco and Zinn-Justin (2006) to show that it implied the ei equation. We describe here the geometric meaning of both ei and fi equations in our slightly extended setting.We also describe the corresponding actions at the level of orbital varieties: while only the ei equations make sense directly on the Joseph polynomials, the fi equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.

New applications of quantum algebraically integrable systems in fluid dynamics

Rational quantum algebraically integrable systems are non-trivial generalizations of Laplacian operators to the case of elliptic operators with variable coefficients. We study corresponding extensions of Laplacian growth connected with algebraically integrable systems, describing viscous free-boundary flows in non-homogenous media. We introduce a class of planar flows related with application of Adler-Moser polynomials and construct solutions for higher-dimensional cases, where the conformal mapping technique is unavailable.

Integrable representations of toroidal Lie algebras co-ordinatized by rational quantum tori

For any positive integers d, n with d>1 and n>1, we fix an n by n complex matrix q=(qij) satisfying qii=1 and qij=qji−1 with all qij roots of unity. Let τ˜(d,q) be the universal central extension of the Lie subalgebra sld(Cq) of gld(Cq) with trace in [Cq,Cq], where Cq is the rational quantum torus associated to q, and let τˆ(d,q) be the Lie algebra by adding the n degree derivations to τ˜(d,q) with respect to the n non-commuting variables in Cq. The Lie algebra τˆ(d,q), called the toroidal Lie algebra co-ordinatized by the rational quantum torus Cq, has an n-dimensional center C. In this paper, we obtain a classification of irreducible integrable modules with finite dimensional weight spaces and with non-zero center action over the toroidal Lie algebra τˆ(d,q).

Categorifying fractional Euler characteristics, Jones–Wenzl projectors and $3j$-symbols

We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan–Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This is the main tool for our categorification of the Jones–Wenzl projector, \Theta -networks and tetrahedron networks. Networks and their evaluations play an important role in the Turaev–Viro construction of 3-manifold invariants. We categorify these evaluations by Ext-algebras of certain simple Harish-Chandra bimodules. The relevance of this construction to categorified colored Jones invariants and invariants of 3-manifolds will be studied in detail in subsequent papers.

Irreducible modules over the derivation algebras of rational quantum tori

Let g be the derivation Lie algebra of a rational quantum torus C q where q = ( q i j ) d × d and all q i j are roots of unity. In Lin and Tan (2004) [LT], a class of uniformly bounded irreducible weight modules over g were constructed, which generalized the construction given by Shen (1986) [Sh], Larsson (1992) [L], and Eswara Rao (1996) [E]. These modules look mysterious because of their very unclear weight sets. In this note, we use very simple methods with a very short proof to re-establish all (even stronger) results in Lin and Tan (2004) [LT] and make these mysterious modules crystal clear. A necessary and sufficient condition for two such modules to be isomorphic is also given.

SOME QUANTUM VERTEX ALGEBRAS OF ZAMOLODCHIKOV–FADDEEV TYPE

This is a continuation of a previous study of quantum vertex algebras of the Zamolodchikov–Faddeev type. In this paper, we focus our attention on the special case associated to diagonal unitary rational quantum Yang–Baxter operators. We prove that the associated weak quantum vertex algebras, if not zero, are irreducible quantum vertex algebras with a normal basis in a certain sense.

The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials

We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra glN. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.

An Algorithmic Construction of Quantum Circuits of High Descriptive Complexity

We discuss an algorithmic construction which, for any finite but universal set of computable quantum gates and a given measurement basis, will produce a rational quantum circuit whose shortest ϵ-approximations from products of instances of the gates have sizes which grow at least exponentially in the input sizes of the circuits and logarithmically in the reciprocal of ϵ. We also discuss the constructive content of the Solovay-Kitaev theorem by considering the algorithmic enumeration of all quantum circuits of a given input size.

On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups

AbstractAnn-dimensional quantum torus is a twisted group algebra of the group ℤn. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rationaln-dimensional quantum tori over any field. Moreover, we show that forn= 2 the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and for $E_{6,7,8}$ is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form being written in a certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for {\it arbitrary} values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.

Form Factors of $\textit{SU}(N)$ Invariant Thirring Model

We obtain a new integral formula for solutions of the rational quantum Knizhnik–Zamolodchikov equation associated with Lie algebra \textit{sl}_N at level zero. Our formula contains the integral representation of form factors of \textit{SU}(N) invariant Thirring model constructed by F. Smirnov. We write down recurrence relations arising from the construction of the form factors. We check that the recurrence relations hold for the form factors of the energy momentum tensor.

Quantum Currents in the Coset Space SU(2)/U(1)**The project supported in part by National Natural Science Foundation of China

We propose a rational quantum deformed nonlocal currents in the homogeneous space , and in terms of it and a free boson field a representation for the Drinfeld currents of Yangian double at a general level is obtained. In the classical limit , the quantum nonlocal currents become parafermion, and the realization of Yangian double becomes the parafermion realization of current algebra.

Q-power function over q-commuting variables and deformed XXX and XXZ chains

We find certain functional identities for the Gauss q-power function of a sum of q-commuting variables. Then we use these identities to obtain two-parameter twists of the quantum affine algebra U_q (\hat{sl}_2) and of the Yangian Y(sl_2). We determine the corresponding deformed trigonometric and rational quantum R-matrices, which then are used in the computation of deformed XXX and XXZ Hamiltonians.

Colouring the rational quantum sphere and the Kochen-Specker theorem

We review and extend recent findings of Godsil and Zaks, who published a constructive colouring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the sphere, exactly one ray is red, one white and one black. They also showed that any consistent colouring of the real sphere requires an additional colour. We discuss some of the consequences for the Kochen-Specker theorem.

Automorphisms of the skew field of rational quantum functions

The automorphisms of the skew field of fractions of the algebra of quantum polynomials in at least three variables are described.

Confluent Hypergeometric Orthogonal Polynomials Related to the Rational Quantum Calogero System with Harmonic Confinement

Two families (type $A$ and type $B$) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type $A$) and the Laguerre polynomials (type $B$), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl.

CURIOSITIES AT c-EFFECTIVE = 1

The moduli space of all rational conformal quantum field theories with effective central charge c eff = 1 is considered. Whereas the space pf unitary theories essentially forms a manifold, the nonunitary ones form a fractal which lies dense in the parameter plane. Moreover, the points of this set are shown to be in one-to-one correspondence with the elements of the modular group for which an action on this set is defined.

On the quantum symmetry of the chiral Ising model

We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of rational quantum field theories. As an example we show that a six-dimensional rational Hopf algebra H can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. H plays the role of the global symmetry algebra of the chiral Ising model in the following sense: (1) a simple field algebra F and a representation π on H π of it is given, which contains the c = 1 2 unitary representations of the Virasoro algebra as subrepresentations; (2) the embedding U: H → B( H π) is such that the observable algebra π( A) − is the invariant subalgebra of B ( H π) with respect to the left adjoint action of H and U(H) is the commutant of π( A ); (3) there exist H-covariant primary fields in B ( H π), which obey generalized Cuntz algebra properties and intertwine between the inequivalent sectors of the observables.

Observation of an even-denominator quantum number in the fractional quantum Hall effect.

An even-denominator rational quantum number has been observed in the Hall resistance of a two-dimensional electron system. At partial filling of the second Landau level ν= 2+(1/2) = 5/2 and at temperatures below 100 mK, a fractional Hall plateau develops at ρ_(xy)=(h/e^2)/(5/2) defined to better than 0.5%. Equivalent even-denominator quantization is absent in the lowest Landau level under comparable conditions.