arXiv subjects

N. Aizawa

Publications and source records attributed to N. Aizawa.

At least 55 records · Page 3Linked to original sources

arXiv2013

Some representations of planar Galilean conformal algebra

Representation theory of an infinite dimensional Galilean conformal algebra introduced by Martelli and Tachikawa is developed. We focus on the algebra defined in (2+1) dimensional spacetime and consider central extension. It is then shown that the Verma modules are irreducible for non-vanishing highest weights. This is done by explicit computation of Kac determinant. We also present coadjoint representations of the Galilean conformal algebra and its Lie group. As an application of them, a coadjoint orbit of the Galilean conformal group is given in a simple case.

math-ph↗

arXiv2012

N = 2 Galilean superconformal algebras with central extension

N = 2 Supersymmetric extensions of Galilean conformal algebra (GCA), specified by spin \ell and dimension of space d, are investigated. Duval and Horvathy showed that the \ell = 1/2 GCA has two types of supersymmetric extensions, called standard and exotic. Recently, Masterov intorduced a centerless super-GCA for arbitrary \ell wchich corresponds to the standard extension. We show that the Masterov's super-GCA has two types of central extensions depending on the parity of 2\ell. We then introduced a novel super-GCA for arbitrary \ell corresponding to the exotic extension. It is shown that the exotic superalgebra also has two types of central extensions depending on the parity of 2\ell. Furthermore, we give a realization of the standard and exotic super-GCA's in terms of their subalgebras. Finally, we present a N = 1 supersmmetric extension of GCA with central extensions.

math-ph↗

arXiv2010

Lowest weight representations of super Schrodinger algebras in low dimensional spacetime

We investigate the lowest weight representations of the super Schrodinger algebras introduced by Duval and Horvathy. This is done by the same procedure as the semisimple Lie algebras. Namely, all singular vectors within the Verma modules are constructed explicitly then irreducibility of the associated quotient modules is studied again by the use of singular vectors. We present the classification of irreducible Verma modules for the super Schrodinger algebras in (1+1) and (2+1) dimensional spacetime with N = 1, 2 extensions.

math-ph↗

arXiv2010

Lowest Weight Representations of Super Schrodinger Algebras in One Dimensional Space

Lowest weight modules, in particular, Verma modules over the N = 1,2 super Schrodinger algebras in (1+1) dimensional spacetime are investigated. The reducibility of the Verma modules is analyzed via explicitly constructed singular vectors. The classification of the irreducible lowest weight modules is given for both massive and massless representations. A vector field realization of the N = 1, 2 super Schrodinger algebras is also presented.

math-ph↗

arXiv2009

Intertwining Operator Realization of Non-Relativistic Holography

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrodinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov-Witten we give the conditions when both boundary fields are physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others.

hep-th↗

arXiv2009

Coherent State on SUq(2) Homogeneous Space

The generalized coherent states for quantum groups introduced by Jurco and Stovicek are studied for the simplest example SU_q(2) in full detail. It is shown that the normalized SU_q(2) coherent states enjoy the property of completeness, and allow a resolution of the unity. This feature is expected to play a key role in application of these coherent states in physical models. The homogeneous space of SU_q(2), i.e. the q-sphere of Podles, is reproduced in complex coordinates by using the coherent states. Differential calculus in the complex form on the homogeneous space is developed. High spin limit of the SU_q(2) coherent states is also discussed.

math.QA↗

arXiv2007

Basic Hypergeometric Functions and Covariant Spaces for Even Dimensional Representations of U_q[osp(1/2)]

Representations of the quantum superalgebra U_q[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch-Gordan decomposition for the superalgebra U_q[osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch-Gordan coefficients are derived, and it is observed that they may be expressed in terms of the $Q$-Hahn polynomials. We next investigate representations of the quantum supergroup OSp_q(1/2) which are not well-defined in the classical limit. Employing the universal T-matrix, the representation matrices are obtained explicitly, and found to be related to the little Q-Jacobi polynomials. Characteristically, the relation Q = -q is satisfied in all cases. Using the Clebsch-Gordan coefficients derived here, we construct new noncommutative spaces that are covariant under the coaction of the even dimensional representations of the quantum supergroup OSp_q(1/2).

math.QA↗

arXiv2007

Fuzzy Torus via q-Parafermion

We note that the recently introduced fuzzy torus can be regarded as a q-deformed parafermion. Based on this picture, classification of the Hermitian representations of the fuzzy torus is carried out. The result involves Fock-type representations and new finite dimensional representations for q being a root of unity as well as already known finite dimensional ones.

hep-th↗

arXiv2006

Universal T-matrix, Representations of OSp_q(1/2) and Little Q-Jacobi Polynomials

We obtain a closed form expression of the universal T-matrix encapsulating the duality of the quantum superalgebra U_q[osp(1/2)] and the corresponding supergroup OSp_q(1/2). The classical q-->1 limit of this universal T-matrix yields the group element of the undeformed OSp(1/2) supergroup. The finite dimensional representations of the quantum supergroup OSp_q(1/2) are readily constructed employing the said universal T-matrix and the known finite dimensional representations of the dually related deformed U_q[osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations and the orthogonality of the representations of the quantum supergroup OSp_q(1/2). It is shown that the entries of these representation matrices are expressed in terms of the little Q-Jacobi polynomials with Q = -q. Two mutually complementary singular maps of the universal T-matrix on the universal R-matrix are also presented.

math.QA↗

arXiv2006

All Link Invariants for Two Dimensional Solutions of Yang-Baxter Equation and Dressings

All polynomial invariants of links for two dimensional solutions of Yang-Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

math.GT↗

arXiv2005

Noncommutative Superspaces Covariant Under $OSp_q(1|2)$ Algebra

Using the corepresentation of the quantum group $ SL_q(2)$ a general method for constructing noncommutative spaces covariant under its coaction is developed. The method allows us to treat the quantum plane and Podleś' quantum spheres in a unified way and to construct higher dimensional noncommutative spaces systematically. Furthermore, we extend the method to the quantum supergroup $ OSp_q(1|2).$ In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed.

math.QA↗

arXiv2005

Generalized boson algebra and its entangled bipartite coherent states

Starting with a given generalized boson algebra U_ (h(1)) known as the bosonized version of the quantum super-Hopf U_q[osp(1/2)] algebra, we employ the Hopf duality arguments to provide the dually conjugate function algebra Fun_ (H(1)). Both the Hopf algebras being finitely generated, we produce a closed form expression of the universal T matrix that caps the duality and generalizes the familiar exponential map relating a Lie algebra with its corresponding group. Subsequently, using an inverse Mellin transform approach, the coherent states of single-node systems subject to the U_ (h(1)) symmetry are found to be complete with a positive-definite integration measure. Nonclassical coalgebraic structure of the U_ (h(1)) algebra is found to generate naturally entangled coherent states in bipartite composite systems.

quant-ph↗

arXiv2005

Quantum Spheres for OSp_q(1/2)

Using the corepresentation of the quantum supergroup OSp_q(1/2) a general method for constructing noncommutative spaces covariant under its coaction is developed. In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed. It is observed that embedding of the supersphere in the OSp_q(1/2) algebra is possible. This realization admits infinitesimal characterization a la Koornwinder. A covariant oscillator realization of the supersphere is also presented.

math.QA↗

arXiv2003

Noncommutative Geometry of Super-Jordanian $OSp_h(2/1)$ Covariant Quantum Space

Extending a recently proposed procedure of construction of various elements of diffential geometry on noncommutative algebras, we obtain these structures on noncommutative superalgebras. As an example, a quantum superspace covariant under the action of super-Jordanian $OSp_h(2/1)$ is studied. It is shown that there exist a two paramater family of torsionless connections, and the curvature computed from this family of connections is bilinear. It is also shown that the connections are not compatible with the metric.

math.QA↗

arXiv2003

Weak Hopf algebras corresponding to $U_q[sl_n]$

We investigate the weak Hopf algebras of Li based on $U_q[sl_n]$ and Sweedler's finite dimensional example. We give weak Hopf algebra isomorphisms between the weak generalisations of $U_q[sl_n]$ which are ``upgraded'' automorphisms of $U_q[sl_n]$ and hence give a classification of these structures as weak Hopf algebras. We also show how to decompose these examples into a direct sum which leads to unexpected isomorphisms between their algebraic structure.

math.QA↗

arXiv2003

Super-Jordanian Quantum Superalgebra ${\cal U}_{h}(osp(2/1))$

A triangular quantum deformation of $ osp(2/1) $ from the classical $r$-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the deformed $ osp(2/1)$. It is also shown that its subalgebra generated by the Borel subalgebra is self-dual.

math.QA↗

arXiv2002

Twisted Conformal Algebra so(4,2)

A new twisted deformation, U_z(so(4,2)), of the conformal algebra of the (3+1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by ten Weyl-Poincare generators, which generalizes non-standard quantum deformations previously obtained for so(2,2) and so(3,2). However, by introducing a conformal null-plane basis it is found that the twist can indeed be supported by an eight-dimensional carrier subalgebra. By construction the Weyl-Poincare subalgebra remains as a Hopf subalgebra after deformation. Non-relativistic limits of U_z(so(4,2)) are shown to be well defined and they give rise to new twisted conformal algebras of Galilean and Carroll spacetimes. Furthermore a difference-differential massless Klein-Gordon (or wave) equation with twisted conformal symmetry is constructed through deformed momenta and position operators. The deformation parameter is interpreted as the lattice step on a uniform Minkowskian spacetime lattice discretized along two basic null-plane directions.

hep-th↗

arXiv2001

Extensions of Peripheric Extended Twists and Inhomogeneous Lie Algebras

Simple extensions of peripheric extended twists, introduced recently by Lyakhovsky and Del Olmo, are presented. Explicit form of twisting elements are given and it is shown that the new twists as well as peripheric extended twists are suitable to deform inhomogeneous Lie algebras such as isu(n), iso(n), (1+n) dimensional Schroedinger algebras and Poincare algebra.

math.QA↗