Search

We formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an ω-algebra defined in this paper. ω-algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping ω-algebra of recoupling Lie algebras and prove a generalized Poincare–Birkhoff–Witt theorem. As an example we consider the algebras over an arbitrary recoupling of Zn graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure.

Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract. One success here comes from “universal deformation formulas” (UDFs), expressions built from a Lie algebra which deform any algebra on which the Lie algebra operates as derivations. The most famous of these is the Moyal product, a special case of a class in which the Lie algebra is Abelian. Another comes from recognition that the Belavin–Drinfel’d solutions to the MCYBE are, in fact, infinitesimal deformations for which, in the case of the special linear groups, it is possible to give explicit formulas for the corresponding quantum Yang–Baxter equations. This review paper discusses, necessarily in brief, these and related topics, including “twisting” as a form of UDF and finding formulas for “preferred deformations” of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.

We compute for any ordering the star exponentials of all polynomials of degree not greater than two on the 2l-dimensional phase space of a quantum system with l degrees of freedom, and we show in the particular case of the Moyal star product that the Weyl transform of the Moyal star exponential of the one-dimensional harmonic oscillator Hamiltonian is the evolution operator of this quantum system.

We derive topological algebras for two-dimensional models admitting a d=[δ,b] decomposition (with b the BRST operator). In general, the topological algebra we obtain is not derived from the twisting of an N=2 supersymmetry algebra. We show, for the particular case of an Abelian model described by a ladder of matter, how to derive N=2 supersymmetry generators from the odd generators of the topological algebra.

A grading of a Lie algebra is called fine if it cannot be further refined. Fine gradings provide basic information about the structure of the algebra. There are eight fine gradings of the simple Lie algebra of type A3 over the complex number field. One of them (root decomposition) is the main tool of the theory and applications in working with A3 and with its representations; one other has also been used in the literature, and the rest have apparently not been recognized so far. An explicit description of all the fine gradings of A3 is given in terms of the four-dimensional [sl(4, C)] and six-dimensional orthogonal [o(6, C)] representations of the algebra. These results should be useful generally for choosing bases which reflect structural properties of the Lie algebra, for defining various sets of additive quantum numbers for systems with such symmetries, and for systematic study of grading preserving contractions of this Lie algebra.

We will consider a supergeneralization of the Duffin–Kemmer–Petiau algebra. It has been shown to be intimately related to the Lie super algebra osp(N,M). We will also discuss its representations.

There are three fine gradings of the simple Lie algebra of type B2 over the complex number field. They provide a basic information about the structure of the algebra. In the paper an explicit description of all fine gradings is given in terms of the four-dimensional symplectic [sp(4, C)] and five-dimensional orthogonal [o(5, C)] representations of the algebra. In addition, the real forms of B2 are considered. It is shown which of the fine gradings survive the restriction to each of the real forms. These results should be useful in defining various sets of additive quantum numbers for systems with such symmetries, for systematic study of grading preserving contractions of this Lie algebra, and generally for choosing bases which reflect structural properties of the Lie algebra.

The inhomogeneous multiboson and multivariable differential realizations of the polynomial angular momentum algebra of quadratic type, L2, which may be seen as the nonlinear extension of an ordinary angular momentum algebra su(2) or its noncompact form su(1,1), are obtained from Fock representations of L2, which are related to master representation of L2 on the space of its universal enveloping algebra U(L2) or to induced representations on their respective quotient spaces U(L2)/Ii with different left ideals Ii.

We give a new derivation of the Manakov and the product case integrability conditions of the Euler equations on Lie algebra so(4). Fourth first integral functionally independent of the three already known integrals is obtained explicitly for all values of the parameters by an algorithmic approach.

Two different methods for the construction of (2+1)-dimensional integrable lattice-field and field Hamiltonian dynamical systems are presented. The first method is based on the so-called central extension procedure applied to the Lie algebra of shift operators and the Lie algebra of pseudodifferential operators. The second method is the so-called operand formalism. Both methods allow a construction of some new integrable nonlinear Hamiltonian lattice-field and field equations in (2+1)-dimensional space.

An aperiodic analog to the Virasoro algebra is introduced and its representation theory is investigated. In particular, highest and lowest weight representations are constructed. An analog to the Kac determinant formula is derived and the implications for unitarity are discussed.

The w∞ algebra is a higher-spin extension of the Virasoro algebra. In this paper, we construct the gauge theory of w symmetry in terms of its representations.

The particles with a scattering matrix R(x) are defined as operators $\Phi_i(z)$ satisfying the relation $ R_{i,j}^{j',i'}(x_1/x_2) \Phi_{i'}(x_1)\Phi_{j'}(x_2)= \Phi_i(x_2)\Phi_j(x_1)$. The algebra generated by those operators is called a Zamolochikov algebra. We construct a new Hopf algebra by adding half of the FRTS construction of a quantum affine algebra with this R(x). Then we double it to obtain a new Hopf algebra such that the full FRTS construction of a quantum affine algebra is a Hopf subalgebra inside. Drinfeld realization of quantum affine algebras is included as an example. This is a further generalization of the constructions in q-alg/9608002., Comment: 9 pages Amslatex

It is argued that geometric algebra, in the form of the multiparticle spacetime algebra, is well suited to the study of multiparticle quantum theory, with advantages over conventional techniques both in ease of calculation and in providing an intuitive geometric understanding of the results. This is illustrated by comparing the geometric algebra approach for a system of two spin-1/2 particles with the nonrelativistic approach of Holland [Phys. Rep. 169, 294 (1988)].

The two-parametric quantum deformation of the algebra of coordinate functions on the (dual) supergroup Gr(1|1) via a contraction of Grp,q(1|1) is presented. Although the quantum superdeterminant of any element of Grp,q(1|1) is not central, in the two parametric Jordanian deformation of Gr(1|1) the quantum superdeterminant belongs to the center. The Hopf algebra structure of Grh1,h2(1|1) is discussed.

A classification of variationally trivial Lagrangians on T*M which are invariant under the Lie algebra of infinitesimal gauge transformations of the principal bundle π:M×U(1)→M, is given. A characterization of Lagrangian densities on T*M which are invariant under the Lie algebra of all infinitesimal automorphisms of M×U(1) is also obtained.

Coordinate-free differential and integral calculus is formulated on the spin groups, regarded as multivector manifolds embedded in Geometric Algebra. The derivative with respect to a point on the manifold is shown to be in the tangent algebra at the inverse point. The curvature of spin(p,q) is calculated and related to the structure constants of its Lie algebra. Volume elements on spin(p,q) are introduced naturally as completely antisymmetrized geometric products of bivectors. They are left invariant like Haar measure and are elements of a multivector-valued Grassman algebra. It is shown that integration in this sense is forbidden for many spin groups. A version of Stokes’s theorem is formulated on spin(p,q).

A modification of the standard * -structure on a quasitriangular Hopf algebra is introduced by means of which, in particular, the quantum deformation Uq(su2) can be endowed with a real structure for q a pure phase. As opposed to the standard definition, its modified version not only works for the quantum group regarded as an algebra, but as a quasitriangular Hopf algebra if q is of modulus one. Applications to two relevant examples are given, which are known to require quantum symmetries: the anisotropic Heisenberg chain and the Landau theory on the torus, thus solving some unitarity problems that have always resisted a satisfactory solution.

We give a classification of all irreducible representations of the Arik–Coon oscillator algebra without the assumption that the number operator has a discrete nondegenerate spectrum.

The interconnection between quantum mechanics and probabilistic classical mechanics for a free relativistic particle is derived in terms of Wigner functions (WF) for both Dirac and Klein-Gordon (K-G) equations. Construction of WF is achieved by first defining a bilocal 4-current and then taking its Fourier transform w.r.t. the relative 4-coordinate. The K-G and Proca cases also lend themselves to a closely parallel treatment provided the Kemmer- Duffin beta-matrix formalism is employed for the former. Calculation of WF is carried out in a Lorentz-covariant fashion by standard `trace' techniques. The results are compared with a recent derivation due to Bosanac., 9 pages, Latex; email: anmitra@csec.ernet.in

It is shown that the quantized enveloping algebra of sl(n) contains a generalized Lie algebra, defined by means of axioms similar to Woronowicz’s. This gives rise to Lie algebra-like generators and relations for the locally finite part of the quantized enveloping algebra, and suggests a canonical Poincare–Birkhoff–Witt basis.

We introduce the new nonhomogeneous graded matrix algebra Matn(R) over the ring R of the pseudodifferential operators of the arbitrary orders. This algebra forms the generalization of the algebra of the standard matrix pseudodifferential operators. We show that the Matn(R) algebra admits the nontrivial central extension. The case n=2 is considered in some detail. As the example we consider the Mat2(W1+∞), Mat2(Virasoro) and Mat2(sl(2,C)) algebras.

Point symmetries are obtained for all equations in the KP hierarchy. The Lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tN. The symmetry algebra is a semidirect sum of a Virasoro algebra and a Kac–Moody one. The “positive” part of this algebra is embedded into the known W∞ algebra of KP symmetries and into the free fermion algebra ĝl(∞). The corresponding action on the tau-function is presented. The negative part of the point symmetries does not fit into the free fermion algebra, but is embedded into a P∞ algebra, based on the algebra of pseudodifferential operators.

An associative algebra AR with exchange properties generalizing the canonical (anti)commutation relations is considered. We introduce a family of involutions in AR and construct the relative Fock representations, examining the positivity of the metric. As an application of the general results, we rigorously prove unitarity of the scattering operator of integrable models in 1+1 space-time dimensions. In this context the possibility of adopting various involutions in the Zamolodchikov–Faddeev algebra is also explored.

Representations of the Lie algebra su(1,1) and of a generalization of the oscillator algebra, b(1), are considered. The paper then introduces polynomials which are related by the coupling (or Clebsch–Gordan) coefficients of the Lie algebra in question; by making a proper choice, these polynomials themselves are related to known special functions. The coupling of two or three representations of the Lie algebra then leads to interesting addition formulas for these special functions. The polynomials appearing here are generalized Laguerre and Jacobi polynomials for the su(1,1) case, and Hermite polynomials for the b(1) algebra.

We construct the anomalous Ward identity for high-order Virasoro gravity and give the general calculation to obtain the anomaly-free condition.

We construct a C‐algebra C associated with an R‐matrix of the 2‐state chiral Potts model. This algebra will be shown to coincide with the Sklyanin algebra at an 8‐torsion point and, in the self‐dual case, with Uq(sl2) at the fourth root of unity.

In a paper by C. N. Pope and K. S. Stelle [Phys. Lett. B 226, 257–263 (1989)], it was shown that the algebra of local area‐preserving diffeomorphisms on the torus is equivalent to su(∞). In this paper we present a few remarks on the universal central extension of this infinite dimensional Lie algebra, and construct, for this case, in analogy to the situation in which the affine Lie algebra has an underlying finite‐dimensional Lie algebra, the nontrivial universal central extension.

The Hamiltonian theory for the two‐component AKNS hierarchy and Yajima–Oikawa hierarchy is considered from the viewpoint of reduction. We show that the second Hamiltonian structures of the former is a Dirac reduction of the sl(3) current algebra, while the latter is related to the classical W(3)4 algebra.

A quantization of a Lie bialgebra structure on a loop extended SU■(2) algebra is considered. An asymptotic expansion for R‐matrix is given.

Using infinite matrices, we construct the generating functional of higher integer conformal spin extensions of the Virasoro algebra and the SU(2) spin 1/2 representation loop algebra. Higher conformal spin symmetries are realized on the set F1/2 of the configurations of the homogeneous Ising chain. This is an infinite‐dimensional vector space reducible into subspaces Fm1/2(l), m∈Z, l∈N, completely specified by the site occupation operator number and the degree of the excitations of the chain states. The algebra of the chain transformations is given by the central extension of the algebra of the site state ones. The anomalous term of the generating functional of the higher integer conformal spin extensions of the Virasoro algebra is calculated and a comment on the unitarity is made. Other properties are discussed.

All local extensions of the chiral algebra of observables in the SU(2)⊗r conformal current algebra models are classified for r≤4 and all levels ki. This method is based on analyzing the polynomial solutions of the corresponding Knizhnik–Zamolodchikov equations and can be applied also for explicit construction of all local extensions corresponding to conformal embeddings of SU(2)⊗r for any r.

In this work, we consider the q-deformation of the Virasoro algebra [M. Chaichian and P. Presnajder, Phys. Lett. B 277, 109 (1992)] expressed in terms of free fermions, and we then realize this algebra, when the deformation parameter is a root of unity, on the lattice in a truncated form in terms of the Clifford algebra of Γ matrices. For this finite size truncation, the commutation relations of the Deformed algebra hold exactly albeit without central extension term. We then study the relations existing between this lattice truncation of the deformed Virasoro algebra at roots of unity and the tower of commuting Temperley-Lieb hamiltonians introduced in a previous work.

A Poisson–Hopf algebra of smooth functions is simultaneously constructed on the two dimensional Euclidean, Poincare, and Heisenberg groups by using a classical r‐matrix which is invariant under contraction. The quantization for this algebra of functions is developed, and its dual Hopf algebra is also computed. Contractions on these quantum groups are studied. It is shown that, within this setting, classical deformations are transformed into quantum ones by Hopf algebra duality and the quantum Heisenberg algebra is derived by means of a (dual) Poisson–Lie quantization that deforms the standard Moyal–Weyl *ℏ‐product.

The high‐order Virasoro algebra (HOVA) is extended to supersymmetric HOVA (SHOVA) and deformed on the quantum supersymmetric hyperplane C1‖1. Then the q‐deformed Jacobi identities and central extension are formulated. Finally, the Hopf algebra structure of q‐deformed SHOVA is defined.

The Wess–Zumino–Witten model in two dimensions is generalized to one in any even dimensions and the Ferretti–Rajeev model in three dimensions to one in any odd dimensions. In this connection some properties of fermion number (topological) currents in odd dimensions are also discussed.

For an arbitrary p ∈ ℕ, a new and computable method is proposed, which can, through a recursive algorithm, calculate the eigenvalues of the string constraint operators corresponding to the τ-function of the p-reduced Kadomtsev-Petviashvili (KP) hierarchy. It shows that not all of these eigenvalues equal 0. In this process, a connection between the W algebra and the p-reduced W algebra is introduced, which can be used to calculate the algebraic structure of the p-reduced W algebra. And it is proved that the p-reduced W algebra includes a Virasoro algebra as its subalgebra. In addition, based on the obtained eigenvalues, it is also showed that the τ-function of the p-reduced KP hierarchy constrained by the string equation is a vacuum vector for a Virasoro algebra for any p ∈ ℕ. When p = 2, it is coincident with the classical fact that the τ-function of the Korteweg-de Vries hierarchy constrained by the string equation is a vacuum vector for a Virasoro algebra.

Irreducible representation and a recurrent formula to calculate reduced coefficients of quantum algebra SUq(4)⊇SUq(2)⊕SUq(2) has been obtained through a tensorlike method.

In this article the new deformed boson algebra is proposed and the nonlinear shift operator for the algebra is obtained.

The covariant differential calculus on the quantum vector space CNq leads to q deformations of the canonical commutation relations and of the Weyl algebra. If the deformation parameter q is of modulus one, integrable (‘‘well‐behaved’’) representations of this twisted canonical commutation relations by self‐adjoint operators on a Hilbert space are defined and studied. All faithful irreducible integrable representations of the real q‐Weyl algebra are classified up to unitary equivalence.

The algebra Ξ of nonlinear (local and nonlocal) differential operators, acting on the ring of analytic functions, is studied. It is shown in particular that this space splits into 3×2 special subalgebras ∑jr, j=0,±1, r=±1. Each subalgebra is completely specified by quantum numbers s and (p,q) describing the conformal spin, and the lowest and the highest degrees, respectively. The algebra ∑++ (and its dual ∑−−) of local (pure nonlocal) differential operators is used to calculate the general expression of the Gelfand–Dickey bracket and the Wn‐symmetry Poisson, one in terms of a set of spin j canonical fields uj, 2≤j≤n and a nonlinear u‐cubic dependent differential operator D(n,i,j;z,u). The explicit form of this operator is worked out. Other remarkable features are also discussed.

In this paper the two parameter deformation of Virasoro algebra is derived. Two parameter deformation of central charge is also obtained. The operator product expansion for (p,q)‐deformed Virasoro algebra is derived.

oscillator Lie algebras with respect to countable bases of eigenfunctions of the Car-tan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings E4 and e,. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vile&in for the q= 1 case.

The quantum group dual to the deformed eq(2) algebra obtained by Celeghini et al. [J. Math. Phys. 31, 2548 (1990)] is constructed.

Nonlocal symmetries of the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy are introduced and it is shown that the symmetry algebra of the AKNS hierarchy is isomorphic to the loop algebra sl(2,C)⊗C[λ, λ−1]. As a special case, the symmetry algebra of the nonlinear Schrodinger equation is determined and is shown to be isomorphic to the loop algebra su(2)⊗R[λ, λ−1] or g⊗R[λ, λ−1] corresponding to the sign of the nonlinear term, where g is a noncompact real form of sl(2,C).

In this paper we study the inter-relationship between the integrable KP hierarchy, nonlinear $\hat{W}_{\infty}$ algebra and conformal noncompact $SL(2,R)/U(1)$ coset model at the classical level. We first derive explicitly the Possion brackets of the second Hamiltonian structure of the KP hierarchy, then use it to define the $\hat{W}_{1+\infty}$ algebra and its reduction $\hat{W}_{\infty}$. Then we show that the latter is realized in the $SL(2,R)/U(1)$ coset model as a hidden current algebra, through a free field realization of $\hat{W}_{\infty}$, in closed form for all higher-spin currents, in terms of two bosons. An immediate consequence is the existence of an infinite number of KP flows in the coset model, which preserve the $\hat{W}_{\infty}$ current algebra.