Mark Gould

University of Queensland | UQ · School of Mathematics and Physics

We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup $U_q[gl(m|n)]$. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients.

We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup $U_q[gl(m|n)]$. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients.

We investigate the prescribed Ricci curvature problem in the class of left-invariant naturally reductive Riemannian metrics on a non-compact simple Lie group. We obtain a number of conditions for the solvability of the underlying equations and discuss several examples.

In a previous paper the generator matrix elements and (dual) vector reduced Wigner coefficients (RWCs) were evaluated via the polynomial identities satisfied by a certain matrix constructed from the R-matrix R and its twisted counterpart . Here we provide an alternative evaluation utilising the R-matrix . This provides a new direct derivation of th...

In a previous paper the generator matrix elements and (dual) vector reduced Wigner coefficients (RWCs) were evaluated via the polynomial identities satisfied by a certain matrix constructed from the $R$-matrix $R$ and its twisted counterpart $R^T=T\circ R$. Here we provide an alternative evaluation utilising the $R$-matrix $\tilde{R} = (R^T)^{-1}$....

One of the most important features of quantum theory is the uncertainty principle. Amount various uncertainty relations, the profound Fine-Grained Uncertainty Relation (FGUR) is used to distinguish the uncertainty inherent in obtaining any combination of outcomes for different measurements. In this paper, we explore this uncertainty relation in rel...

Consider a compact Lie group $G$ and a closed subgroup $H<G$. Suppose $\mathcal M$ is the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G/H$. We obtain a sufficient condition for the existence of $g\in\mathcal M$ and $c>0$ such that the Ricci curvature of $g$ equals $cT$ for a given $T\in\mathcal M$. This condition is also nec...

In this paper fundamental Wigner coefficients are determined algebraically by considering the eigenvalues of certain generalized Casimir invariants. Here this method is applied in the context of both type 1 and type 2 unitary representations of the Lie superalgebra gl(mjn). Extensions to the non-unitary case are investigated. A symmetry relation be...

The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible $gl(m|n)$ modules. In particular, we give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, together with their phases on finite dimensional type 2 unitary irre...

We explore the entropic uncertainty relation in the curved background outside a Schwarzschild black hole, and find that Hawking radiation introduce a nontrivial modification on the uncertainty bound for particular observer, therefore could be witnessed by proper uncertainty game experimentally. We first investigate an uncertainty game between a fre...

We present an overview of characteristic identities for Lie algebras and superalgebras. We outline methods that employ these characteristic identities to deduce matrix elements of finite dimensional representations. To demonstrate the theory, we look at the examples of the general linear Lie algebras and Lie superalgebras.

We utilise characteristic identities to construct eigenvalue formulae for invariants and reduced matrix elements corresponding to irreducible representations of osp(m|n). In presenting these results, we further develop our programme of constructive representation theory via characteristic identities.

Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our results with existing works...

The uncertainty principle bounds our ability to simultaneously predict two incompatible observables of a quantum particle. Assisted by a quantum memory to store the particle, this uncertainty could be reduced and quantified by a new Entropic Uncertainty Relation (EUR). In this Letter, we explore how the relativistic motion of the system would affec...

We construct explicit formulae for the eigenvalues of certain invariants of the Lie superalgebra gl(m|n) using characteristic identities. We discuss how such eigenvalues are related to reduced Wigner coefficients and the reduced matrix elements of generators, and thus provide a first step to a new algebraic derivation of matrix element formulae for...

It is demonstrated how one may derive quantum supersymmetric integrable models on an open chain corresponding to (affinizable) irreducible representations of a quantum superalgebra. A new procedure for constructing the transfer matrix is obtained and an expression for the Hamiltonian is given. Applications to Uq(gl(m|n)) invariant systems are discu...

A method is developed for systematically constructing trigonometric and rational solutions of the Yang-Baxter equation using the representation theory of quantum supergroups. New quantum R-matrices are obtained by applying the method to the vector representations of quantum osp(1/2) and gl(m/n).

Motivated by application of current superalgebras in the study of disordered systems such as the random XY and Dirac models, we investigate gl(2\2) current superalgebra at general level k. We construct its free field representation and corresponding Sugawara energy-momentum tensor in the non-standard basis. Three screen currents of the first kind a...

We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. The category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.

Formulae for hermitian operators representing covalent, ionic, and total bond indices are derived. The eigenstates of these operators come in pairs, and can be considered as bonding, anti-bonding and lone-pair orbitals. The form of these operators is derived by generalising the rule that the bond order be defined as the net number of bonding electr...

Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S-2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the twosphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite e...

Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so contains a universal R-matrix in the tensor product algebra which satisfies the Yang - Baxter equation. Applying the vector representation p, which acts on the vector module V, to one side of a universal R-matrix gives a Lax operator. In this paper a Lax operator is constructed f...

We introduce quasi-Hopf $*$-algebras i.e. quasi-Hopf algebras equipped with a conjugation (star) operation. The definition of quasi-Hopf $*$-algebras proposed ensures that the class of quasi-Hopf $*$-algebras is closed under twisting and additionally, that any Hopf $*$-algebra becomes a quasi-Hopf $*$-algebra via twisting. The basic properties of t...

The main problem with current approaches to quantum computing is the difficulty of establishing and maintaining entanglement. A Topological Quantum Computer (TQC) aims to overcome this by using different physical processes that are topological in nature and which are less susceptible to disturbance by the environment. In a (2+1)-dimensional system,...

For each quantum superalgebra U-q[osp(m parallel to n)] with m > 2, an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary irreducible highest weight module is also calculated. (c) 2005 American Institute of Physics.

The Perk--Schultz model may be expressed in terms of the solution of the Yang--Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra $U_q[sl(m|n)]$, with a multiparametric co-product action as given by Reshetikhin. Here we present analogous explicit expressions fo...

For each quantum superalgebra $U_q[osp(m|n)]$ with $m>2$, an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary irreducible highest weight module is also calculated.

The Drinfeld twist for the opposite quasi-Hopf algebra, H-COP, is determined and is shown to be related to the (second) Drinfeld twist on a quasi-Hopf algebra. The twisted form of the Drinfeld twist is investigated. In the quasi-triangular case, it is shown that the Drinfeld u-operator arises from the equivalence of H-COP to the quasi-Hopf algebra...

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang--Baxter equation. Applying the vector representation, which acts on the...

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the...

The XXZ Gaudin model with generic integrable boundaries specified by generic non-diagonal K-matrices is studied. The commuting families of Gaudin operators are diagonalized by the algebraic Bethe ansatz method. The eigenvalues and the corresponding Bethe ansatz equations are obtained. (C) 2004 Elsevier B.V. All rights reserved.

In this article, we demonstrate a complementarity between the quasi-spin SU(2) algebra of the Hubbard model and the pseudo-orthogonal group O(m,m), where n = 2m is the number of lattice sites. It is shown that all N-electron states for the one-dimensional Hubbard model, corresponding to given values of spin and quasi-spin, give rise to an irreducib...

Representations of the non-semisimple superalgebra $gl(2|2)$ in the standard basis are investigated by means of the vector coherent state method and boson-fermion realization. All finite-dimensional irreducible typical and atypical representations and lowest weight (indecomposable) Kac modules of $gl(2|2)$ are constructed explicitly through the exp...

Motivated by application of current superalgebras in the study of disordered systems such as the random XY and Dirac models, we investigate $gl(2|2)$ current superalgebra at general level $k$. We construct its free field representation and corresponding Sugawara energy-momentum tensor in the non-standard basis. Three screen currents of the first ki...

Based on Haldane's spherical geometrical formalism of two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of $S^2$ and the two-dimensional quantum Hall fluids is exhibited. If the number of particles $N$ is infinitely large, two-dimensional quantum Hall physics can be precisely described in terms of the noncommutat...

In this review we demonstrate how the algebraic Bethe ansatz is used for the calculation of the energy spectra and form factors (operator matrix elements in the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As examples we apply the theory to several models of current interest in the study of Bose-Einstein condensates, which...

Nine classes of integrable open boundary conditions, further extending the one-dimensional U q (gl(2|2)) extended Hubbard model, have been constructed previously by means of the boundary ℤ 2 -graded quantum inverse scattering method. The boundary systems are now solved by using the algebraic Bethe ansatz method, and the Bethe ansatz equations are o...

We study the conformal field theories corresponding to current superalgebras $osp(2|2)^{(1)}_k$ and $osp(2|2)^{(2)}_k$. We construct the free field realizations, screen currents and primary fields of these current superalgebras at general level $k$. All the results for $osp(2|2)^{(2)}_k$ are new, and the results for the primary fields of $osp(2|2)^...

A new algebraic Bethe ansatz scheme is proposed to diagonalise classes of integrable models relevant to the description of Bose-Einstein condensates in dilute alkali gases. This is achieved by introducing the notion of Z-graded representations of the Yang-Baxter algebra. Comment: 14 pages, latex, no figures

Supersymmetric t–J Gaudin models with open boundary conditions are investigated by means of the algebraic Bethe ansatz method. Off-shell Bethe ansatz equations of the boundary Gaudin systems are derived, and used to construct and solve the KZ equations associated with sl(2|1)(1) superalgebra.

A model is introduced for two reduced BCS systems which are coupled through the transfer of Cooper pairs between the systems. The model may thus be used in the analysis of the Josephson effect arising from pair tunneling between two strongly coupled small metallic grains. At a particular coupling strength the model is integrable and explicit result...

A new parafermionic algebra associated with the homogeneous space $A^{(2)}_2/U(1)$ and its corresponding $Z$-algebra have been recently proposed. In this paper, we give a free boson representation of the $A^{(2)}_2$ parafermion algebra in terms of seven free fields. Free field realizations of the parafermionic energy-momentum tensor and screening c...

This is a reply to the comment by P Schlottmann and A A Zvyagin.

A pairing model for nucleons, introduced by Richardson in 1966, which describes proton–neutron pairing as well as proton–proton and neutron–neutron pairing, is re-examined in the context of the quantum inverse scattering method. Specifically, this shows that the model is integrable by enabling the explicit construction of the conserved operators. W...

A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and its collaborators. This method has been referred to as the Tensor Product Graph Method. Here we describe applic...

Free field and twisted parafermionic representations of twisted $su(3)^{(2)}_k$ current algebra are obtained. The corresponding twisted Sugawara energy-momentum tensor is given in terms of three (β,γ) pairs and two scalar fields and also in terms of twisted parafermionic currents and one scalar field. Two screening currents of the first kind are pr...

Free field and twisted parafermionic representations of twisted su(3)(k)((2)) current algebra are obtained. The corresponding twisted Sugawara energy-momentum tensor is given in terms of three (beta, gamma) pairs and two scalar fields and also in terms of twisted parafermionic currents and one scalar field. Two screening currents of the first kind...

The Gaudin models based on the face-type elliptic quantum groups and the XYZ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding face-type Knizhnik–Zamolodchikov equations and their solutions are given.

This is a reply to a comment by P. Schlottmann and A.A. Zvyagin.

A new type of nonlocal currents (quasi-particles), which we call twisted parafermions, and its corresponding twisted Z-algebra are found. The system consists of one spin-1 bosonic field and six nonlocal fields of fractional spins. Jacobi-type identities for the twisted parafermions are derived, and a new conformal field theory is constructed from t...

Supersymmetric t-J Gaudin models with both periodic and open boundary conditions are constructed and diagonalized by means of the algebraic Bethe ansatz method. Off-shell Bethe ansatz equations of the Gaudin systems are derived, and used to construct and solve the KZ equations associated with $sl(2|1)^{(1)}$ superalgebra.

Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and...

Integrable Kondo impurities in two cases of one-dimensional q-deformed t-J models are studied by means of the boundary Z2-graded quantum inverse scattering method. The boundary K matrices depending on the local magnetic moments of the impurities are presented as nontrivial realizations of the reflection equation algebras in an impurity Hilbert spac...

Three kinds of integrable Kondo impurity additions to one-dimensional q-deformed extended Hubbard models are studied by means of the boundary Z(2)-graded quantum inverse scattering method. The boundary K matrices depending on the local magnetic moments of the impurities are presented as nontrivial realisations of the reflection equation algebras in...

The Gaudin models based on the face-type elliptic quantum groups and the $XYZ$ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding face-type Knizhnik-Zamolodchikov equations and their solutions are given.

A new type of nonlocal currents (quasi-particles), which we call twisted parafermions, and its corresponding twisted $Z$-algebra are found. The system consists of one spin-1 bosonic field and six nonlocal fields of fractional spins. Jacobi-type identities for the twisted parafermions are derived, and a new conformal field theory is constructed from...

Motivated by application of twisted current algebra in description of the entropy of Ads(3) black hole, we investigate the simplest twisted current algebra sl(3, c)(k)((2)). Free field representation of the twisted algebra, and the corresponding twisted Sugawara energy-momentum tensor are obtained by using three (beta, gamma) pairs and two scalar f...

We propose a general method for constructing boundary integrable Gaudin models associated with (twisted) affine algebras ${\cal G}^{(k)} (k=1, 2)$, where ${\cal G}$ is a simple Lie algebra or superalgebra. Many new integrable Gaudin models with boundaries are constructed using this approach.

The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The H...

Integrable extended Hubbard models arising from symmetric group solutions are examined in the framework of the graded Quantum Inverse Scattering Method. The Bethe ansatz equations for all these models are derived by using the algebraic Bethe ansatz method. Comment: 15 pages, RevTex, No figures, to be published in J. Phys. A

The integrable open-boundary conditions for the Bariev model of three coupled one-dimensional XY spin chains are studied in the framework of the boundary quantum inverse scattering method. Three kinds of diagonal boundary K-matrices leading to nine classes of possible choices of boundary fields are found and the corresponding integrable boundary te...

Motivated by applications of twisted current algebras in description of the entropy of $Ads_3$ black hole, we investigate the simplest twisted current algebra $sl(3,{\bf C})^{(2)}_k$. Free field representation of the twisted algebra and the corresponding twisted Sugawara energy-momentum tensor are obtained by using three $(\beta,\gamma)$ pairs and...

The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The H...

Bosonized q-vertex operators related to the four-dimensional evaluation modules of the quantum affine superalgebra Uq[sl(2|^1)] are constructed for arbitrary level k=α, where α≠0,-1 is a complex parameter appearing in the four-dimensional evaluation representations. They are intertwiners among the level-α highest weight Fock-Wakimoto modules. Scree...

The minimal irreducible representations of Uq[ gl( m| n)], i.e. those irreducible representations that are also irreducible under Uq[ osp( m| n)] are investigated and shown to be affinizable to give irreducible representations of the twisted quantum affine superalgebra Uq[ gl( m| n) (2)]. The Uq[ osp( m| n)] invariant R-matrices corresponding to th...

This paper has been withdrawn by the authors, due an error in Bethe Ansatz equations (16).

The graded-fermion algebra and quasispin formalism are introduced and applied to obtain the gl(m∣n)↓osp(m∣n) branching rules for the “two-column” tensor irreducible representations of gl(m∣n), for the case m ⩽ n(n>2). In the case m<n, all such irreducible representations of gl(m∣n) are shown to be completely reducible as representations of osp(m∣n)...

Three kinds of integrable Kondo problems in one-dimensional extended Hubbard models are studied by means of the boundary graded quantum inverse scattering method. The boundary K matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras acting in a (2s alpha + 1)-di...

The minimal irreducible representations of $U_q[gl(m|n)]$, i.e. those irreducible representations that are also irreducible under $U_q[osp(m|n)]$ are investigated and shown to be affinizable to give irreducible representations of the twisted quantum affine superalgebra $U_q[gl(m|n)^{(2)}]$. The $U_q[osp(m|n)]$ invariant R-matrices corresponding to...

The construction of link polynomials associated with finite dimensional representations of ribbon quasi-Hopf algebras is discussed in terms of the formulation of an appropriate Markov trace. We then show that this Markov trace is invariant under twisting of the quasi-Hopf structure, which in turn implies twisting invariance of the associated link p...

Integrable Kondo impurities in the one-dimensional supersymmetric U model of strongly correlated electrons are studied by means of the boundary graded quantum inverse scattering method. The boundary K-matrices depending on the local magnetic moments of the impurities are presented as non-trivial realizations of the reflection equation algebras in a...

A full set of Casimir operators for the Lie superalgebra gl(m/∞) is constructed and shown to be well defined in the category OFS generated by the highest-weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Cha...

Linear superalgebraic equations giving rise to solutions of the graded classical Yang-Baxter equation are developed and solved explicitly. The connection of these equations with the theory of Lie bi-superalgebras is pointed out, and the possibility of using the solutions of the graded classical Yang-Baxter equation to construct integrable supersymm...

Polynomial identities satisfied by certain matrices a, with entries from the group algebra of a finite group G, are derived in the irreducible representations of G. In the special case of the symmetric group Sn the identities obtained parallel those previously encountered by Green (1971) for the infinitesimal generators of the general linear group....

Generalized Gelfand invariants of quantum groups are explicitly constructed, using a general procedure given in an earlier publication together with the Kirillov-Reshetikhin formula (1990) for universal R-matrices. As examples, invariants of Uq(so(5)) are considered.

The gl(n|1) down arrow gl(n)(+)gl(1) branching rules are determined for all finite-dimensional irreducible typical and atypical representations of gl(n mod 1), using a recently introduced induced module construction for atypical modules, and confirm those found recently by Palev (1988) by another method. The lowest weights and characters of the irr...

Quantum Lie algebras are non-associative algebras which are embedded into the quantized enveloping algebras of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The quantum Lie product on is induced by the quantum adjoint action of . We construct the quantum Lie algebras associated to and . We...

A q-analogue of Bargmann space is defined, using the properties of coherent states associated with a pair of q-deformed bosons. The space consists of a class of entire functions of a complex variable z, and has a reproducing kernel. On this space, the q-boson creation and annihilation operators are represented as multiplication by z and q-different...

Polynomial identities satisfied by the infinitesimal generators of a semi-simple Lie group are employed to construct projection operators which project a tensor product representation V( lambda )(X)V (V( lambda ) finite dimensional and irreducible, V an infinite dimensional representation admitting an infinitesimal character) onto a primary subrepr...

A new modified induced module construction is presented for all finite-dimensional irreducible typical and atypical modules of a type-I basic classical Lie superalgebra. The method is illustrated with some low-dimensional representations of gl(m mod n) and all representations of gl(2 mod 1).

A fully explicit formula for the eigenvalues of Casimir invariants for is given which applies to all unitary irreps. This is achieved by making some interesting observations on atypicality indices for irreps occurring in the tensor product of unitary irreps of the same type. These results have applications in the determination of link polynomials a...

In this work we investigate several important aspects of the structure theory of the recently introduced quasi-Hopf superalgebras (QHSAs), which play a fundamental role in knot theory and integrable systems. In particular we introduce the opposite structure and prove in detail (for the graded case) Drinfeld's result that the coproduct \(\) induced...

We show how to construct, starting from a quasi-Hopf (super)algebra, central elements or Casimir invariants. We show that these central elements are invariant under quasi-Hopf twistings. As a consequence, the elliptic quantum (super)groups, which arise from twisting the normal quantum (super)groups, have the same Casimir invariants as the correspon...

An integrable Kondo problem in the one-dimensional supersymmetric extended Hubbard model is studied by means of the boundary graded quantum inverse scattering method. The boundary $K$ matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras in a two-dimensional im...

A new model for correlated electrons is presented which is integrable in one-dimension. The symmetry algebra of the model is the Lie superalgebra gl(2\1) which depends on a continuous free parameter. This symmetry algebra contains the eta pairing algebra as a subalgebra which is used to show that the model exhibits Off-Diagonal Long-Range Order in...

We introduce the quasi-Hopf superalgebras which are $Z_2$ graded versions of Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing...

An integrable Kondo problem in the one-dimensional supersymmetric t-J model is studied by means of the boundary supersymmetric quantum inverse scattering method. The boundary K matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras in a two-dimensional impurity...

Integrable Kondo impurities in two cases of the one-dimensional t-J model are studied by means of the boundary Z(2)-graded quantum inverse scattering method. The boundary K-matrices depending on the local magnetic moments of the impurities are presented as non-trivial realizations of the reflection equation algebras in an impurity Hilbert space. Fu...

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for . We present a practical method for the determinat...

This is the first in a series of three articles which aimed to derive the matrix elements of the U(2n) generators in a multishell spin-orbit basis. This is a basis appropriate to many-electron systems which have a natural partitioning of the orbital space and where also spin-dependent terms are included in the Hamiltonian. The method is based on a...

This is the second in a series of articles whose ultimate goal is the evaluation of the matrix elements (MEs) of the U(2n) generators in a multishell spin-orbit basis. This extends the existing unitary group approach to spin-dependent configuration interaction (CI) and many-body perturbation theory calculations on molecules to systems where there i...

A class of integrable boundary terms for the eight-state supersymmetric U model are presented by solving the graded reflection equations. The boundary model is solved by using the coordinate Bethe ansatz method and the Bethe ansatz equations are obtained.

We describe the realization of the super Reshetikhin-Semenov-Tian-Shansky (RS) algebra in quantum affine superalgebras, thus generalizing the approach of Frenkel-Reshetikhin to the supersymmetric (and twisted) case. The algebraic homomorphism between the super RS algebra and the Drinfeld current realization of quantum affine superalgebras is establ...

This is the third and final article in a series directed toward the evaluation of the U(2n) generator matrix elements (MEs) in a multishell spin/orbit basis. Such a basis is required for many-electron systems possessing a partitioned orbital space and where spin-dependence is important. The approach taken is based on the transformation properties o...

17B37 Quantum groups (quantized enveloping algebras) and related deformations (See also 16W35, 20G42, 81R50, 82B23) 81Q60 Supersymmetric quantum mechanics 81R50 Quantum groups and related algebraic methods (See also 16W35, 17B37) 82B23 Exactly solvable models; Bethe ansatz

A class of integrable boundary terms for the eight-state supersymmtric $U$ model are presented by solving the graded reflection equations. The boundary model is solved by using the coordinate Bethe ansatz method and the Bethe ansatz equations are obtained.

An integrable eight state supersymmtric $U$ model is proposed, which is a fermion model with correlated single-particle and pair hoppings as well as uncorrelated triple-particle hopping. It has an $gl(3|1)$ supersymmetry and contains one symmetry-preserving free parameter. The model is solved and the Bethe ansatz equations are obtained. Comment: So...

We describe the twisted affine superalgebra and its quantized version . We investigate the tensor product representation of the four-dimensional grade star representation for the fixed-point subsuperalgebra . We work out the tensor product decomposition explicitly and find that the decomposition is not completely reducible. Associated with this fou...