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An efficient method for the evaluation of the matrix elements of U(2n) spin-dependent generators in a fully spin adapted Gelfand–Tsetlin basis is given. This is done by evaluating the matrix elements of the U(2n) generators in a Yamanouchi–Kotani basis whose orbital part is equivalent, up to phase factors, to the Gelfand–Tsetlin basis. This allows the expression for the matrix elements to be separated into products of creation and annihilation operators, which are evaluated using Wick’s theorem, and products of SU(2) Clebsch–Gordan coefficients, whose spin graphs are factorized into easily evaluated segment diagrams. The matrix elements of a single U(2n) generator reduce to a sum of products of segment values. These values are given in formula form involving 3-j and 6-j symbols and in table form, where the formulas have been evaluated for all the nonvanishing segments. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

This paper is concerned with matrix elements of spin-dependent operators. We present a detailed derivation of the matrix elements of the operators Δ(n)ij and Δ(n+1)ij shown in the article by Gould and Paldus [J. Chem. Phys. 92, 7394 (1990]) to lead to the matrix elements of all spin-dependent operators within the Gel’fand spin-adapted basis; as required for spin-dependent CI and the calculation of relativistic energy level shifts. Besides being useful for spin-dependent CI calculations, involving the incorporation of spin–orbit and spin–spin type electronic interactions, our results can be used for the calculation of spin-dependent properties coming from a wave function with well defined spin. Such wave functions can now be routinely computed using any one of numerous unitary group based program systems, hence the usefulness of our formalism for chemical properties. In particular we give an application of this formalism to the calculation of the spin density of a molecule using a spin-adapted wave function. In a future publication we will show how our formalism involving the Δ(n) operator leads to compact formulas for the shifts in the electronic energy spectrum of a molecule due to relativistic effects. We briefly describe our numerical implementation of these new matrix elements as used for the calculation of the spin density of a molecule. [ABSTRACT FROM AUTHOR]

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 irreducible representations which include the contravariant tensor representations and an additional class of essentially typical representations. Remarkably, we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non-vanishing matrix elements up to a phase. [ABSTRACT FROM AUTHOR]

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 existingworks that deal with only subsets of the class of type 1 unitary representations, all of which only present explicit matrix elements for elementary generators. Our work therefore provides an important extension to existing methods, and thus highlights the strength of our techniques which exploit the characteristic identities. [ABSTRACT FROM AUTHOR]

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 explicit construction of all gl(2)+gl(2) particle states (multiplets) in terms of boson and fermion creation operators in the super-Fock space. This gives a unified and complete treatment of finite-dimensional representations of gl(2|2) in explicit form, essential for the construction of primary fields of the corresponding current superalgebra at arbitrary level. [ABSTRACT FROM AUTHOR]

A new algebraic Bethe ansatz scheme is proposed to diagonalize classes of integrable models relevant to the description of Bose–Einstein condensation in dilute alkali gases. This is achieved by introducing the notion of Z-graded representations of the Yang–Baxter algebra. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Introduces the quasi-Hopf superalgebras which are z[sub 2]-graded versions of Drinfeld's quasi-Hopf algebras. Realization of the elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras; Types of elliptic quantum supergroups; Construction of the vertex type twistor in a nonstandard system of simple roots.

Investigates the antisymmetric tensor irreducible representations of graded-fermion algebra. Determination of the parities for the components appearing in the twisted tensor product graphs; Construction of the twisted quantum affine superalgebra and R-matrices; Reduction of irreducible representations by using the quasispin formalism.

Examines the application of a general eigenvalue formula for the eigenvalues of Casimir invariants to the construction of link polynomials associated with any finite dimensional unitary irrep for the algebras. Theorems concerning the computation of q-supertraces; Method for constructing link polynomials; Nonequivalent two-variable link polynomials determined in fully explicit form.

Let Uq(G) be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation. These eigenvalue formulas are shown to be absolutely convergent when the deformation parameter q is such that |q|>=1. It is proven that the universal R-matrix R of Uq(G) satisfies the celebrated conjugation relation R°=TR with T the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight Uq(G)-module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin’s and Gould’s forms to the present affine case. Casimir invariants acting on a specified module are also constructed and their eigenvalues, again absolutely convergent for |q|>=1, computed by means of the spectral decomposition formula. © 1994 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An explicit algebraic evaluation is given for all q-tensor operators (with unit norm) belonging to the quantum group Uq(u(n)) and having extremal operator shift patterns acting on arbitrary Uq(u(n)) irreps. These rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns. A more conceptual derivation of these results is discussed using fundamental properties of the q-6j coefficients. [ABSTRACT FROM AUTHOR]

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 all generators of the algebra. [ABSTRACT FROM AUTHOR]

Presents bosonized q-vertex operators related to the four-dimensional evaluation modules of the quantum affine superalgebra U (subscript q[sl(2|1)]. Intertwiners among the level-alpha highest weight Fock-Wakimoto modules; Integral formulas for N-point functions of type I and type II q-vertex operators; Use of the free field realization of the given superalgebra.

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 corresponding quantum (super)groups. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]