Quantum doubles from a class of noncocommutative weak Hopf algebras

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products is constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf algebra is built. Examples of quantum doubles from a Clifford monoid as well as a noncommutative and noncocommutative weak Hopf algebra are given, generalizing quantum doubles from a group and a noncommutative and noncocommutative Hopf algebra, respectively. Moreover, some characterizations of quantum doubles of finite dimensional biperfect weak Hopf algebras are obtained. (C) 2004 American Institute of Physics.

Braided m-Lie algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of End(F)M, where M is a Yetter-Drinfeld module over B with dimB < infinity. In particular, generalized classical braided m-Lie algebras sl(q,f)(GM(G)(A),F) and osp(q,l)(GM(G)(A),M,F) of generalized matrix algebra GMG(A) are constructed and their connection with special generalized matrix Lie superalgebra sl(s,f)(GM(Z2)(A(s)),F) and orthosymplectic generalized matrix Lie super algebra osp(s,l) (GM(Z2)(A(s)),M-s,F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.

Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m is an element of {3, 5} only, and that the algebras arising from {1, 2}-perfect m-cycle systems form a variety for m is an element of {3, 5, 7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety. (c) 2006 Elsevier B.V. All rights reserved.

Entangled subspaces and quantum symmetries

Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum j with a spin 1/2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.

Pseudo memory effects, majorization and entropy in quantum random walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.

Coherent state construction of representations of osp(2|2) and primary fields of osp(2|2) conformal field theory

Representations of the superalgebra osp(2/2)(k)((1)) and current superalgebra. osp(2/2)k in the standard basis are investigated. All finite-dimensional typical and atypical representations of osp(2/2) are constructed by the vector coherent state method. Primary fields of the non-unitary conformal field theory associated with osp(2/2)(k)((1)) in the standard basis are obtained for arbitrary level k. (C) 2004 Elsevier B.V. All rights reserved.

Solution of the dual reflection equation for A(n-1)((1)) solid-on-solid model

We obtain a diagonal solution of the dual reflection equation for the elliptic A(n-1)((1)) solid-on-solid model. The isomorphism between the solutions of the reflection equation and its dual is studied. (C) 2004 American Institute of Physics.

Fungible dynamics: There are only two types of entangling multiple-qubit interactions

What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? It has been shown that all two-body Hamiltonian evolutions can be simulated using any fixed two-body entangling n-qubit Hamiltonian and fast local unitaries. By entangling we mean that every qubit is coupled to every other qubit, if not directly, then indirectly via intermediate qubits. We extend this study to the case where interactions may involve more than two qubits at a time. We find necessary and sufficient conditions for an arbitrary n-qubit Hamiltonian to be dynamically universal, that is, able to simulate any other Hamiltonian acting on n qubits, possibly in an inefficient manner. We prove that an entangling Hamiltonian is dynamically universal if and only if it contains at least one coupling term involving an even number of interacting qubits. For odd entangling Hamiltonians, i.e., Hamiltonians with couplings that involve only an odd number of qubits, we prove that dynamic universality is possible on an encoded set of n-1 logical qubits. We further prove that an odd entangling Hamiltonian can simulate any other odd Hamiltonian and classify the algebras that such Hamiltonians generate. Thus, our results show that up to local unitary operations, there are only two fundamentally different types of entangling Hamiltonian on n qubits. We also demonstrate that, provided the number of qubits directly coupled by the Hamiltonian is bounded above by a constant, our techniques can be made efficient.

Primary Fields and Screening Currents of gl (2/2) non-unitary conformal field theory

The non-semisimple gl(2)k current superalgebra in the standard basis and the corresponding non-unitary conformal field theory are investigated. Infinite families of primary fields corresponding to all finite-dimensional irreducible typical and atypical representations of gl(212) and three (two even and one odd) screening currents of the first kind are constructed explicitly in terms of ten free fields. (C) 2004 Elsevier B.V All rights reserved.

Eigenvalues of Casimir invariants for Uq[osp(m|n)]

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.

Some twisted results

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 induced by twisting H with the R-matrix. The Altschuler-Coste u-operator arises in a similar way and is shown to be closely related to the Drinfeld u-operator. The quasi-cocycle condition is introduced and is shown to play a central role in the uniqueness of twisted structures on quasi-Hopf algebras. A generalization of the dynamical quantum Yang-Baxter equation, called the quasi-dynamical quantum Yang-Baxter equation, is introduced.

On the construction of correlation functions for the integrable supersymmetric fermion models

We review the recent progress on the construction of the determinant representations of the correlation functions for the integrable supersymmetric fermion models. The factorizing F-matrices (or the so-called F-basis) play an important role in the construction. In the F-basis, the creation (and the annihilation) operators and the Bethe states of the integrable models are given in completely symmetric forms. This leads to the determinant representations of the scalar products of the Bethe states for the models. Based on the scalar products, the determinant representations of the correlation functions may be obtained. As an example, in this review, we give the determinant representations of the two-point correlation function for the U-q(gl(2 vertical bar 1)) (i.e. q-deformed) supersymmetric t-J model. The determinant representations are useful for analyzing physical properties of the integrable models in the thermodynamical limit.

A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons

We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al. [J. Dukelsky, G.G. Dussel, C. Esebbag, S. Pittel, Phys. Rev. Lett. 93 (2004) 050403]. Here we show that there is a second integrable manifold, established using the boundary quantum inverse scattering method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrodinger operators. For the solution we derive here the potential of the Schrodinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map into a Hilbert space of PE-symmetric wavefunctions defined on a contour in the complex plane. (c) 2006 Elsevier B.V. All rights reserved.

Generalized Perk-Schultz models: solutions of the Yang-Baxter equation associated with quantized orthosymplectic superalgebras

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 (gl(m/n)], with a multiparametric coproduct action as given by Reshetikhin. Here, we present analogous explicit expressions for solutions of the Yang-Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras U-q[osp(m/n)]. In this manner, we obtain generalizations of the Perk-Schultz model.