The structure of quantum Lie algebras for the classical series, B-l, C-l and D-l

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 adjoint x adjoint --> adjoint We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B-l, C-l and D-l. In the quantum case the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.

Exactly solvable quantum spin ladders associated with the orthogonal and symplectic Lie algebras

We extend the results of spin ladder models associated with the Lie algebras su(2(n)) to the case of the orthogonal and symplectic algebras o(2(n)), sp(2(n)) where n is the number of legs for the system. Two classes of models are found whose symmetry, either orthogonal or symplectic, has an explicit n dependence. Integrability of these models is shown for an arbitrary coupling of XX-type rung interactions and applied magnetic field term.

Quantum Lie algebras, their existence, uniqueness and q-antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules L-h(g) of the quantized enveloping algebras U-h(g). On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra g(h) independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras L-h(g) are isomorphic to an abstract quantum Lie algebra g(h). In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras L-h(g) associated to the same g are isomorphic, 2) the quantum Lie product of any Ch(B) is q-antisymmetric. We also describe a construction of L-h(g) which establishes their existence.

Integrability and exact solution for coupled BCS systems associated with the su(4) Lie algebra

We introduce an integrable model for two coupled BCS systems through a solution of the Yang-Baxter equation associated with the Lie algebra su(4). By employing the algebraic Bethe ansatz, we determine the exact solution for the energy spectrum. An asymptotic analysis is conducted to determine the leading terms in the ground state energy, the gap and some one point correlation functions at zero temperature. (C) 2002 Published by Elsevier Science B.V.

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.

4-Engel groups are locally nilpotent

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

Traditional architecture in the Pacific

Architecture of the Pacific covers a region of more than third of the earth’s surface. The sparse Pacific population spreads over some 30 000 islands, which graduate in size from small atolls to the largest island, Australia, a continent. Pacific architecture can be studied as four cultural units: Micronesia, Polynesia, Melanesia, and Australasia (Australia and New Zealand). While many of the islands of Micronesia lie above the Equator, the remaining Pacific islands are in the southern hemisphere. With the exception of Australia, most of the islands have a warm and humid tropical climate with high rainfalls and lush vegetation. Some islands lie in the cyclonic and earthquake belts. Two distinct racial groups settled the region. The indigenous people, the Micronesians, Melanesians, Polynesians, Australian Aborigines and New Zealand Maoris, migrated from Asia thousands of years ago. The second group, the recent immigrants, were Europeans, who occupied the region during the last two centuries, and pockets of Asians brought in by colonial administrations as labourers during the early twentieth century.

A Step-by-Step Approach to Transit Oriented Development Project Delivery

A major challenge in successfully implementing transit-oriented development (TOD) is having a robust process that ensures effective appraisal, initiation and delivery of multi-stakeholder TOD projects. A step-by step project development process can assist in the methodic design, evaluation, and initiation of TOD projects. Successful TOD requires attention to transit, mixed-use development and public space. Brisbane, Australia provides a case-study where recent planning policies and infrastructure documents have laid a foundation for TOD, but where barriers lie in precinct level planning and project implementation. In this context and perhaps in others, the research effort needs to shift toward identification of appropriate project processes and strategies. This paper presents the outcomes of research conducted to date. Drawing on the mainstream approach to project development and financial evaluation for property projects, key steps for potential use in successful delivery of TOD projects have been identified, including: establish the framework; location selection; precinct context review; preliminary precinct design; the initial financial viability study; the decision stage; establishment of project structure; land acquisition; development application; and project delivery. The appropriateness of this mainstream development and appraisal process will be tested through stakeholder research, and the proposed process will then be refined for adoption in TOD projects. It is suggested that the criteria for successful TOD should be broadened beyond financial concerns in order to deliver public sector support for project initiation.

Integrable multiparametric quantum spin chains

Using Reshetikhin's construction for multiparametric quantum algebras we obtain the associated multiparametric quantum spin chains. We show that under certain restrictions these models can be mapped to quantum spin chains with twisted boundary conditions, We illustrate how this general formalism applier; to construct multiparametric versions of the supersymmetric t-J and Li models.

Off-diagonal long-range order in a supersymmetric integrable model of correlated electrons

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 any number of dimensions.

NMR structure of C-terminally tagged gramicidin channels

A biotin group was covalently attached to the C terminus of gramicidin A (gA) through a linker arm comprising a glycine residue with either one (gAXB) or two caproyl groups (gAXXB). High-resolution two-dimensional NMR spectroscopy was used to determine the structure of these modified gA analogues and [Lys(16)]gramicidin A (gA-Lys) in sodium dodecyl-d(25) sulphate micelles. Gated gA ion channels based on linking a receptor group to these gA analogues have been used recently as a component in a sensing device. The conformations of the gA backbones and amino acid side chains of lysinated gA and biotinylated gA in detergent micelles were found to be almost identical to that of native gA, i.e. that of an N-terminal to N-terminal (head to head) dimer formed by two right-handed, single-stranded beta(6.3) helices. The biotin tail of the gAXB and gAXXB and the lysine extremity of gA-Lys appeared to lie outside the micelle. Thus it appears that the covalent attachment of functional groups to the C terminus of gA does not disrupt the peptide's helical configuration. Further, single channel measurements of all three gA analogues showed that functioning ion channels were preserved within a membrane environment. (C) 1999 Elsevier Science B.V. All rights reserved.

Eigenvalues of Casimir operators for gl(m/infinity)

A full set of Casimir operators for the Lie superalgebra gl(m/infinity) is constructed and shown to be well defined in the category O-FS 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. Characteristic identities satisfied by certain (infinite) matrices with entries from gl(m/infinity) are also determined.