Noncommutative Symmetries and Gravity

Spacetime geometry is twisted (deformed) into noncommutative spacetime geometry, where functions and tensors are now star-multiplied. Consistently, spacetime diffeomorhisms are twisted into noncommutative diffeomorphisms. Their deformed Lie algebra structure and that of infinitesimal Poincare' transformations is defined and explicitly constructed. This allows to construct a noncommutative theory of gravity.Comment: 26 pages. Lectures given at the workshop `Noncommutative Geometry in Field and String Theories', Corfu Summer Institute on EPP, September 2005, Corfu, Greece. Version 2: Marie Curie European Reintegration Grant MERG-CT-2004-006374 acknowledge

Glial D-Serine Gates NMDA Receptors at Excitatory Synapses in Prefrontal Cortex.

N-methyl-D-aspartate receptors (NMDARs) subserve numerous neurophysiological and neuropathological processes in the cerebral cortex. Their activation requires the binding of glutamate and also of a coagonist. Whereas glycine and D-serine (D-ser) are candidates for such a role at central synapses, the nature of the coagonist in cerebral cortex remains unknown. We first show that the glycine-binding site of NMDARs is not saturated in acute slices preparations of medial prefrontal cortex (mPFC). Using enzymes that selectively degrade either D-ser or glycine, we demonstrate that under the present conditions, D-ser is the principle endogenous coagonist of synaptic NMDARs at mature excitatory synapses in layers V/VI of mPFC where it is essential for long-term potentiation (LTP) induction. Furthermore, blocking the activity of glia with the metabolic inhibitor, fluoroacetate, impairs NMDAR-mediated synaptic transmission and prevents LTP induction by reducing the extracellular levels of D-serine. Such deficits can be restored by exogenous D-ser, indicating that the D-amino acid mainly originates from glia in the mPFC, as further confirmed by double-immunostaining studies for D-ser and anti-glial fibrillary acidic protein. Our findings suggest that D-ser modulates neuronal networks in the cerebral cortex by gating the activity of NMDARs and that altering its levels is relevant to the induction and potentially treatment of psychiatric and neurological disorders

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 also 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.Comment: 25 pages, amslatex, eepic. Final version for publication in J. Phys. A. Minor misprints in eqs. 5.11 and 5.12 correcte

The Serre spectral sequence of a noncommutative fibration for de Rham cohomology

For differential calculi on noncommutative algebras, we construct a twisted de Rham cohomology using flat connections on modules. This has properties similar, in some respects, to sheaf cohomology on topological spaces. We also discuss generalised mapping properties of these theories, and relations of these properties to corings. Using this, we give conditions for the Serre spectral sequence to hold for a noncommutative fibration. This might be better read as giving the definition of a fibration in noncommutative differential geometry. We also study the multiplicative structure of such spectral sequences. Finally we show that some noncommutative homogeneous spaces satisfy the conditions to be such a fibration, and in the process clarify the differential structure on these homogeneous spaces. We also give two explicit examples of differential fibrations: these are built on the quantum Hopf fibration with two different differential structures.Comment: LaTeX, 33 page

Morita base change in Hopf-cyclic (co)homology

In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus.Comment: Final version to appear in Lett. Math. Phy

q-Analogue of Am−1⊕An−1⊂Amn−1A_{m-1}\oplus A_{n-1}\subset A_{mn-1}

A natural embedding $A_{m-1}\oplus A_{n-1}\subset A_{mn-1}$ for the corresponding quantum algebras is constructed through the appropriate comultiplication on the generators of each of the $A_{m-1}$ and $A_{n-1}$ algebras. The above embedding is proved in their $q$-boson realization by means of the isomorphism between the $\mathcal{A}_q^{-}$ (mn)$\sim {\otimes} ^n \mathcal{A}_q^{-}$(m)$\sim {\otimes}^m\mathcal{A}_q^{-}$(n) algebras.Comment: 11 pages, no figures. In memory of professor R. P. Rousse

Weak Localization Effect in Superconductors by Radiation Damage

Large reductions of the superconducting transition temperature $T_{c}$ and the accompanying loss of the thermal electrical resistivity (electron-phonon interaction) due to radiation damage have been observed for several A15 compounds, Chevrel phase and Ternary superconductors, and $\rm{NbSe_{2}}$ in the high fluence regime. We examine these behaviors based on the recent theory of weak localization effect in superconductors. We find a good fitting to the experimental data. In particular, weak localization correction to the phonon-mediated interaction is derived from the density correlation function. It is shown that weak localization has a strong influence on both the phonon-mediated interaction and the electron-phonon interaction, which leads to the universal correlation of $T_{c}$ and resistance ratio.Comment: 16 pages plus 3 figures, revtex, 76 references, For more information, Plesse see http://www.fen.bilkent.edu.tr/~yjki

Hopf Categories

We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.Comment: 47 pages; final version to appear in Algebras and Representation Theor

Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions

A general procedure to get the explicit solution of the equations of motion for N-body classical Hamiltonian systems equipped with coalgebra symmetry is introduced by defining a set of appropriate collective variables which are based on the iterations of the coproduct map on the generators of the algebra. In this way several examples of N-body dynamical systems obtained from q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2) Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of Ruijsenaars type arising from the same (non co-boundary) q-deformation of the (1+1) Poincare' algebra. Also, a unified interpretation of all these systems as different Poisson-Lie dynamics on the same three dimensional solvable Lie group is given.Comment: 19 Latex pages, No figure

Hom-quantum groups I: quasi-triangular Hom-bialgebras

We introduce a Hom-type generalization of quantum groups, called quasi-triangular Hom-bialgebras. They are non-associative and non-coassociative analogues of Drinfel'd's quasi-triangular bialgebras, in which the non-(co)associativity is controlled by a twisting map. A family of quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular bialgebra, such as Drinfel'd's quantum enveloping algebras. Each quasi-triangular Hom-bialgebra comes with a solution of the quantum Hom-Yang-Baxter equation, which is a non-associative version of the quantum Yang-Baxter equation. Solutions of the Hom-Yang-Baxter equation can be obtained from modules of suitable quasi-triangular Hom-bialgebras.Comment: 21 page