Quantum and Braided Lie Algebras

We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space \CL equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$and a Yang-Baxter operator$\Psi:\CL\tens\CL\to \CL\tens\CL$obeying some axioms. We show that such an object has an enveloping braided-bialgebra$U(\CL)$. We show that every generic$R$-matrix leads to such a braided Lie algebra with$[\ ,\ ]$given by structure constants$c^{IJ}{}_K$determined from$R$. In this case$U(\CL)=B(R)$the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural right-regular action of a braided-Lie algebra$\CL$by braided vector fields, the braided-Killing form and the quadratic Casimir associated to$\CL$. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations$U_q(g)$are understood as the enveloping algebras of such underlying braided Lie algebras with$[\ ,\ ]$on$\CL\subset U_q(g)$ given by the quantum adjoint action.Comment: 56 page

Equivalence of quotient Hilbert modules

Let \cl{M} be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(\Omega)$, where \Omega \subseteq \bb{C}^m is a bounded domain. Let \cl{M}_0\subseteq \cl{M} be the submodule of functions vanishing to order $k$ on a hypersurface \cl{Z} \subseteq \Omega. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modules \cl{Q}=\cl{M} \ominus \cl{M}_0. The invariants are given explicitly in the particular case of $k = 2$.Comment: 11 page

Characterization of constitutive and acid-induced outwardly rectifying chloride currents in immortalized mouse distal tubular cells

Thiazides block Na+ reabsorption while enhancing Ca2 + reabsorption in the kidney. As previously demonstrated in immortalized mouse DCT (MDCT) cells, chlorothiazide application induced a robust plasma membrane hyperpolarization, which increased Ca2 + uptake. This essential thiazide-induced hyperpolarization was prevented by the Cl− channel inhibitor 5-Nitro-2-(3-phenylpropylamino) benzoic acid (NPPB), implicating NPPB-sensitive Cl− channels, however the nature of these Cl− channels has been rarely described in the literature. Here we show that MDCT cells express a dominant, outwardly rectifying Cl− current at extracellular pH 7.4. This constitutive Cl− current was more permeable to larger anions (Eisenman sequence I; I− > Br− ≥ Cl−) and was substantially inhibited by > 100 mM [Ca2 +]o, which distinguished it from ClC-K2/Barttin. Moreover, the constitutive Cl− current was blocked by NPPB, along with other Cl− channel inhibitors (DIDS, FFA). Subjecting the MDCT cells to an acidic extracellular solution (pH < 5.5) induced a substantially larger outwardly rectifying NPPB-sensitive Cl− current. This acid-induced Cl− current was also anion permeable (I− > Br− > Cl−), but was distinguished from the constitutive Cl− current by its rectification characteristics, ion sensitivities, and response to FFA. In addition, we have identified similar outwardly rectifying and acid-sensitive currents in immortalized cells from the inner medullary collecting duct (mIMCD-3 cells). Expression of an acid-induced Cl− current would be particularly relevant in the acidic IMCD (pH < 5.5). To our knowledge, the properties of these Cl− currents are unique and provide the mechanisms to account for the Cl− efflux previously speculated to be present in MDCT cells

Conjugacy classes and finite pp-groups

Let $G$ be a finite $p$-group, where $p$ is a prime number, and $a\in G$. Denote by \Cl(a)=\{gag^{-1}\mid g\in G\} the conjugacy class of $a$ in $G$. Assume that |\Cl(a)|=p^n. Then \Cl(a)\Cl(a^{-1})=\{xy\mid x\in \Cl(a), y\in \Cl(a^{-1})\} is the union of at least $n(p-1)+1$ distinct conjugacy classes of $G$

Nonequilibrium gating and voltage dependence of the ClC-0 Cl- channel.

The gating of ClC-0, the voltage-dependent Cl- channel from Torpedo electric organ, is strongly influenced by Cl- ions in the external solution. Raising external Cl- over the range 1-600 mM favors the fast-gating open state and disfavors the slow-gating inactivated state. Analysis of purified single ClC-0 channels reconstituted into planar lipid bilayers was used to identify the role of Cl- ions in the channel's fast voltage-dependent gating process. External, but not internal, Cl- had a major effect on the channel's opening rate constant. The closing rate was more sensitive to internal Cl- than to external Cl-. Both opening and closing rates varied with voltage. A model was derived that postulates (a) that in the channel's closed state, Cl- is accessible to a site located at the outer end of the conduction pore, where it binds in a voltage-independent fashion, (b) that this closed conformation can open, whether liganded by Cl- or not, in a weakly voltage-dependent fashion, (c) that the Cl(-)-liganded closed channel undergoes a conformational change to a different closed state, such that concomitant with this change, Cl- ion moves inward, conferring voltage-dependence to this step, and (d) that this new Cl(-)-liganded closed state opens with a very high rate. According to this picture, Cl- movement within the pre-open channel is the major source of voltage dependence, and charge movement intrinsic to the channel protein contributes very little to voltage-dependent gating of ClC-0. Moreover, since the Cl- activation site is probably located in the ion conduction pathway, the fast gating of ClC-0 is necessarily coupled to ion conduction, a nonequilibrium process

Phase Space Description of the Leading Order Quark and Gluon Production from a Space-Time Dependent Chromofield

We derive source terms for the production of quarks and gluons from the QCD vacuum in the presence of a space-time dependent external chromofield A_{cl} to the order of S^{(1)}. We found that the source terms for the parton production processes A_{cl} -> q\bar{q} and A_{cl},A_{cl}A_{cl} -> gg also include the annihilation processes q\bar{q} -> A_{cl} and gg -> A_{cl},A_{cl}A_{cl}. The source terms we derive are applicable for the description of the production of partons with momentum p larger rhan gA which itself must be larger than \Lambda_{QCD}. We observe that these source terms for the production of partons from a space-time dependent chromofield can be used to study the production and equilibration of the quark-gluon plasma during the very early stages of an ultrarelativistic heavy-ion collision.Comment: 30 pages latex (single spaced), 7 eps figures, Revised Version, To appear in Physical Review