Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras

As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^\alpha that are characterized by a root space of non-degenerate one-forms \alpha. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.Comment: LaTeX. 34 pages.v2. deleted some distracting paragraphs in the introduction to bring more out the main results of the paper. typos corrected and references adde

Mechanisms of CO2/H+ Sensitivity of Astrocytes.

Ventral regions of the medulla oblongata of the brainstem are populated by astrocytes sensitive to physiological changes in PCO2/[H+]. These astrocytes respond to decreases in pH with elevations in intracellular Ca2+ and facilitated exocytosis of ATP-containing vesicles. Released ATP propagates Ca2+ excitation among neighboring astrocytes and activates neurons of the brainstem respiratory network triggering adaptive increases in breathing. The mechanisms linking increases in extracellular and/or intracellular PCO2/[H+] with Ca2+ responses in chemosensitive astrocytes remain unknown. Fluorescent imaging of changes in [Na+]i and/or [Ca2+]i in individual astrocytes was performed in organotypic brainstem slice cultures and acute brainstem slices of adult rats. It was found that astroglial [Ca2+]i responses triggered by decreases in pH are preceded by Na+ entry, markedly reduced by inhibition of Na+/HCO3- cotransport (NBC) or Na+/Ca2+ exchange (NCX), and abolished in Na+-free medium or by combined NBC/NCX blockade. Acidification-induced [Ca2+]i responses were also dramatically reduced in brainstem astrocytes of mice deficient in the electrogenic Na+/HCO3- cotransporter NBCe1. Sensitivity of astrocytes to changes in pH was not affected by inhibition of Na+/H+ exchange or blockade of phospholipase C. These results suggest that in pH-sensitive astrocytes, acidification activates NBCe1, which brings Na+ inside the cell. Raising [Na+]i activates NCX to operate in a reverse mode, leading to Ca2+ entry followed by activation of downstream signaling pathways. Coupled NBC and NCX activities are, therefore, suggested to be responsible for functional CO2/H+ sensitivity of astrocytes that contribute to homeostatic regulation of brain parenchymal pH and control of breathing.Significance statementBrainstem astrocytes detect physiological changes in pH, activate neurons of the neighboring respiratory network, and contribute to the development of adaptive respiratory responses to the increases in the level of blood and brain PCO2/[H+]. The mechanisms underlying astroglial pH sensitivity remained unknown and here we show that in brainstem astrocytes acidification activates Na+/HCO3- cotransport, which brings Na+ inside the cell. Raising [Na+]i activates the Na+/Ca2+ exchanger to operate in a reverse mode leading to Ca2+ entry. This identifies a plausible mechanism of functional CO2/H+ sensitivity of brainstem astrocytes, which play an important role in homeostatic regulation of brain pH and control of breathing