Contractions of Filippov algebras

We introduce in this paper the contractions $\mathfrak{G}_c$ of $n$-Lie (or Filippov) algebras $\mathfrak{G}$ and show that they have a semidirect structure as their $n=2$ Lie algebra counterparts. As an example, we compute the non-trivial contractions of the simple $A_{n+1}$ Filippov algebras. By using the \.In\"on\"u-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the $\mathfrak{G}=A_{n+1}$ simple case) the Lie algebras Lie$\,\mathfrak{G}_c$ (the Lie algebra of inner endomorphisms of $\mathfrak{G}_c$) with certain contractions $(\mathrm{Lie}\,\mathfrak{G})_{IW}$ and $(\mathrm{Lie}\,\mathfrak{G})_{W-W}$ of the Lie algebra Lie$\,\mathfrak{G}$ associated with $\mathfrak{G}$.Comment: plain latex, 36 pages. A few misprints corrected. This v3 is actually v2 (v1 had been replaced by itself by error). To appear in J. Math. Phy

Effect of ion irradiation on the surface energy of deposited coatings

We investigated multi-element coatings exposed to argon ion bombardment. The coatings were irradiated using a multi-ampere hollow-cathode ion source. The arc current was 1 A, and the potential of the substrate was maintained equal to 300 V. The surface tension (surface energy) of the coatings was measured before and after irradiation through the size-dependence of the microhardness and electrical resistivity of coatings on their thickness. Ion irradiation was found to affect the surface energy of the coatings in different ways. This is due to both the structure of the coating and its elemental composition

Thermodynamic models of radiation-induced processes in solids

A thermodynamic model is proposed to qualitatively describe the radiation-induced processes in solids: temperature dependence of the X-ray radio luminescence output, dependence of these processes on the excitation density, energy accumulating in a solid under exposure to ionizing radiation and its temperature dependence. The proposed model and the formula derived can be used to develop radiation-resistant and radiation-sensitive materials

Effect of ultraviolet and x-ray radiation on optical properties of epoxy polymers dyed with organic phosphors

Highly purified industrial bisphenol and cycloaliphatic epoxy oligomers of ED-24 and UP-612 brands were used to produce optically transparent products. UV radiation of a low-pressure mercury lamp with 80% of the light energy at 254 nm was used to study photodegradation. X-ray apparatus with 0.7BSV- Ag tube was used as an ionizing radiation source to investigate the effect of X-rays on the spectra of organic dyes in epoxy polymer. The threshold value of the energy generated by ruby laser which indicated the degradation in the test samples recorded by light scattering method was determined to study radiation resistance of epoxy polymers. Basically, all the dyes exhibited high resistance to UV light. The observation of the absorption spectra showed that on average, a third of the dye molecules in the matrix experienced photobleaching within 200 hour exposure. The exception was coumarin 1, which was completely decolourized after 40 hours of exposure. X-ray irradiation of the samples for two hours results in the change in the optical density equivalent to that caused by 40 hour exposure to UV irradiation. However, in the first case, the matrix optical density is proportional to the irradiation time, and in the second case, it remains stable upon further UV irradiation. The comparison of photoaging processes in dyed and undyed epoxy polymers showed that the investigated organic dyes do not have a sensitizing effect on the matrix. The stability of the optical properties of the epoxy matrices exposed to the effects of different factors was found to depend on the nature of epoxy polymer and the technique of its production. The results of these effects are significantly different in the character of the change in the optical density and mechanisms of chemical transformations in polymer

Orbifold Singularities, Lie Algebras of the Third Kind (LATKes), and Pure Yang-Mills with Matter

We discover the unique, simple Lie Algebra of the Third Kind, or LATKe, that stems from codimension 6 orbifold singularities and gives rise to a kind of Yang-Mills theory which simultaneously is pure and contains matter. The root space of the LATKe is 1-dimensional and its Dynkin diagram consists of one point. The uniqueness of the LATKe is a vacuum selection mechanism.Comment: 42 pages; version appearing in JM

Topics on n-ary algebras

We describe the basic properties of two n-ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (or n-Lie) algebras (FAs), and comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all $n\geq 2$ FAs, n=2 being the standard Lie algebra case. When the n-bracket of the FAs is no longer required to be fully skewsymmetric one is led to the n-Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz algebras, those with an anticommutative n-bracket, we study the class of n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first n-1 entires of the n-Leibniz bracket.Comment: 11 page

Multiple M2-branes and Generalized 3-Lie algebras

We propose a generalization of the Bagger-Lambert-Gustavsson action as a candidate for the description of an arbitrary number of M2-branes. The action is formulated in terms of N=2 superfields in three dimensions and corresponds to an extension of the usual superfield formulation of Chern-Simons matter theories. Demanding gauge invariance of the resulting theory does not imply the total antisymmetry of the underlying 3-Lie algebra structure constants. We relax this condition and propose a class of examples for these generalized 3-Lie algebras. We also discuss how to associate various ordinary Lie algebras.Comment: 1+19 pages, version published in Phys. Rev.

Effect of ultraviolet and x-ray radiation on optical properties of epoxy polymers dyed with organic phosphors

Highly purified industrial bisphenol and cycloaliphatic epoxy oligomers of ED-24 and UP-612 brands were used to produce optically transparent products. UV radiation of a low-pressure mercury lamp with 80% of the light energy at 254 nm was used to study photodegradation. X-ray apparatus with 0.7BSV- Ag tube was used as an ionizing radiation source to investigate the effect of X-rays on the spectra of organic dyes in epoxy polymer. The threshold value of the energy generated by ruby laser which indicated the degradation in the test samples recorded by light scattering method was determined to study radiation resistance of epoxy polymers. Basically, all the dyes exhibited high resistance to UV light. The observation of the absorption spectra showed that on average, a third of the dye molecules in the matrix experienced photobleaching within 200 hour exposure. The exception was coumarin 1, which was completely decolourized after 40 hours of exposure. X-ray irradiation of the samples for two hours results in the change in the optical density equivalent to that caused by 40 hour exposure to UV irradiation. However, in the first case, the matrix optical density is proportional to the irradiation time, and in the second case, it remains stable upon further UV irradiation. The comparison of photoaging processes in dyed and undyed epoxy polymers showed that the investigated organic dyes do not have a sensitizing effect on the matrix. The stability of the optical properties of the epoxy matrices exposed to the effects of different factors was found to depend on the nature of epoxy polymer and the technique of its production. The results of these effects are significantly different in the character of the change in the optical density and mechanisms of chemical transformations in polymer

Cohomology of Filippov algebras and an analogue of Whitehead's lemma

We show that two cohomological properties of semisimple Lie algebras also hold for Filippov (n-Lie) algebras, namely, that semisimple n-Lie algebras do not admit non-trivial central extensions and that they are rigid i.e., cannot be deformed in Gerstenhaber sense. This result is the analogue of Whitehead's Lemma for Filippov algebras. A few comments about the n-Leibniz algebras case are made at the end.Comment: plain latex, no figures, 29 page

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