Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras

In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.Comment: 16 pages, LaTeX, no figure

Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier

In quantum mechanics some properties are maximally incompatible, such as the position and momentum of a particle or the vertical and horizontal projections of a 2-level spin. Given any definite state of one property the other property is completely random, or unbiased. For N-level systems, the 6-level ones are the smallest for which a tomographically efficient set of N+1 mutually unbiased bases (MUBs) has not been found. To facilitate the search, we numerically extend the classification of unbiased bases, or Hadamards, by incrementally adjusting relative phases in a standard basis. We consider the non-unitarity caused by small adjustments with a second order Taylor expansion, and choose incremental steps within the 4-dimensional nullspace of the curvature. In this way we prescribe a numerical integration of a 4-parameter set of Hadamards of order 6.Comment: 5 pages, 2 figure

Nilpotent Classical Mechanics

The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates $\eta$. Necessary geometrical notions and elements of generalized differential $\eta$-calculus are introduced. The so called $s-$geometry, in a special case when it is orthogonally related to a traceless symmetric form, shows some resemblances to the symplectic geometry. As an example of an $\eta$-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the $R$-symmetry known for the Graded Superfield Oscillator (GSO) is present also here for the supersymmetric $\eta$-system. The generalized Poisson bracket for $(\eta,p)$-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.Comment: 23 pages, no figures. Corrected version. 2 references adde

Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter

Each of the local isometry groups arising in 3d gravity can be viewed as the group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement, and use it as a unifying framework for studying Poisson structures associated with the local isometry groups. We show that, in all cases except for Euclidean signature with positive cosmological constant, the local isometry groups are equipped with the Poisson-Lie structure of a classical double. We calculate the dressing action of the factor groups on each other and find, amongst others, a simple and unified description of the symplectic leaves of SU(2) and SL(2,R). We also compute the Poisson structure on the dual Poisson-Lie groups of the local isometry groups and on their Heisenberg doubles; together, they determine the Poisson structure of the phase space of 3d gravity in the so-called combinatorial description.Comment: 34 pages, minor corrections, references adde

New invariants for entangled states

We propose new algebraic invariants that distinguish and classify entangled states. Considering qubits as well as higher spin systems, we obtained complete entanglement classifications for cases that were either unsolved or only conjectured in the literature.Comment: published versio

Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them

Among the simple finite dimensional Lie algebras, only sl(n) possesses two automorphisms of finite order which have no common nonzero eigenvector with eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow one to generate all of sl(n) under bracketing. It seems that Sylvester was the first to mention these generators, but he used them as generators of the associative algebra of all n times n matrices Mat(n). These generators appear in the description of elliptic solutions of the classical Yang-Baxter equation, orthogonal decompositions of Lie algebras, 't Hooft's work on confinement operators in QCD, and various other instances. Here I give an algorithm which both generates sl(n) and explicitly describes a set of defining relations. For simple (up to center) Lie superalgebras, analogs of Sylvester generators exist only for sl(n|n). The relations for this case are also computed.Comment: 14 pages, 6 figure

Restricted infinitesimal deformations of restricted simple Lie algebras

We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic different from 2 and 3.Comment: 15 pages; final version, to appear in Journal of Algebra and Its Application

Bases for qudits from a nonstandard approach to SU(2)

Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1+p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1+p mutually unbiased bases in C^p. Repeated application of the formula can be used for generating mutually unbiased bases in C^d with d = p^e (e > or = 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly discussed in the case d = p^e.Comment: From a talk presented at the 13th International Conference on Symmetry Methods in Physics (Dubna, Russia, 6-9 July 2009) organized in memory of Prof. Yurii Fedorovich Smirnov by the Bogoliubov Laboratory of Theoretical Physics of the JINR and the ICAS at Yerevan State University

Grape selection for resistance to biotic and abiotic environmental factors

Most of the viticultural regions of the USSR are located under conditions of limiting biotic and abiotic factors, with frosts, drought, fungal diseases, phylloxera, mites, grape berry moths and some others being of primary importance. The main breeding organizations have been creating for more than 40 years new table and wine cultivars with complex resistance according to long-term programs. These cultivars are own-rooted and capable of wintering in outdoor culture with a limited amount of spray treatments, if any. In crossing, Amur grape and its hybrids, cultivars Seibel and Seyve Villard and some others are used as donors of resistance. Using biophysical and cytoembryological methods, gametes are treated with physical and chemical mutagenic factors in order to increase the variability range of F(1) seedlings, aiming at higher efficiency of selection. The process of selection is accelerated if seedlings are grown hydroponically. Analysis of the F(1) hybrid population determines the nature of the inheritance of valuable agricultural characters and the selection of pairs. The in vitro method is used when seedlings are grown from non-vital seeds, callus embryoids and in accelerated propagation of valuable genotypes providing virus and bacteria elimination. More than 50 cultivars with complex resistance have been bred during 35 years. More than 10 of them have been recommended for culture (Moldova, Lyana, Vostorg, Sukholimanski biely, Pervenets Magaracha, and others), while the remainder are being tested in different viticultural regions of the Soviet Union

МИКРОВОЛНОВОЙ БЕСКОНТАКТНЫЙ МЕТОД ИЗМЕРЕНИЯ УРОВНЯ ВЕЩЕСТВ

The analysis and comparison of existing contactless methods of substance level measurement is carried out. Advantages and lacks of optical, ultrasonic and microwave measurement methods are considered in details. The conclusion about expediency of a certain use contactless method of level measurement depending on set conditions is made. Microwave methods of substance level measurement are allocated as the most universal and perspective. А microwave contactless measuring instrument of substances level is considered.Проведен анализ и сравнение существующих бесконтактных методов измерения уровня вещества. Подробно рассмотрены преимущества и недостатки оптических, ультразвуковых и микроволновых методов измерения. Сделано заключение о целесообразности использования определенного бесконтактного метода измерения уровня в зависимости от заданных условий. Микроволновые методы измерения уровня вещества выделены как наиболее универсальные и перспективные. Рассмотрен СВЧ бесконтактный измеритель уровня веществ