82 research outputs found

### Some results on the Zeeman topology

In a 1967 paper, Zeeman proposed a new topology for Minkowski spacetime,
physically motivated but much more complicated than the standard one. Here a
detailed study is given of some properties of the Zeeman topology which had not
been considered at the time. The general setting refers to Minkowski spacetime
of any dimension k+1. In the special case k=1, a full characterization is
obtained for the compact subsets of spacetime; moreover, the first homotopy
group is shown to be nontrivial.Comment: Part of my Laurea thesis. REVTeX4. Minor changes from previous
versio

### 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

### 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

### A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable

This is an old paper put here for archeological purposes. It is proved that a
finite-dimensional Lie algebra over a field of characteristic p>5, that can be
written as a vector space (not necessarily direct) sum of two nilpotent
subalgebras, is solvable. The same result (but covering also the cases of low
characteristics) was established independently by V. Panyukov (Russ. Math.
Surv. 45 (1990), N4, 181-182), and the homological methods utilized in the
proof were developed later in arXiv:math/0204004. Many inaccuracies in the
English translation are corrected, otherwise the text is identical to the
published version.Comment: v2: minor change

### 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

### 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

### 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

### 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

### 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

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