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Journal of Generalized Lie Theory and Applications

ISSN: 1736-4337

Open Access

Volume 2, Issue 3 (2008)

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A Horn type spectral problem 1

Stanislav POPOVYCH

The famous Horn’s problem is about the possible eigenvalue list of a sum of two Hermitian matrices with prescribed eigenvalue lists. The Spectral Problem is to describe possible spectra for an irreducible finite family of Hermitian operators with the sum being a scalar operator. In case when spectra consist of finite number of points the complexity of the problem depends on properties of some rooted tree. We will consider the cases for which the explicit answer on the Spectral Problem can be obtained.

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Hypersurfaces of paraquaternionic space forms 1

In this paper we obtain some properties for real lightlike hypersurfaces of paraquaternionic space forms and give an example.

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Simplicial Hochschild cochains as an Amitsur complex 1

Lars KADISON

It is shown how the cochain complex of the relative Hochschild A-valued cochains of a depth two extension A|B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End BAB over the centralizer R = AB with grouplike element 1S. This specializes to finite dimensional algebras, Hopf-Galois extensions and H-separable extensions.

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On triple systems and extended Dynkin diagrams of Lie superalgebras 1

Noriaki KAMIYA and Susumu OKUBO

Our aim is to give a characterization of extended Dynkin diagrams of Lie superalgebras by means of concept of triple systems.

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Models solvable by Bethe Ansatz 1

Petr P. KULISH

Diagonalization of integrable spin chain Hamiltonians by the quantum inverse scattering method gives rise to the connection with representation theory of different (quantum) algebras. Extending the Schur-Weyl duality between sl2 and the symmetric group SN from the case of the isotropic spin 1/2 chain (XXX-model) to a general spin chains related to the Temperley-Lieb algebra TLN(q) one finds a new quantum algebra Uq(n) with the representation ring equivalent to the sl2 one.

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Quasi-Lie deformations on the algebra F[t]/(tN) 1

Daniel LARSSON , Gunnar SIGURDSSON , and Sergei D. SILVESTROV

This paper explores the quasi-deformation scheme devised by Hartwig, Larsson and Silvestrov as applied to the simple Lie algebra sl2(F). One of the main points of this method is that the quasi-deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when the quasi-deformation method is applied to sl2(F) via representations by twisted derivations on the algebra F[t]/(tN) one obtains interesting new multi-parameter families of almost quadratic algebras.

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A class of unitary irreducible representations of the Lie superalgebra osp(1|2n) 1

Stijn LIEVENS, Nedialka I. STOILOVA and Joris Van der JEUGT

Using the equivalence of the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) to the defining triple relations of n pairs of parabose operators b± i we construct a class of unitary irreducible (infinite-dimensional) lowest weight representations V (p) of osp(1|2n). We introduce an orthogonal basis of V (p) in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role and we present explicit actions of the osp(1|2n) generators.

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Weak topological functors 1

Patrik LUNDSTROM

We introduce weak topological functors and show that they lift and preserve weak limits and weak colimits. We also show that if A ! B is a topological functor and J is a category, then the induced functor AJ ! BJ is topological. These results are applied to a generalization of Wyler’s top categories and in particular to functor categories of fuzzy maps, fuzzy relations, fuzzy topological spaces and fuzzy measurable spaces.

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On a correspondence between ideals and commutativity in algebraic crossed products 1

Johan OINERT and Sergei D. SILVESTROV

In this paper we will give an overview of some recent results which display a connection between commutativity and the ideal structures in algebraic crossed products.

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Note on 2d binary operadic harmonic oscillator 1

Eugen PAAL and Juri VIRKEPU

It is explained how the time evolution of the operadic variables may be introduced. As an example, a 2-dimensional binary operadic Lax representation for the harmonic oscillator is constructed.

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A generalization of the Kantor-Koecher-Tits construction 1

Jakob PALMKVIST

The Kantor-Koecher-Tits construction associates a Lie algebra to any Jordan algebra. We generalize this construction to include also extensions of the associated Lie algebra. In particular, the conformal realization of so(p + 1, q + 1) generalizes to so(p + n, q + n), for arbitrary n, with a linearly realized subalgebra so(p, q). We also show that the construction applied to 3 × 3 matrices over the division algebras R, C, H, O gives rise to the exceptional Lie algebras f4, e6, e7, e8, as well as to their affine, hyperbolic and further extensions.

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Reflection equation algebra in braided geometry 1

Dimitri GUREVICH , Pavel PYATOV , and Pavel SAPONOV

Braided geometry is a sort of the noncommutative geometry related to a braiding. The central role in this geometry is played by the reflection equation algebra associated with a braiding of the Hecke type. Using this algebra, we introduce braided versions of the Lie algebras gl(n) and sl(n). We further define braided analogs of the coadjoint orbits and the vector fields on a q-hyperboloid which is the simplest example of a ”braided orbit”. Besides, we present a braided version of the Cayley-Hamilton identity generalizing the result of Kantor and Trishin on the super-matrix characteristic identities.

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Local envelopes on CR manifolds 1

Egmont PORTEN

We study the problem whether CR functions on a sufficiently pseudoconcave CR manifold M extend locally across a hypersurface of M. The sharpness of the main result will be discussed by way of a counter-example.

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A non-abeliann nonlinear Schr Ã?Å?Ã?Ë?odinger equation and countable superposition of solitons 1

Cornelia SCHIEBOLD

We study solutions of an operator-valued NLS and apply our results to construct countable superpositions of solitons for the scalar NLS.
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A standard example in noncommutative deformation theory 1

Arvid SIQVELAND

In this paper we generalize the commutative generalized Massey products to the noncommutative deformation theory given by O. A. Laudal. We give an example illustrating the generalized Burnside theorem, one of the starting points in this noncommutative algebraic geometry.

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On constructing Hermitian unitary matrices with prescribed moduli 1

Alicja SMOKTUNOWICZ and Wojciech TADEJ

The problem of reconstructing the phases of a unitary matrix with prescribed moduli is of a broad interest to people working in many applications, e.g in the circuit theory, phase shift analysis, multichannel scattering, computer science (e.g in the theory of error correcting codes, design theory). We propose efficient algorithms for computing Hermitian unitary matrices for given symmetric bistochastic matrices A(n × n) for n = 3 and n = 4. We mention also some results for matrices of arbitrary size n.

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A Peirce decomposition for (-1,-1)-Freudenthal-Kantor triple systems

Noriaki KAMIYA

In this paper, we study a Peirce decomposition for (-1,-1)-Freudenthal-Kantor triple sys- tems and give several examples.

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On the structure of left and right F-, SM-, and E-quasigroups

Victor SHCHERBACOV

It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given; for example, nite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isotopic to the direct product of a group and a left S-loop (this is an answer to Belousov \1a" problem). Any left E-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(1) problem). As corollary it is obtained that any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.

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Connection on module over a graded q-differential algebra 1

Viktor ABRAMOV and Olga LIIVAPUU

We study a concept of a q-connection on a left module, where q is a primitive Nth root of unity. This concept is based on a notion of a graded q-differential algebra whose differential d satisfies dN = 0. We propose a notion of a graded q-differential algebra with involution and making use of this notion we introduce and study a concept of a q-connection consistent with a Hermitian structure of a left module. Assuming module to be a finitely generated free module we define the components of q-connection and show that these components with respect to different basises are related by gauge transformation. We also derive the relation for components of a q-connection consistent with Hermitian structure of a module.

Editorial Pages: 0 - 0

Algebra, Geometry, and Mathematical Physics

Goteborg

The intention of the AGMF worskhops is to further understanding of the fundamental role of non-commutative and non-associative structures in Mathematics, Theoretical and Mathematical Physics. The Goteborg meeting was dedicated to memory of Professor Isaiah Kantor (1936- 2006).

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Journal of Generalized Lie Theory and Applications

Efim ZELMANOV

Leo Landau once said that all physicists can be divided into physicists-composers and physicistsperformers. Issai Kantor is, in my opinion, a mathematician-composer. Here are two his compositions on the theme of the Jordan algebras.

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The purpose of this note is to show that many of the examples of quantum vertex operators do not satisfy vertex operator associativity.

Iana I. ANGUELOVA

The purpose of this note is to show that many of the examples of quantum vertex operators do not satisfy vertex operator associativity.

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Gauge dependence in the nonlinearly realized massive SU(2) gauge theory 1

Daniele BETTINELLI, Ruggero FERRARI and Andrea QUADRI

The implementation of the ’t Hooft ®-gauge in the symmetrically subtracted massive gauge theory based on the nonlinearly realized SU(2) gauge group is discussed. The gauge independence of the self-mass of the gauge bosons is proven by cohomological techniques.

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An algebraic approach to the center problem for ODEs 1

Alexander BRUDNYI

The classical Poincar´e Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem is reduced to a center problem for certain ODE . We present an algebraic approach to the center problem based on the study of the group of paths determined by the coefficients of the ODE.

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Ideals of the enveloping algebra U(osp(1, 2)) 1

C. BURDIK , M. HAVLI CEK , O. NAVRATIL , and S. POSTA

We explore the general form of two sided ideals of the enveloping algebra of the Lie superalgebra osp(1, 2). We begin by disclosing the internal structure of U(osp(1, 2)) computing the decomposition of adjoint representation. The classification of the ideals we reach is done via presenting generators for the each ideal and by showing that each ideal is generated uniquely.

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A multiplicative product of distributions and a global formulation of the confined Schr Ã?Å?Ã?Ë?odinger equation 1

A concise derivation of a new multiplicative product of Schwartz distributions is presented. The new product ? is defined in the vector space A of piecewise smooth functions on IR and all their (distributional) derivatives; it is associative, satisfies the Leibnitz rule and reproduces the usual product of functions for regular distributions. The algebra (A,+, ?) yields a sufficiently general setting to address some interesting problems. As an application we consider the problem of deriving a global formulation for quantum confined systems.
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From the Kadomtsev-Petviashvili equation halfway to Ward�¢����s chiral model 1

Aristophanes DIMAKIS and Folkert MULLER-HOISSEN

The “pseudodual” of Ward’s modified chiral model is a dispersionless limit of the matrix Kadomtsev-Petviashvili (KP) equation. This relation allows to carry solution techniques from KP over to the former model. In particular, lump solutions of the su(m) model with rather complex interaction patterns are reached in this way. We present a new example.
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Hopf algebras, random walks and quantum master equations 1

Hopf algebras of functions and operators are utilized to develop a mathematical construction scheme for building algebraic random walks. The main construction treats systems of covariance formed by translation operator and its associated operator valued measures on e.g. the circle and the line, and derives an algebraic quantum random walk by means of completely positive trace preserving maps. Asymptotic limit of the action of such maps is shown to lead to quantum master equations of Lindblad type.

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An example of noncommutative deformations 1

Eivind ERIKSEN

We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory.

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A characterization of a class of 2-groups by their defining relations 1

Tatjana GRAMUSHNJAK

Let n,m be integers such that n ¸ 3, m > 0 and Ck a cyclic group of order k. All groups which can be presented as a semidirect product (C2n+m × C2n) h C2 are described.

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