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Journal of Generalized Lie Theory and Applications

ISSN: 1736-4337

Open Access

Volume 3, Issue 4 (2009)

Review Article Pages: 0 - 0

Symmetric bundles and representations of Lie triple systems

Wolfgang BERTRAM and Manon DIDRY

We de ne symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying re ection space, and we investigate the corresponding forgetful functor both from the point of view of di erential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing \unusual" symmetric bundle structures on the tangent bundles of certain symmetric spaces.

Research Article Pages: 0 - 0

Unital algebras of Hom-associative type and surjective or injective twistings

Yael FREGIER, Aron GOHR, and Sergei SILVESTROV

In this paper, we introduce a common generalizing framework for alternative types of Hom-associative algebras. We show that the observation that unital Hom-associative algebras with surjective or injective twisting map are already associative has a generalization in this new framework. We also show by construction of a counterexample that another such generalization fails even in a very restricted particular case. Finally, we discuss an application of these observations by answering in the negative the question whether nonassociative algebras with unit such as the octonions may be twisted by the composition trick into Hom-associative algebras.

Research Article Pages: 0 - 0

Arithmetic Witt-hom-Lie algebras

Daniel LARSSON

This paper is concerned with explaining and further developing the rather technical de nition of a hom-Lie algebra given in a previous paper which was an adaption of the ordinary de nition to the language of number theory and arithmetic geometry. To do this we here introduce the notion of Witt-hom-Lie algebras and give interesting arithmetic applications, both in the Lie algebra case and in the hom-Lie algebra case. The paper ends with a discussion of a few possible applications of the developed hom-Lie language.

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A connection whose curvature is the Lie bracket

Kent E. MORRISON

Let G be a Lie group with Lie algebra g. On the trivial principal G-bundle over g there is a natural connection whose curvature is the Lie bracket of g. The exponential map of G is given by parallel transport of this connection. If G is the di eomorphism group of a manifold M, the curvature of the natural connection is the Lie bracket of vector elds on M. In the case that G = SO(3) the motion of a sphere rolling on a plane is given by parallel transport of a pullback of the natural connection by a map from the plane to so(3). The motion of a sphere rolling on an oriented surface in R3 can be described by a similar connection.

Research Article Pages: 0 - 0

On algebraic curves for commuting elements in q-Heisenberg algebras

Johan RICHTER and Sergei SILVESTROV

In the present article we continue investigating the algebraic dependence of commuting elements in q-deformed Heisenberg algebras. We provide a simple proof that the 0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that it coincides with the centralizer (commutant) of any one of its elements di erent from the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for proving algebraic dependence and obtaining corresponding algebraic curves for commuting elements in the q-deformed Heisenberg algebra by computing a certain determinant with entries depending on two commuting variables and one of the generators. The coe cients in front of the powers of the generator in the expansion of the determinant are polynomials in the two variables de ning some algebraic curves and annihilating the two commuting elements. We show that for the elements from the 0-chain subalgebra exactly one algebraic curve arises in the expansion of the determinant. Finally, we present several examples of computation of such algebraic curves and also make some observations on the properties of these curves.

Research Article Pages: 0 - 0

Matrix Bosonic realizations of a Lie colour algebra with three generators and ve relations of Heisenberg Lie type

Gunnar SIGURDSSON a and Sergei D. SILVESTROV

We describe realizations of a Lie colour algebra with three generators and ve relations by matrices of power series in noncommuting indeterminates satisfying Heisenberg's canonical commutation relation of quantum mechanics. The obtained formulas are used to construct new operator representations of this Lie colour algebra using canonical representation of the Heisenberg commutation relation and creation and annihilation operators of the quantum mechanical harmonic oscillator.

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