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On Lie algebras in braided categories

Bodo Pareigis (1997)

Banach Center Publications

The category of group-graded modules over an abelian group G is a monoidal category. For any bicharacter of G this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have n -ary multiplications between various graded components. They possess universal enveloping algebras that are Hopf algebras in the given category. Their biproducts with the group ring are noncommutative...

On maximal subalgebras of central simple Malcev algebras.

Alberto C. Elduque Palomo (1986)

Extracta Mathematicae

In this paper the structure of the maximal elements of the lattice of subalgebras of central simple non-Lie Malcev algebras is considered. Such maximal subalgebras are studied in two ways: first by using theoretical results concerning Malcev algebras, and second by using the close connection between these simple non-Lie Malcev algebras and the Cayley-Dickson algebras, which have been extensively studied (see [4]).

On normal abelian subgroups in parabolic groups

Gerhard Röhrle (1998)

Annales de l'institut Fourier

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical P u , and A a closed connected subgroup of P u which is normalized by P . We show that P acts on A with finitely many orbits provided A is abelian. This generalizes a well-known finiteness result, namely the case when A is central in P u . We also obtain an analogous result for the adjoint action of P on invariant linear subspaces of the Lie algebra of P u which are abelian Lie algebras. Finally, we discuss a connection...

On one-sided division infinite-dimensional normed real algebras.

José Antonio Cuenca Mira (1992)

Publicacions Matemàtiques

In this note we introduce the concept of Cayley homomorphism which is closely related with those of composition algebra and normalized orthogonal multiplication. The key result shows the existence of certain types of Cayley homomorphisms for infinite dimension. As an application we prove the existence of left division infinite-dimensional complete normed real algebras with left unity.

On quantum and classical Poisson algebras

Janusz Grabowski, Norbert Poncin (2007)

Banach Center Publications

Results on derivations and automorphisms of some quantum and classical Poisson algebras, as well as characterizations of manifolds by the Lie structure of such algebras, are revisited and extended. We prove in particular a somewhat unexpected fact that the algebras of linear differential operators acting on smooth sections of two real vector bundles of rank 1 are isomorphic as Lie algebras if and only if the base manifolds are diffeomorphic, whether or not the line bundles themselves are isomorphic....

On quantum weyl algebras and generalized quons

WŁadysŁaw Marcinek (1997)

Banach Center Publications

The model of generalized quons is described in an algebraic way as certain quasiparticle states with statistics determined by a commutation factor on an abelian group. Quantization is described in terms of quantum Weyl algebras. The corresponding commutation relations and scalar product are also given.

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