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Bases in Lie and Quantum Algebras
Applications of algebras in physics are related to the connection of
measurable observables to relevant elements of the algebras, usually the
generators. However, in the determination of the generators in Lie algebras
there is place for some arbitrary conventions. The situation is much more
involved in the context of quantum algebras, where inside the quantum universal
enveloping algebra, we have not enough primitive elements that allow for a
privileged set of generators and all basic sets are equivalent. In this paper
we discuss how the Drinfeld double structure underlying every simple Lie
bialgebra characterizes uniquely a particular basis without any freedom,
completing the Cartan program on simple algebras. By means of a perturbative
construction, a distinguished deformed basis (we call it the analytical basis)
is obtained for every quantum group as the analytical prolongation of the above
defined Lie basis of the corresponding Lie bialgebra. It turns out that the
whole construction is unique, so to each quantum universal enveloping algebra
is associated one and only one bialgebra. In this way the problem of the
classification of quantum algebras is moved to the classification of
bialgebras. In order to make this procedure more clear, we discuss in detail
the simple cases of su(2) and su_q(2).Comment: 16 pages, Proceedings of the 5th International Symposium on Quantum
Theory and Symmetries QTS5 (July 22-28, 2007, Valladolid (Spain)
Skew-adjoint maps and quadratic Lie algebras
The procedure of double extension of vector spaces endowed with
non-degenerate bilinear forms allows us to introduce the class of generalized
\mbK-oscillator algebras over any arbitrary field \mbK. Starting from basic
structural properties of such algebras and the canonical forms of skew-adjoint
endomorphisms, we will proceed to classify the subclass of quadratic nilpotent
algebras and characterize those algebras in the class with quadratic dimension
2. This will enable us to recover the well-known classification of real
oscillator algebras, also known as Lorentzian algebras, given by Alberto Medina
in 1985.Comment: 24 page
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