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The Weyl realizations of Lie algebras, and left-right duality. (English) Zbl 1342.58005
Summary: We investigate dual realizations of non-commutative spaces of Lie algebra type in terms of formal power series in the Weyl algebra. To each realization of a Lie algebra we associate a star-product on the symmetric algebra $$S(\mathfrak{g})$$ and an ordering on the enveloping algebra $$U(\mathfrak{g})$$. Dual realizations of are defined in terms of left-right duality of the star-products on $$S(\mathfrak{g})$$. It is shown that the dual realizations are related to an extension problem for $$\mathfrak{g}$$ by shift operators whose action on $$U(\mathfrak{g})$$ describes left and right shift of the generators of $$U(\mathfrak{g})$$ in a given monomial. Using properties of the extended algebra, in the Weyl symmetric ordering we derive closed form expressions for the dual realizations of $$\mathfrak{g}$$ in terms of two generating functions for the Bernoulli numbers. The theory is illustrated by considering the $$\kappa$$-deformed space.
©2016 American Institute of Physics

##### MSC:
 58B34 Noncommutative geometry (à la Connes) 11L15 Weyl sums 53D55 Deformation quantization, star products 11B68 Bernoulli and Euler numbers and polynomials
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