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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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