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Realization of bi-covariant differential calculus on the Lie algebra type noncommutative spaces. (English) Zbl 1370.83064
Summary: This paper investigates bi-covariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra \(\mathfrak{g}_0\), we construct a Lie superalgebra \(\mathfrak{g} = \mathfrak{g}_0 \oplus \mathfrak{g}_1\) containing noncommutative coordinates and one-forms. We show that \(\mathfrak{g}\) can be extended by a set of generators \(T_{AB}\) whose action on the enveloping algebra \(U(\mathfrak{g})\) gives the commutation relations between monomials in \(U(\mathfrak{g}_0)\) and one-forms. Realizations of noncommutative coordinates, one-forms, and the generators \(T_{AB}\) as formal power series in a semicompleted Weyl superalgebra are found. In the special case \(\dim(\mathfrak{g}_0) = \dim(\mathfrak{g}_1)\), we also find a realization of the exterior derivative on \(U(\mathfrak{g}_0)\). The realizations of these geometric objects yield a bi-covariant differential calculus on \(U(\mathfrak{g}_0)\) as a deformation of the standard calculus on the Euclidean space.
©2017 American Institute of Physics

83C65 Methods of noncommutative geometry in general relativity
17A70 Superalgebras
17B35 Universal enveloping (super)algebras
83C45 Quantization of the gravitational field
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