alexa Leibniz rules for enveloping algebras and a diagrammatic expansion
Mathematics

Mathematics

Journal of Generalized Lie Theory and Applications

Author(s): Stjepan Meljanac, Zoran Skoda

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Given a nite-dimensional Lie algebra, and a representation by derivations on the completed symmetric algebra of its dual, a number of interesting twisted constructions appear: certain twisted Weyl algebras, deformed Leibniz rules, quantized \star" product. We rst illuminate a number of interrelations be- tween these constructions and then proceed to study a special case in certain precise sense corresponding to the symmetric ordering. It has been known earlier that this case is related to the computations with Hausdor series, for example an expression for the star product is in such terms. For the deformed Leib- niz rule, hence a coproduct, we present here a new nonsymmetric expression, which is then expanded into a sum of expressions labelled by a class of planar trees, and for a given tree evaluated by Feynman-like rules. These expressions are graded by a bidegree and we show recursion formulas for the component of xed bidegree, and compare the recursion to the recursions for Hausdor series, including a nontrivial comparison of initial conditions. This way we show a di- rect correspondence between the Hausdor series and the expression for twisted coproduct. The expansion labelled by planar series in particular gives a new way to reorganize the Hausdor series

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This article was published in Elsevier and referenced in Journal of Generalized Lie Theory and Applications

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