Algebraic deformation quantization of Leibniz algebras

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. We construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra.Our motivation is deformation quantization of Leibniz algebras in the sense of [6 Dherin, B., Wagemann, F. (2015). Deformation quantization of Leibniz algebras. Adv. Math. 270 :21–48.[Crossref] , [Google Scholar]]. Namely, the canonical rack bialgebras we have constructed for any Leibniz algebra lead to a simple explicit formula of the rack-star-product on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann in [6 Dherin, B., Wagemann, F. (2015). Deformation quantization of Leibniz algebras. Adv. Math. 270 :21–48.[Crossref] , [Google Scholar]]. We clarify this framework setting up a general deformation theory for rack bialgebras and show that the rack-star-product turns out to be a deformation of the trivial rack bialgebra product.

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