Geodesic
Universal property of skew $PBW$ extensions
Algebra and discrete mathematics, Tome 20 (2015) no. 1, pp. 1-12.

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In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of non-commutative rings. Skew $PBW$ extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others. As a corollary we will give a new short proof of the Poincaré-Birkhoff-Witt theorem about the bases of enveloping algebras of finite-dimensional Lie algebras.
Keywords: skew polynomial rings, skew $PBW$ extensions, $PBW$ bases, quantum algebras. 
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Juan Pablo Acosta; Oswaldo Lezama. Universal property of skew $PBW$ extensions. Algebra and discrete mathematics, Tome 20 (2015) no. 1, pp. 1-12. http://geodesic.mathdoc.fr/item/ADM_2015_20_1_a1/

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