In the non-commutative situation, it is called Groebner-Shirshov basis. See, for example, this survey paper and references there. The case of universal enveloping algebra was considered by Bergman in G. M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29, 178-218(1978). I do not know about representation theory, but certainly this is one of the main tools of studying finitely presented rings and algebras over a field. The problem, of course, is that the Groebner-Shirshov basis (unlike Groebner basis in the commutative case) does not always exist.
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