It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good reference for the general theory of SAGBI/Gröbner bases for non-commutative setting. Actually, I am interesting in the following subjects:


(1) Where can I find the general construction of these bases for non-commutative setting?


(2) Is there some reference for the specific case of the Universal enveloping algebra of a finite-dimensional semi-simple Lie algebra over $\mathbb{C}$?


(3) Does exist a reference using this bases in representation theory? For instance, in the study of universal objects defined by generators and relations.


THANKS!



Gröbner/SAGBI bases for non-commutative setting
In the non-commutative situation, it is called Groebner-Shirshov basis. See, for example, the survey paper "[Groebner-Shirshov Bases: Some New Results](https://arxiv.org/abs/0804.1344 "Proceedings of the 2nd international congress of Algebras and Combinatorics, World Scientific, 2008, 35-56, doi:10.1142/9789812790019_0003. zbMATH review at https://zbmath.org/1207.16023")" by L. A. Bokut and Yuqun Chen, and references there. The case of universal enveloping algebra was considered by Bergman in G. M. Bergman, [The diamond lemma for ring theory](https://doi.org/10.1016/0001-8708(78)90010-5 "zbMATH review at https://zbmath.org/0326.16019"), 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 a finite Groebner-Shirshov basis (unlike Groebner basis in the commutative case) does not always exist.


