Gröbner–Shirshov bases and PBW theorems
Leonid A. Bokut, Yuqun Chen

References
1.	G. M. Bergman, “The diamond lemma for ring theory”, Adv. in Math., 29 (1978), 178–218
2.	G. M. Bergman, Privite communication, 2013
3.	L. A. Bokut, “On one property of the Boone group”, Algebra Logika, 5:5 (1966), 5–23 (in Russian)
4.	L. A. Bokut, “On the Novikov groups”, Algebra Logika, 6:1 (1967), 25–38 (in Russian)
5.	L. A. Bokut, “Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras”, Izv. Akad. Nauk SSSR Ser. Mat., 36:6 (1972), 1173–1219 (in Russian)
6.	L. A. Bokut, “Imbeddings into simple associative algebras”, Algebra Logika, 15:2 (1976), 117–142 (in Russian)
7.	L. A. Bokut, “Gröbner–Shirshov bases for braid groups in Artin–Garside generators”, J. Symbolic Computation, 43 (2008), 397–405
8.	L. A. Bokut, “Gröbner–Shirshov bases for the braid group in the Birman–Ko–Lee generators”, J. Algebra, 321 (2009), 361–379
9.	L. A. Bokut, V. V. Chainikov, K. P. Shum, “Markov and Artin normal form theorem for braid groups”, Commun. Algebra, 35 (2007), 2105–2115
10.	L. A. Bokut, V. Chainikov, “Gröbner–Shirshov bases of Adjan extension of the Novikov group”, Discrete Mathematics, 308 (2008), 4916–4930
11.	L. A. Bokut, Y. Q. Chen, “Gröbner–Shirshov bases for Lie algebras: after A. I. Shirshov”, Southeast Asian Bull. Math., 31 (2007), 1057–1076
12.	L. A. Bokut, Y. Q. Chen, “Gröbner–Shirshov bases: some new results”, Advance in Algebra and Combinatorics, Proceedings of the Second International Congress in Algebra and Combinatorics, eds. K. P. Shum, E. Zelmanov, Jiping Zhang, Li Shangzhi, World Scientific, 2008, 35–56
13.	L. A. Bokut, Y. Q. Chen, Y. S. Chen, “Composition-Diamond lemma for tensor product of free algebras”, J. Algebra, 323 (2010), 2520–2537
14.	L. A. Bokut, Y. Q. Chen, Y. S. Chen, “Gröbner–Shirshov bases for Lie algebras over a commutative algebra”, J. Algebra, 337 (2011), 82–102
15.	L. A. Bokut, Y. Q. Chen, W. P. Chen, J. Li, Gröbner–Shirshov bases for plactic monoids, Preprint, 2012
16.	L. A. Bokut, Y. Q. Chen, X. M. Deng, “Gröbner–Shirshov bases for Rota–Baxter algebras”, Siberian Math. J., 51:6 (2010), 978–988
17.	L. A. Bokut, Y. Q. Chen, J. P. Huang, “Gröbner–Shirshov bases for $L$-algebras”, Internat. J. Algebra Comput., 23 (2013), 547–571
18.	L. A. Bokut, Y. Q. Chen, Y. Li, “Gröbner–Shirshov bases for Vinberg–Koszul–Gerstenhaber right-symmetric algebras”, J. Math. Sci., 166 (2010), 603–612
19.	L. A. Bokut, Y. Q. Chen, Y. Li, “Anti-commutative Gröbner–Shirshov basis of a free Lie algebra”, Science in China Series A: Mathematics, 52 (2009), 244–253
20.	L. A. Bokut, Y. Q. Chen, Y. Li, “Gröbner–Shirshov bases for categories”, Nankai Series in Pure, Applied Mathematics and Theoretical Physical, Operads and Universal Algebra, 9 (2012), 1–23
21.	L. A. Bokut, Y. Q. Chen, Y. Li, “Lyndon–Shirshov words and anti-commutative algebras”, J. Algebra, 378 (2013), 173–183
22.	L. A. Bokut, Y. Q. Chen, C. H. Liu, “Gröbner–Shirshov bases for dialgebras”, Internat. J. Algebra Comput., 20 (2010), 391–415
23.	L. A. Bokut, Y. Q. Chen, Q. H. Mo, “Gröbner–Shirshov bases and embeddings of algebras”, Internat. J. Algebra Comput., 20 (2010), 875–900
24.	L. A. Bokut, Y. Q. Chen, Q. H. Mo, “Gröbner–Shirshov bases for semirings”, J. Algebra, 378 (2013), 47–63
25.	L. A. Bokut, Y. Q. Chen, J. J. Qiu, “Gröbner–Shirshov bases for associative algebras with multiple operations and free Rota–Baxter algebras”, J. Pure Applied Algebra, 214 (2010), 89–100
26.	L. A. Bokut, Y. Q. Chen, K. P. Shum, “Some new results on Gröbner–Shirshov bases”, Proceedings of International Conference on Algebra 2010, Advances in Algebraic Structures, 2012, 53–102
27.	L. A. Bokut, Y. Q. Chen, G. L. Zhang, Composition-Diamond lemma for associative $n$-conformal algebras, arXiv: 0903.0892
28.	L. A. Bokut, Y. Q. Chen, X. G. Zhao, “Gröbner–Shirshov beses for free inverse semigroups”, Internat. J. Algebra Comput., 19 (2009), 129–143
29.	L. A. Bokut, Y. Fong, W.-F. Ke, “Composition-Diamond lemma for associative conformal algebras”, J. Algebra, 272 (2004), 739–774
30.	L. A. Bokut, Y. Fong, W.-F. Ke, P. S. Kolesnikov, “Gröbner and Gröbner–Shirshov bases in algebra and conformal algebras”, Fundamental and Applied Mathematics, 6:3 (2000), 669–706
31.	L. A. Bokut, S.-J. Kang, K.-H. Lee, P. Malcolmson, “Gröbner–Shirshov bases for Lie superalgebras and their universal enveloping algebras”, J. Algebra, 217 (1999), 461–495
32.	L. A. Bokut, P. S. Kolesnikov, “Gröbner–Shirshov bases: from their incipiency to the present”, J. Math. Sci., 116 (2003), 2894–2916
33.	L. A. Bokut, P. S. Kolesnikov, “Gröbner–Shirshov bases, conformal algebras and pseudo-algebras”, J. Math. Sci., 131 (2005), 5962–6003
34.	L. A. Bokut, P. Malcolmson, “Gröbner–Shirshov bases for Lie and associative algebras”, Collection of Abstracts, ICAC'97, Hong Kong, 1997, 139–142
35.	L. A. Bokut, P. Malcolmson, “Gröbner–Shirshov bases for relations of a Lie algebra and its enveloping algebra”, Algebras and combinatorics, Papers from the international congress, ICAC'97 (Hong Kong, August 1997), eds. Shum Kar-Ping et al., Springer, Singapore, 1999, 47–54
36.	L. A. Bokut, K. P. Shum, “Relative Gröbner–Shirshov bases for algebras and groups”, St. Petersbg. Math. J., 19:6 (2008), 867–881
37.	B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, Ph. D. thesis, University of Innsbruck, Austria, 1965
38.	B. Buchberger, “An algorithmical criteria for the solvability of algebraic systems of equations”, Aequationes Math., 4 (1970), 374–383
39.	P. Cartier, “Remarques sur le théorème de Birkhoff–Witt”, Annali della Scuola Norm. Sup. di Pisa, Série III, 12 (1958), 1–4
40.	Y. Q. Chen, “Gröbner–Shirshov basis for Schreier extensions of groups”, Commun. Algebra, 36 (2008), 1609–1625
41.	Y. Q. Chen, “Gröbner–Shirshov basis for extensions of algebras”, Algebra Colloq., 16 (2009), 283–292
42.	Y. S. Chen, Y. Q. Chen, “Gröbner–Shirshov bases for matabelian Lie algebras”, J. Algebra, 358 (2012), 143–161
43.	Y. Q. Chen, Y. S. Chen, Y. Li, “Composition-Diamond lemma for differential algebras”, The Arabian Journal for Science and Engineering, 34 (2009), 135–145
44.	Y. Q. Chen, W. S. Chen, R. I. Luo, “Word problem for Novikov's and Boone's group via Gröbner–Shirshov bases”, Southeast Asian Bull. Math., 32 (2008), 863–877
45.	Y. Q. Chen, Y. S. Chen, C. Y. Zhong, “Composition-Diamond lemma for modules”, Czechoslovak Math. J., 60 (2010), 59–76
46.	Y. Q. Chen, Y. Li, “Some remarks for the Akivis algebras and the Pre-Lie algebras”, Czechoslovak Math. J., 61:136 (2011), 707–720
47.	Y. Q. Chen, Y. Li, Q. Y. Tang, Gröbner–Shirshov bases for some Lie algebras, Preprint, 2012
48.	Y. Q. Chen, Q. H. Mo, “Artin-Markov normal form for braid group”, Southeast Asian Bull. Math., 33 (2009), 403–419
49.	Y. Q. Chen, Q. H. Mo, “Embedding dendriform algebra into its universal enveloping Rota–Baxter algebra”, Proc. Am. Math. Soc., 139 (2011), 4207–4216
50.	Y. Q. Chen, Q. H. Mo, “Gröbner–Shirshov bases for free partially commutative Lie algebras”, Commun. Algebra, 41 (2013), 3753–3761
51.	Y. Q. Chen, J. J. Qiu, “Gröbner–Shirshov basis for the Chinese monoid”, Journal of Algebra and its Applications, 7 (2008), 623–628
52.	Y. Q. Chen, H. S. Shao, K. P. Shum, “On Rosso–Yamane theorem on PBW basis of $U_q(A_N)$”, CUBO A Mathematical Journal, 10 (2008), 171–194
53.	Y. Q. Chen, M. M. Yang, A Gröbner–Shirshov basis for free idempoten semigroup, Preprint, 2012
54.	Y. Q. Chen, C. Y. Zhong, “Gröbner–Shirshov basis for HNN extensions of groups and for the alternative group”, Commun. Algebra, 36 (2008), 94–10
55.	Y. Q. Chen, C. Y. Zhong, “Gröbner–Shirshov basis for some one-relator groups”, Algebra Colloq., 19 (2011), 99–116
56.	Y. Q. Chen, C. Y. Zhong, “Gröbner–Shirshov bases for braid groups in Adyan-Thurston generators”, Algebra Colloq., 20 (2013), 309–318
57.	P. M. Cohn, “A remark on the Birkhoff-Witt theorem”, Journal London Math. Soc., 38 (1963), 197–203
58.	Marshall Hall (Jr.), The Theory of Groups, The Macmillan Company, 1959
59.	E. N. Poroshenko, “Bases for partially commutative Lie algebras”, Algebra Logika, 50:5 (2011), 405–417 (in Russian)
60.	J. J. Qiu, Y. Q. Chen, “Composition-Diamond lemma for $\lambda$-differential associative algebras with multiple operators”, Journal of Algebra and its Applications, 9 (2010), 223–239
61.	J. J. Qiu, “Gröbner–Shirshov bases for commutative algebras with multiple operators and free commutative Rota–Baxter algebras”, Asian-European Jour. Math. (to appear)
62.	A. I. Shirshov, “On the representation of Lie rings in associative rings”, Uspekhi Mat. Nauk N.S., 8:5(57) (1953), 173–175 (in Russian)
63.	SIGSAM Bull., 33 (1999), 3–6
64.	Bokut L. A., Latyshev V., Shestakov I., Zelmanov E., Bremner Trs. M., Kochetov M. (Eds.), Selected works of A. I. Shirshov, Birkhäuse, Basel–Boston–Berlin, 2009