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Fundam. Prikl. Mat., 2000, Volume 6, Issue 3, Pages 669–706 (Mi fpm497)  

This article is cited in 29 scientific papers (total in 29 papers)

Gröbner and Gröbner–Shirshov bases in algebra and conformal algebras

L. A. Bokut'a, Yu. Fongb, W.-F. Keb, P. S. Kolesnikova

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b National Cheng Kung University

Abstract: In this paper the Gröbner–Shirshov bases theory is regularly presented for commutative, non-commutative, Lie and conformal algebras. The general form of Composition-Diamond lemma for conformal relations is stated. We have made a review of some results obtained with Gröbner–Shirshov bases of usual and conformal algebras. It is proved that every finitely generated commutative conformal algebra is Noetherian, an analogue of Specht problem is considered for commutative conformal algebras.

Full text: PDF file (1663 kB)

Bibliographic databases:
UDC: 512.55+512.62
Received: 01.09.2000

Citation: L. A. Bokut', Yu. Fong, W. Ke, P. S. Kolesnikov, “Gröbner and Gröbner–Shirshov bases in algebra and conformal algebras”, Fundam. Prikl. Mat., 6:3 (2000), 669–706

Citation in format AMSBIB
\Bibitem{BokFonKe00}
\by L.~A.~Bokut', Yu.~Fong, W.~Ke, P.~S.~Kolesnikov
\paper Gr\"obner and Gr\"obner--Shirshov bases in algebra and conformal algebras
\jour Fundam. Prikl. Mat.
\yr 2000
\vol 6
\issue 3
\pages 669--706
\mathnet{http://mi.mathnet.ru/fpm497}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1801321}
\zmath{https://zbmath.org/?q=an:0990.17007}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Poroshenko, EN, “Grobner-Shirshov bases for the Kac-Moody algebras of the type A(n)((1))”, Communications in Algebra, 30:6 (2002), 2617  crossref  mathscinet  zmath  isi
    2. R. M. Garipov, “Klassifikatsiya po izomorfizmu kristallograficheskikh grupp na psevdoevklidovoi ploskosti. I. Obschii sluchai”, Sib. zhurn. industr. matem., 6:4 (2003), 11–31  mathnet  mathscinet  zmath
    3. Kolesnikov, P, “Associative enveloping pseudoalgebras of finite Lie pseudoalgebras”, Communications in Algebra, 31:6 (2003), 2909  crossref  mathscinet  zmath  isi
    4. Bokut L., Fong Y., Shiao L.S., “Grobner-Shirshov bases for algebras, groups, and semigroups”, Proceedings of the Third International Algebra Conference, 2003, 17–32  crossref  mathscinet  zmath  isi
    5. P. S. Kolesnikov, “Gröbner–Shirshov Bases for Universal Enveloping Conformal Algebras of Simple Conformal Lie Superalgebras of Type $W_N$”, Algebra and Logic, 43:2 (2004), 109–122  mathnet  crossref  mathscinet  zmath
    6. D. A. Mikhailov, A. A. Nechaev, “Solving systems of polynomial equations over Galois–Eisenstein rings with the use of the canonical generating systems of polynomial ideals”, Discrete Math. Appl., 14:1 (2004), 41–73  mathnet  crossref  crossref  mathscinet  zmath
    7. Bokut, LA, “Composition-Diamond Lemma for associative conformal algebras”, Journal of Algebra, 272:2 (2004), 739  crossref  mathscinet  zmath  isi
    8. Drensky, V, “Grobner bases of ideals invariant under endomorphisms”, Journal of Symbolic Computation, 41:7 (2006), 835  crossref  mathscinet  zmath  isi
    9. Kolesnikov, PS, “Simple associative conformal algebras of linear growth”, Journal of Algebra, 295:1 (2006), 247  crossref  mathscinet  zmath  isi
    10. Bokut L., Chibrikov E.S., “Lyndon-Shirshov words, Grobner-Shirshov bases, and free Lie algebras”, Non-Associative Algebra and its Applications, Monographs and Textbooks in Pure and Applied Mathematics, 246, 2006, 17–39  mathscinet  zmath  isi
    11. Kolesnikov, P, “On the wedderburn principal theorem in conformal algebras”, Journal of Algebra and Its Applications, 6:1 (2007), 119  crossref  mathscinet  zmath  isi
    12. L. A. Bokut, Yuqun Chen, Yu Li, “Gröbner–Shirshov bases for Vinberg–Koszul–Gerstenhaber right-symmetric algebras”, J. Math. Sci., 166:5 (2010), 603–612  mathnet  crossref  mathscinet  elib  elib
    13. I. A. Dolguntseva, “Triviality of the second cohomology group of the conformal algebras $\mathrm{Cend}_n$ and $\mathrm{Cur}_n$”, St. Petersburg Math. J., 21:1 (2010), 53–63  mathnet  crossref  mathscinet  zmath  isi
    14. Bokut L.A., Chen Yu., Qiu J., “Grobner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras”, Journal of Pure and Applied Algebra, 214:1 (2010), 89–100  crossref  mathscinet  zmath  isi
    15. A. N. Koryukin, “Gröbner–Shirshov bases of the Lie algebra $D^+_n$”, St. Petersburg Math. J., 22:4 (2011), 573–614  mathnet  crossref  mathscinet  zmath  isi
    16. L. A. Bokut, Yu. Chen, X. Deng, “Gröbner–Shirshov bases for Rota–Baxter algebras”, Siberian Math. J., 51:6 (2010), 978–988  mathnet  crossref  mathscinet  isi
    17. E. N. Poroshenko, “Bases for partially commutative Lie algebras”, Algebra and Logic, 50:5 (2011), 405–417  mathnet  crossref  mathscinet  zmath  isi
    18. Chen Yu., Li Yu., “Some Remarks on the Akivis Algebras and the Pre-Lie Algebras”, Czechoslovak Math J, 61:3 (2011), 707–720  crossref  mathscinet  zmath  isi  elib
    19. Bokut L.A., Chen Yu., Chen Y., “Grobner-Shirshov bases for Lie algebras over a commutative algebra”, J Algebra, 337:1 (2011), 82–102  crossref  mathscinet  zmath  isi  elib
    20. Leonid A. Bokut, Yuqun Chen, “Gröbner–Shirshov bases and PBW theorems”, Zhurn. SFU. Ser. Matem. i fiz., 6:4 (2013), 417–427  mathnet
    21. Bokut L.A. Chen Yu., “Grobner-Shirshov Bases and Their Calculation”, Bull. Math. Sci., 4:3 (2014), 325–395  crossref  mathscinet  zmath  isi  elib
    22. Bokut L.A., Chen Yu., Chen W., Li J., “New Approaches To Plactic Monoid Via Grobner-Shirshov Bases”, J. Algebra, 423 (2015), 301–317  crossref  mathscinet  zmath  isi  elib
    23. Qiu J., Chen Yu., “Grobner-Shirshov Bases For Lie Omega-Algebras and Free Rota-Baxter Lie Algebras”, J. Algebra. Appl., 16:10 (2017), 1750190  crossref  mathscinet  zmath  isi  scopus
    24. Zhang G., Chen Yu., “A New Composition-Diamond Lemma For Dialgebras”, Algebr. Colloq., 24:2 (2017), 323–350  crossref  mathscinet  zmath  isi  scopus
    25. Ni L., Chen Yu., “A New Composition-Diamond Lemma For Associative Conformal Algebras”, J. Algebra. Appl., 16:5 (2017), 1750094  crossref  mathscinet  zmath  isi  scopus
    26. Li Y., Mo Q.H., “Some New Results For Leibniz Algebras and Non-Associative Algebras”, Southeast Asian Bull. Math., 41:1 (2017), 45–54  mathscinet  isi
    27. Bokut L.A., Chen Yu., Obul A., “Some New Results on Grobner-Shirshov Bases For Lie Algebras and Around”, Int. J. Algebr. Comput., 28:8, SI (2018), 1403–1423  crossref  mathscinet  zmath  isi  scopus
    28. Tuniyaz R., Bokut L.A., Xiryazidin M., Obul A., “Grobner-Shirshov Bases For Free Gelfand-Dorfman-Novokov Algebras and For Right Ideals of Free Right Leibniz Algebras”, Commun. Algebr., 46:10 (2018), 4392–4402  crossref  mathscinet  zmath  isi  scopus
    29. R. A. Kozlov, “Kogomologii Khokhshilda assotsiativnoi konformnoi algebry $\mathrm{Cend}_{1,x}$”, Algebra i logika, 58:1 (2019), 52–68  mathnet  crossref
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