Geodesic
Gr\"obner--Shirshov bases of the Lie algebra $D^+_n$
Algebra i analiz, Tome 22 (2010) no. 4, pp. 76-136.

Voir la notice de l'article provenant de la source Math-Net.Ru

Over a field of characteristic 0, the reduced Gröbner–Shirshov bases (RGShB) are computed in the positive part $D_n^+$ of the simple finite-dimensional Lie algebra $D_n$ for the canonical generators corresponding to simple roots, under an arbitrary ordering of these generators (i.e., an aritrary basis among the $n!$ bases is fixed and analyzed). In this setting, the RGShBs were previously computed by the author for the Lie algebras $A_n^+$, $B_n^+$, and $C_n^+$. For one ordering of the generators, the RGShBs of these algebras were calculated by Bokut and Klein (1996).
Keywords: Gröbner–Shirshov, bases Lie algebras. 
@article{AA_2010_22_4_a3,
     author = {A. N. Koryukin},
     title = {Gr\"obner--Shirshov bases of the {Lie} algebra $D^+_n$},
     journal = {Algebra i analiz},
     pages = {76--136},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2010_22_4_a3/}
}
TY  - JOUR
AU  - A. N. Koryukin
TI  - Gr\"obner--Shirshov bases of the Lie algebra $D^+_n$
JO  - Algebra i analiz
PY  - 2010
SP  - 76
EP  - 136
VL  - 22
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2010_22_4_a3/
LA  - ru
ID  - AA_2010_22_4_a3
ER  - 
%0 Journal Article
%A A. N. Koryukin
%T Gr\"obner--Shirshov bases of the Lie algebra $D^+_n$
%J Algebra i analiz
%D 2010
%P 76-136
%V 22
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2010_22_4_a3/
%G ru
%F AA_2010_22_4_a3
A. N. Koryukin. Gr\"obner--Shirshov bases of the Lie algebra $D^+_n$. Algebra i analiz, Tome 22 (2010) no. 4, pp. 76-136. http://geodesic.mathdoc.fr/item/AA_2010_22_4_a3/

[1] Shirshov A. I., “Nekotorye algoritmicheskie problemy dlya algebr Li”, Sib. mat. zh., 3:2 (1962), 292–296 | Zbl

[2] Buchberger B., An algorithm for finding a bases for the residue class ring of a zero-dimensional polynomial ideal, Ph. D. Thesis, Univ. Innsburg, Avstria, 1965 (German)

[3] Shirshov A. I., “Ob odnoi gipoteze teorii algebr Li”, Sib. mat. zh., 3:2 (1962), 297–301 | Zbl

[4] Bokut L. A., “Vlozhenie algebr Li v algebraicheski zamknutye algebry Li”, Doklady Akademii Nauk SSSR, 154:5 (1962), 963–964 | MR

[5] Buchberger B., Winkler F. (eds.), Gröbner bases and applications, London Math. Soc. Lecture Notes Ser., 251, Cambridge Univ. Press, Cambridge, 1998 | MR

[6] Gerdt V. P., Lassner W., “Isomorphism verification for complex and real Lie algebras by Gröbner basis technique”, Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics (Acireale, 1992), eds. N. H. Ibragimov et al., Kluwer Acad. Publ., Dordrecht, 1993, 245–254 | MR | Zbl

[7] Bokut L. A., Klein A. A., “Serre relations and Gröbner–Shirshov bases for simple Lie algebras, I”, Internat. J. Algebra Comput., 6:4 (1996), 389–400 | DOI | MR | Zbl

[8] Bokut L. A., Klein A. A., “Serre relations and Gröbner–Shirshov bases for simple Lie algebras, II”, Internat. J. Algebra Comput., 6:4 (1996), 401–412 | DOI | MR | Zbl

[9] Borkut L. A., Klein A. A., “Gröbner–Shirshov bases for exceptional Lie algebras. I”, Ring Theory (Miskolc, 1996), J. Pure Appl. Algebra, 133 (1998), 51–57 | DOI | MR

[10] Bokut L. A., Klein A. A., “Gröbner–Shirshov bases for exceptional Lie algebras $E_6$, $E_7$, $E_8$”, Algebras and Combinatorics (Hong Kong, 1997), Springer, Singapore, 1999, 37–46 | MR | Zbl

[11] Bokut L. A., Malcolmson P., “Gröbner–Shirshov bases for quantum enveloping algebras”, Israel J. Math., 96 (1996), 97–113 | DOI | MR | Zbl

[12] Bokut L. A., Malcolmson P., “Gröbner–Shirshov bases for relations of Lie algebra and its enveloping algebra”, Algebras and Combinatorics (Hong Kong, 1997), Springer, Singapore, 1999, 47–54 | MR | Zbl

[13] Gateva-Ivanova T., Latyshev V., “On recognisable properties of associative algebras. Commutational aspects of commutative algebra”, J. Symbolic Comput., 6:2–3 (1998), 371–388 | MR

[14] Mikhalëv A. A., “Lemma o sliyanii i problema ravenstva dlya tsvetnykh superalgebr Li”, Vestn. MGU. Ser. I Mat. Mekh., 1989, no. 5, 88–91 ; “Подалгебры свободных $p$-супералгебр Ли”, Мат. заметки, 43:2 (1988), 178–191 | MR | Zbl | MR | Zbl

[15] Mikhalëv A. A., “Tekhnika kompozitsii A. I. Shirshova v superalgebrakh Li (nekommutativnye bazisy Grëbnera)”, Tr. Semin. im. I. G. Petrovskogo, 18, 1995, 277–289 | MR | Zbl

[16] Mikhalev A. A., “The composition lemma for color Lie superalgebras and for Lie $p$-superalgebras”, Proc. of the Internat. Conf. on Algebra (Novosibirsk, 1989), Contemp. Math., 131, Amer. Math. Soc., Providence, RI, 1992, 91–104 | MR

[17] Bokut L. A., Kang S.-J., Lee K.-H., Malcolmson P., “Gröbner–Shirshov bases for Lie superalgebras and their universal enveloping algebras”, J. Algebra, 217 (1999), 461–495 | DOI | MR | Zbl

[18] Kang S.-J., Lee K.-H., “Gröbner–Shirshov bases for representation theory”, J. Korean Math. Soc., 37 (2000), 55–72 | MR | Zbl

[19] Kang S.-J., Lee K.-H., “Gröbner–Shirshov bases for irreducible $sl_{n+1}$-modules”, J. Algebra, 232:1 (2000), 1–20 | MR | Zbl

[20] Chibrikov E. S., “O svobodnykh konformnykh algebrakh Li”, Vestn. Novosib. gos. un-ta. Ser. Mat., mekh., inf., 4:1 (2004), 65–83

[21] Roitman M., “On free conformal and vertex algebras”, J. Algebra, 217 (1999), 496–527 | DOI | MR | Zbl

[22] Roitman M., “Universal enveloping conformal algebras”, Selecta Math. (to appear)

[23] Bokut L. A., Fong Y., Ke W.-F., “Free associative conformal algebras”, Proc. of the 2nd Tainan–Moscow Algebra and Combinatorics Workshop (Tainan, 1997), Springer-Verlag, Hong Kong, 2000, 13–25 | Zbl

[24] Bokut L. A., Fong Y., Ke W.-F., “Composition–Diamond lemma for associative conformal algebras”, J. Algebra, 272:2 (2004), 739–774 | MR | Zbl

[25] Vasiliev N. N., Pavlov D., “Enumeration of finite monomial orders and Young tableux”, Internat. Conf. Polynom. Comput. Algebra, 2008, 71–76 | MR

[26] Bokut L. A., Ke V.-F., Kolesnikov P. S., Fong Yu., “Bazisy Grëbnera i Grëbnera–Shirshova v algebre i konformnykh algebrakh”, Fundam. i prikl. mat., 6:3 (2000), 669–706 | MR | Zbl

[27] Bokut L. A., Kolesnikov P. S., “Bazisy Grëbnera–Shirshova ot zarozhdeniya do nashikh dnei”, Zap. nauch. semin. POMI, 272, 2000, 26–67 | MR | Zbl

[28] Poroshenko E. N., Bazisy Grëbnera–Shirshova affinnykh neskruchennykh algebr Katsa–Mudi, Diss. na soisk. uchënoi st. kand. fiz.-mat. nauk, Novosibirsk, 2002

[29] Poroshenko E. N., “Groebner–Shirshov bases for the Kac–Moody algebras of the type $A_n^{(1)}$”, Comm. Algebra, 30 (2002), 2617–2637 | DOI | MR | Zbl

[30] Koryukin A. N., “Bazisy Grëbnera–Shirshova algebry Li $A_n$”, Algebra i logika, 44:2 (2005), 131–147 | MR | Zbl

[31] Koryukin A. N., “Bazisy Grëbnera–Shirshova algebry Li $B^+_n$”, Algebra i analiz, 20:1 (2008), 93–137 | MR

[32] Koryukin A. N., “Bazisy Grëbnera–Shirshova algebry Li $C_n^+$, I”, Nauch. vestn. NGTU, 2006, no. 4(25), 139–154; “II”, 2007, No 1(26), 89–99; “III”, 2007, No 2(27), 107–117; “IV”, 2007, No 3(28), 111–120; “V”, 2007, No 4(29), 63–76; “VI”, 2008, No 1(30), 121–130

[33] Koryukin A. N., Shum K. P., “Redutsirovannye bazisy algebry Li $D_n^+$”, Sib. zh. industr. mat., 9:4 (2006), 90–104 | MR

[34] Lalonde P., Ram A., “Standard Lyndon bases of Lie algebras and enveloping algebras”, Trans. Amer. Math. Soc., 347:5 (1995), 1821–1830 | DOI | MR | Zbl

[35] Cox David, Little John, O'Shea Donal, Ideals, varieties, and algorithms, Springer-Verlag, New York, 1997, 536 pp. | MR

[36] Shirshov A. I., “O svobodnykh koltsakh Li”, Mat. sb., 45(87):2 (1958), 113–122 | MR | Zbl

[37] Shirshov A. I., Selected works of A. I. Shirshov, eds. L. A. Bokut, V. Latyshev, I. Shestakov, E. Zelmanov, Birkhäuser, Basel, 2009, 242 pp. | MR | Zbl

[38] Lyndon R., “On Burnside's problem”, Trans. Amer. Math. Soc., 77 (1954), 202–215 | MR | Zbl

[39] Chen K. T., Fox R. H., Lyndon R. C., “Free differential calculus. IV. The quotient groups of the lower central series”, Ann. of Math. (2), 68 (1958), 81–95 | DOI | MR | Zbl

[40] Bokut L. A., Chen Yuqun, Gröbner–Shirshov bases for Lie algebras: after A. I. Shirshov, 8 Apr. 2008, 18 pp., arXiv: ; Southeast Asian Bull. Math., 31:6 (2007), 1057–1076 0804.1254v1[math.RA] | MR | Zbl

[41] Humphreys J. E., Introduction to Lie algebras and representation theory, Grad. Texts in Math., 9, Springer-Verlag, New York–Berlin, 1972 | MR | Zbl

[42] Dzhekobson N., Algebry Li, Mir, M., 1964 | MR

[43] Burbaki H., Gruppy i algebry Li. Gruppy Kokstera i sistemy Titsa. Gruppy, porozhdennye otrazheniyami sistemy kornei, Mir, M., 1972 | MR | Zbl