| Title:
|
Free associative algebras, noncommutative Gröbner bases, and universal associative envelopes for nonassociative structures (English) |
| Author:
|
Bremner, Murray R. |
| Language:
|
English |
| Journal:
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Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
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0010-2628 (print) |
| ISSN:
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1213-7243 (online) |
| Volume:
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55 |
| Issue:
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3 |
| Year:
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2014 |
| Pages:
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341-379 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems. (English) |
| Keyword:
|
free associative algebras |
| Keyword:
|
Gröbner bases |
| Keyword:
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composition (diamond) lemma |
| Keyword:
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universal associative envelopes |
| Keyword:
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Lie algebras and triple systems |
| Keyword:
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PBW theorem |
| Keyword:
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Jordan algebras and triple systems |
| Keyword:
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trilinear operations |
| Keyword:
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computer algebra |
| MSC:
|
16S10 |
| MSC:
|
16S30 |
| MSC:
|
16W10 |
| MSC:
|
16Z05 |
| MSC:
|
17A30 |
| MSC:
|
17A40 |
| MSC:
|
17A42 |
| MSC:
|
17B35 |
| MSC:
|
17B60 |
| MSC:
|
17C05 |
| MSC:
|
17C50 |
| MSC:
|
68W30 |
| idZBL:
|
Zbl 06391547 |
| idMR:
|
MR3225614 |
| . |
| Date available:
|
2015-01-19T10:53:40Z |
| Last updated:
|
2016-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143812 |
| . |
| Reference:
|
[1] Abramson M.P.: Historical background to Gröbner's paper.ACM Commun. Comput. Algebra 43 (2009), no. 1-2, 22–23. MR 2571829 |
| Reference:
|
[2] Adams W.W., Loustaunau P.: An Introduction to Gröbner Bases.American Mathematical Society, Providence, RI, 1994. Zbl 0803.13015, MR 1287608 |
| Reference:
|
[3] Aguiar M., Loday J.-L.: Quadri-algebras.J. Pure Appl. Algebra 191 (2004), no. 3, 205–221. Zbl 1097.17002, MR 2059613, 10.1016/j.jpaa.2004.01.002 |
| Reference:
|
[4] Aymon M., Grivel P.-P.: Un théorème de Poincaré-Birkhoff-Witt pour les algèbres de Leibniz.Comm. Algebra 31 (2003), no. 2, 527–544. Zbl 1020.17002, MR 1968912, 10.1081/AGB-120017324 |
| Reference:
|
[5] Baader F., Nipkow T.: Term Rewriting and All That.Cambridge University Press, Cambridge, 1998. Zbl 0948.68098, MR 1629216 |
| Reference:
|
[6] Bai C., Liu L., Ni X.: Some results on L-dendriform algebras.J. Geom. Phys. 60 (2010), no. 6-8, 940–950. MR 2647294 |
| Reference:
|
[7] Bashir S.: Automorphisms of simple anti-Jordan pairs.Ph.D. thesis, University of Ottawa, Canada, 2008. MR 2712919 |
| Reference:
|
[8] Becker T., Weispfenning V.: Gröbner Bases: A Computational Approach to Commutative Algebra.Springer, New York, 1993. Zbl 0772.13010, MR 1213453 |
| Reference:
|
[9] Benkart G., Roby T.: Down-up algebras.J. Algebra 209 (1998), no. 1, 305–344; Addendum: “Down-up algebras”, J. Algebra 213 (1999), no. 1, 378. Zbl 0922.17006, MR 1652138, 10.1006/jabr.1998.7511 |
| Reference:
|
[10] Bergman G.M.: The diamond lemma for ring theory.Adv. in Math. 29 (1978), no. 2, 178–218. Zbl 0326.16019, MR 0506890, 10.1016/0001-8708(78)90010-5 |
| Reference:
|
[11] Birkhoff G.: On the structure of abstract algebras.Proc. Cambridge Philos. Soc. 31 (1935), no. 4, 433–454. Zbl 0013.00105 |
| Reference:
|
[12] Birkhoff G., Whitman P.M.: Representation of Jordan and Lie algebras.Trans. Amer. Math. Soc. 65 (1949), 116–136. Zbl 0032.25102, MR 0029366, 10.1090/S0002-9947-1949-0029366-6 |
| Reference:
|
[13] Bokut L.A.: Imbeddings into simple associative algebras.Algebra i Logika 15 (1976), no. 2, 117–142. MR 0506423, 10.1007/BF01877233 |
| Reference:
|
[14] Bokut L.A.: The method of Gröbner-Shirshov bases.Siberian Adv. Math. 9 (1999), no. 3, 1–16. Zbl 0937.17004, MR 1796985 |
| Reference:
|
[15] Bokut L.A., Chen Y.: Gröbner-Shirshov bases for Lie algebras: after A.I. Shirshov.Southeast Asian Bull. Math. 31 (2007), no. 6, 1057–1076. Zbl 1150.17008, MR 2386984 |
| Reference:
|
[16] Bokut L.A., Chen Y., Deng X.: Gröbner-Shirshov bases for Rota-Baxter algebras.Sibirsk. Mat. Zh. 51 (2010), no. 6, 1237–1250. Zbl 1235.16021, MR 2797594 |
| Reference:
|
[17] Bokut L.A., Chen Y., Huang J.: Gröbner-Shirshov bases for $L$-algebras.Internat. J. Algebra Comput. 23 (2013), no. 3, 547–571. Zbl 1282.17004, MR 3048111, 10.1142/S0218196713500094 |
| Reference:
|
[18] Bokut L.A., Chen Y., Li Y.: Gröbner-Shirshov bases for Vinberg-Koszul-Gerstenhaber right-symmetric algebras.Fundam. Prikl. Mat. 14 (2008), no. 8, 55–67. MR 2744933 |
| Reference:
|
[19] Bokut L.A., Chen Y., Liu C.: Gröbner-Shirshov bases for dialgebras.Internat. J. Algebra Comput. 20 (2010), no. 3, 391–415. Zbl 1245.17001, MR 2658418, 10.1142/S0218196710005753 |
| Reference:
|
[20] Bokut L.A., Chen Y., Qiu J.: Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras.J. Pure Appl. Algebra 214 (2010), no. 1, 89–100. Zbl 1213.16014, MR 2561769, 10.1016/j.jpaa.2009.05.005 |
| Reference:
|
[21] Bokut L.A., Chibrikov E.S.: Lyndon-Shirshov words, Gröbner-Shirshov bases, and free Lie algebras.Non-associative Algebra and Its Applications, pp. 17–39, Chapman & Hall/CRC, Boca Raton, 2006. MR 2203694 |
| Reference:
|
[22] Bokut L.A., Kolesnikov P.S.: Gröbner-Shirshov bases: from their incipiency to the present.J. Math. Sci. (N.Y.) 116 (2003), no. 1, 2894–2916. MR 1811792, 10.1023/A:1023490323855 |
| Reference:
|
[23] Bokut L.A., Kukin G.P.: Algorithmic and Combinatorial Algebra.Kluwer Academic Publishers Group, Dordrecht, 1994. Zbl 0826.17002, MR 1292459 |
| Reference:
|
[24] Bokut L.A., Shum K.P.: Gröbner and Gröbner-Shirshov bases in algebra: an elementary approach.Southeast Asian Bull. Math. 29 (2005), no. 2, 227–252. Zbl 1133.16037, MR 2217531 |
| Reference:
|
[25] Borges-Trenard M.A., Borges-Quintana M., Mora T.: Computing Gröbner bases by FGLM techniques in a non-commutative setting.J. Symbolic Comput. 30 (2000), no. 4, 429–449. Zbl 0996.16033, MR 1784751, 10.1006/jsco.1999.0415 |
| Reference:
|
[26] Bremner M.R.: How to compute the Wedderburn decomposition of a finite-dimensional associative algebra.Groups Complex. Cryptol. 3 (2011), no. 1, 47–66. Zbl 1250.16018, MR 2806081, 10.1515/gcc.2011.003 |
| Reference:
|
[27] Bremner M.R.: Algebras, dialgebras, and polynomial identities.Serdica Math. J. 38 (2012), 91–136. MR 3014494 |
| Reference:
|
[28] Bremner M.R., Hentzel I.R.: Identities for generalized Lie and Jordan products on totally associative triple systems.J. Algebra 231 (2000), no. 1, 387–405. Zbl 0999.17044, MR 1779606, 10.1006/jabr.2000.8372 |
| Reference:
|
[29] Bremner M.R., S. Madariaga S.: Polynomial identities for tangent algebras of monoassociative loops.Comm. Algebra 42 (2014), no. 1, 203–227. MR 3169565, 10.1080/00927872.2012.709567 |
| Reference:
|
[30] Bremner M.R., Peresi L.A.: Classification of trilinear operations.Comm. Algebra 35 (2007), no. 9, 2932–2959. Zbl 1172.17003, MR 2356309, 10.1080/00927870701353126 |
| Reference:
|
[31] Bremner M.R., Peresi L.A.: An application of lattice basis reduction to polynomial identities for algebraic structures.Linear Algebra Appl. 430 (2009), no. 2-3, 642–659. Zbl 1173.17001, MR 2469318 |
| Reference:
|
[32] Bremner M.R., Peresi L.A.: Polynomial identities for the ternary cyclic sum.Linear Multilinear Algebra 57 (2009), no. 6, 595–608. Zbl 1188.17003, MR 2543721, 10.1080/03081080802267748 |
| Reference:
|
[33] Buchberger B.: An Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero Dimensional Polynomial Ideal.translated from the 1965 German original by Michael P. Abramson, J. Symbolic Comput. 41 (2006), no. 3-4, 475–511. Zbl 1158.01307, MR 2202562 |
| Reference:
|
[34] Buchberger B.: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems.Aequationes Math. 4 (1970), 374–383. Zbl 0212.06401, MR 0268178, 10.1007/BF01844169 |
| Reference:
|
[35] Buchberger B.: History and basic features of the critical-pair/completion procedure.Rewriting Techniques and Applications (Dijon, 1985), J. Symbolic Comput. 3 (1987), nos. 1–2, 3–38. Zbl 0645.68094, MR 0893184 |
| Reference:
|
[36] Buchberger B.: Comments on the translation of my Ph.D. thesis: “An Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero Dimensional Polynomial Ideal”.J. Symbolic Comput. 41 (2006), no. 3-4, 471–474. MR 2202561 |
| Reference:
|
[37] Bueso J., Gómez-Torrecillas J., Verschoren A.: Algorithmic Methods in Noncommutative Algebra: Applications to Quantum Groups.Kluwer Academic Publishers, Dordrecht, 2003. MR 2006329 |
| Reference:
|
[38] Carlsson R.: $n$-ary algebras.Nagoya Math. J. 78 (1980), 45–56. Zbl 0466.16017, MR 0571436 |
| Reference:
|
[39] Casas J.M., Insua M.A., Ladra M.: Poincaré-Birkhoff-Witt theorem for Leibniz $n$-algebras.J. Symbolic Comput. 42 (2007), no. 11-12, 1052–1065. Zbl 1131.17001, MR 2368072, 10.1016/j.jsc.2007.05.003 |
| Reference:
|
[40] Chen Y., Mo Q.: Embedding dendriform algebra into its universal enveloping Rota-Baxter algebra.Proc. Amer. Math. Soc. 139 (2011), no. 12, 4207–4216. MR 2823066, 10.1090/S0002-9939-2011-10889-X |
| Reference:
|
[41] Chen Y., Wang B.: Gröbner-Shirshov bases and Hilbert series of free dendriform algebras.Southeast Asian Bull. Math. 34 (2010), no. 4, 639–650. Zbl 1232.17002, MR 2768676 |
| Reference:
|
[42] Chibrikov E.S.: On free Sabinin algebras.Comm. Algebra 39 (2011), no. 11, 4014–4035. Zbl 1251.17016, MR 2855109, 10.1080/00927872.2010.515637 |
| Reference:
|
[43] Church A.: A set of postulates for the foundation of logic.Ann. of Math. (2) 33 (1932), no. 2, 346–366. Zbl 0008.28902, MR 1503059, 10.2307/1968337 |
| Reference:
|
[44] Church A., Rosser J.B.: Some properties of conversion.Trans. Amer. Math. Soc. 39 (1936), no. 3, 472–482. Zbl 0014.38504, MR 1501858, 10.1090/S0002-9947-1936-1501858-0 |
| Reference:
|
[45] Clifton J.M.: A simplification of the computation of the natural representation of the symmetric group $S_n$.Proc. Amer. Math. Soc. 83 (1981), no. 2, 248–250. Zbl 0443.20013, MR 0624907 |
| Reference:
|
[46] Cohen A.M., Gijsbers D.A.H.: Documentation on the GBNP Package.available at: tt{http://www.win.tue.nl/{ asciitilde}amc/pub/grobner/doc.html}. |
| Reference:
|
[47] Cox D., Little J., O'Shea D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra.Springer, New York, 1992. Zbl 1118.13001, MR 1189133 |
| Reference:
|
[48] de Graaf W.A.: Lie Algebras: Theory and Algorithms.North-Holland Publishing Co., Amsterdam, 2000. Zbl 1122.17300, MR 1743970 |
| Reference:
|
[49] Dotsenko V., Khoroshkin A.: Gröbner bases for operads.Duke Math. J. 153 (2010), no. 2, 363–396. Zbl 1208.18007, MR 2667136, 10.1215/00127094-2010-026 |
| Reference:
|
[50] Dotsenko V., Khoroshkin A.: Quillen homology for operads via Gröbner bases.Doc. Math. 18 (2013), 707–747. Zbl 1278.18018, MR 3084563 |
| Reference:
|
[51] Dotsenko V., Vallette B.: Higher Koszul duality for associative algebras.Glasg. Math. J. 55 (2013), no. A, 55–74. Zbl 1284.18026, MR 3110804, 10.1017/S0017089513000505 |
| Reference:
|
[52] Eisenbud D., Peeva I., Sturmfels B.: Non-commutative Gröbner bases for commutative algebras.Proc. Amer. Math. Soc. 126 (1998), no. 3, 687–691. Zbl 0898.16015, MR 1443825, 10.1090/S0002-9939-98-04229-4 |
| Reference:
|
[53] Elgendy H.A.: Polynomial Identities and Enveloping Algebras for $n$-ary Structures.Ph.D. thesis, University of Saskatchewan, Canada, 2012. MR 3152544 |
| Reference:
|
[54] Elgendy H.A.: Universal associative envelopes of nonassociative triple systems.Comm. Algebra 42 (2014), no. 4, 1785–1810. MR 3169671, 10.1080/00927872.2012.749409 |
| Reference:
|
[55] Elgendy H.A., Bremner M.R.: Universal associative envelopes of $(n{+}1)$-dimensional $n$-Lie algebras.Comm. Algebra 40 (2012), no. 5, 1827–1842. Zbl 1260.17005, MR 2924485, 10.1080/00927872.2011.558549 |
| Reference:
|
[56] Ene V., Herzog J.: Gröbner Bases in Commutative Algebra.American Mathematical Society, Providence, RI, 2012. Zbl 1242.13001, MR 2850142 |
| Reference:
|
[57] Evans T.: The word problem for abstract algebras.J. London Math. Soc. 26 (1951), 64–71. Zbl 0042.03303, MR 0038958, 10.1112/jlms/s1-26.1.64 |
| Reference:
|
[58] Faulkner J.R.: Identity classification in triple systems.J. Algebra 94 (1985), no. 2, 352–363. Zbl 0596.17002, MR 0792960, 10.1016/0021-8693(85)90189-9 |
| Reference:
|
[59] Faulkner J.R., Ferrar J.C.: Simple anti-Jordan pairs.Comm. Algebra 8 (1980), no. 11, 993–1013. Zbl 0447.17003, MR 0577825, 10.1080/00927878008822505 |
| Reference:
|
[60] Fröberg R.: An Introduction to Gröbner Bases.John Wiley & Sons, Ltd., Chichester, 1997. Zbl 0997.13500, MR 1483316 |
| Reference:
|
[61] Gerritzen L.: On infinite Gröbner bases in free algebras.Indag. Math. (N.S.) 9 (1998), no. 4, 491–501. Zbl 0930.16016, MR 1691989, 10.1016/S0019-3577(98)80029-3 |
| Reference:
|
[62] Gerritzen L.: Hilbert series and non-associative Gröbner bases.Manuscripta Math. 103 (2000), no. 2, 161–167. Zbl 0961.17002, MR 1796312 |
| Reference:
|
[63] Gerritzen L.: Tree polynomials and non-associative Gröbner bases.J. Symbolic Comput. 41 (2006), no. 3-4, 297–316. Zbl 1158.17300, MR 2202553, 10.1016/j.jsc.2003.09.005 |
| Reference:
|
[64] Gerritzen L., Holtkamp R.: On Gröbner bases of noncommutative power series.Indag. Math. (N.S.) 9 (1998), no. 4, 503–519. Zbl 0930.16017, MR 1691990, 10.1016/S0019-3577(98)80030-X |
| Reference:
|
[65] Glennie C.M.: Some identities valid in special Jordan algebras but not valid in all Jordan algebras.Pacific J. Math. 16 (1966), 47–59. Zbl 0134.26903, MR 0186708, 10.2140/pjm.1966.16.47 |
| Reference:
|
[66] Green E.L.: An introduction to noncommutative Gröbner bases.Computational Algebra, pp. 167–190, Dekker, New York, 1994. Zbl 0807.16002, MR 1245952 |
| Reference:
|
[67] Green E.L.: Noncommutative Gröbner bases, and projective resolutions.Computational Methods for Representations of Groups and Algebras, pp. 29–60, Birkhäuser, Basel, 1999. Zbl 0957.16033, MR 1714602 |
| Reference:
|
[68] Green E.L., Heath L.S., Keller B.J.: Opal: a system for computing noncommutative Gröbner bases.Rewriting Techniques and Applications, pp. 331–334, Lecture Notes in Computer Science, 1232, Springer, 1997. MR 1605520 |
| Reference:
|
[69] Green E.L., Mora T., Ufnarovski V.: The non-commutative Gröbner freaks.Symbolic Rewriting Techniques (Ascona, 1995), pp. 93–104, Birkhäuser, Basel, 1998. Zbl 1020.16017, MR 1624647 |
| Reference:
|
[70] Gröbner W.: Über die algebraischen Eigenschaften der Integrale von linearen Differentialgleichungen mit konstanten Koeffizienten.Monatsh. Math. Phys. 47 (1939), no. 1, 247–284. Zbl 0021.22505, MR 1550816, 10.1007/BF01695500 |
| Reference:
|
[71] Gröbner W.: On the algebraic properties of integrals of linear differential equations with constant coefficients.translated from the German by Michael Abramson, ACM Commun. Comput. Algebra 43 (2009), no. 1-2, 24–46. MR 2571830 |
| Reference:
|
[72] Guo L., Sit W., Zhang R.: Differential type operators and Gröbner-Shirshov bases.J. Symbolic Comput. 52 (2013), 97–123. MR 3018130, 10.1016/j.jsc.2012.05.014 |
| Reference:
|
[73] Hestenes M.R.: A ternary algebra with applications to matrices and linear transformations.Arch. Rational Mech. Anal. 11 (1962), 138–194. Zbl 0201.37001, MR 0150166, 10.1007/BF00253936 |
| Reference:
|
[74] Hodge T.L., Parshall B.J.: On the representation theory of Lie triple systems.Trans. Amer. Math. Soc. 354 (2002), no. 11, 4359–4391. Zbl 1012.17001, MR 1926880, 10.1090/S0002-9947-02-03050-7 |
| Reference:
|
[75] Hou D., Bai C.: J-dendriform algebras.Front. Math. China 7 (2012), no. 1, 29–49. MR 2876897, 10.1007/s11464-011-0160-7 |
| Reference:
|
[76] Hou D., Ni X., Bai C.: Pre-Jordan algebras.Math. Scand. 112 (2013), no. 1, 19–48. MR 3057597 |
| Reference:
|
[77] Insua M.A., Ladra M.: Gröbner bases in universal enveloping algebras of Leibniz algebras.J. Symbolic Comput. 44 (2009), no. 5, 517–526. Zbl 1163.17004, MR 2499928, 10.1016/j.jsc.2007.07.020 |
| Reference:
|
[78] Jacobson N.: Structure and Representations of Jordan Algebras.American Mathematical Society, Providence, R.I., 1968. Zbl 0218.17010, MR 0251099 |
| Reference:
|
[79] Kang S.-J., Lee D.-I., Lee K.-H., Park H.: Linear algebraic approach to Gröbner-Shirshov basis theory.J. Algebra 313 (2007), no. 2, 988–1004. Zbl 1180.17005, MR 2329580, 10.1016/j.jalgebra.2007.02.001 |
| Reference:
|
[80] Keller B.J.: Algorithms and Orders for Finding Noncommutative Gröbner Bases.Ph.D. thesis, Virginia Polytechnic Institute and State University, 1997. MR 2696478 |
| Reference:
|
[81] Keller B.J.: Alternatives in implementing noncommutative Gröbner basis systems.Symbolic Rewriting Techniques, pp. 105–126, Birkhüser, Basel, 1998. Zbl 0927.16042, MR 1624651 |
| Reference:
|
[82] Kleene S.C.: Proof by cases in formal logic.Ann. of Math. (2) 35 (1934), no. 3, 529–544. Zbl 0010.14602, MR 1503178, 10.2307/1968749 |
| Reference:
|
[83] Knuth D.E., Bendix P.B.: Simple word problems in universal algebras.Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pp. 263–297, Pergamon, Oxford, 1970. Zbl 0188.04902, MR 0255472 |
| Reference:
|
[84] Li H.: Gröbner Bases in Ring Theory.World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. MR 2894019 |
| Reference:
|
[85] Lister W.G.: A structure theory of Lie triple systems.Trans. Amer. Math. Soc. 72 (1952), 217–242. Zbl 0046.03404, MR 0045702, 10.1090/S0002-9947-1952-0045702-9 |
| Reference:
|
[86] Lister W.G.: Ternary rings.Trans. Amer. Math. Soc. 154 (1971), 37–55. Zbl 0502.17002, MR 0272835, 10.1090/S0002-9947-1971-0272835-6 |
| Reference:
|
[87] Loday J.-L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz.Enseign. Math. (2) 39 (1993), no. 3-4, 269–293. Zbl 0806.55009, MR 1252069 |
| Reference:
|
[88] Loday J.-L.: Algèbres ayant deux opérations associatives (digèbres).C.R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 2, 141–146. Zbl 0845.16036, MR 1345436 |
| Reference:
|
[89] Loday J.-L.: Dialgebras.Dialgebras and Related Operads, pp. 7–66, Lecture Notes in Math., 1763, Springer, Berlin, 2001. Zbl 0999.17002, MR 1860994 |
| Reference:
|
[90] Loday J.-L., Pirashvili T.: Universal enveloping algebras of Leibniz algebras and (co)homology.Math. Ann. 296 (1993), no. 1, 139–158. Zbl 0821.17022, MR 1213376, 10.1007/BF01445099 |
| Reference:
|
[91] Loday J.-L., Vallette B.: Algebraic Operads.Grundlehren der Mathematischen Wissenschaften, 346, Springer, Heidelberg, 2012. Zbl 1260.18001, MR 2954392 |
| Reference:
|
[92] Loos O.: Lectures on Jordan Triples.University of British Columbia, Canada, 1971. Zbl 0337.17006, MR 0325717 |
| Reference:
|
[93] Loos O.: Assoziative Tripelsysteme.Manuscripta Math. 7 (1972), 103–112. Zbl 0237.16007, MR 0304446, 10.1007/BF01679707 |
| Reference:
|
[94] Macaulay F.S.: The Algebraic Theory of Modular Systems.Cambridge University Press, 1916, available online: tt{http://archive.org/details/algebraictheoryo00macauoft}. Zbl 0802.13001, MR 1281612 |
| Reference:
|
[95] Madariaga S.: Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras.J. Symbolic Comput. 60 (2014), 1–14. MR 3131375, 10.1016/j.jsc.2013.10.016 |
| Reference:
|
[96] Marché C.: Normalized rewriting: a unified view of Knuth-Bendix completion and Gröbner bases computation.Symbolic Rewriting Techniques (Ascona, 1995), pp. 193–208, Progr. Comput. Sci. Appl. Logic, 15, Birkhäuser, Basel, 1998. Zbl 0915.68100, MR 1624584 |
| Reference:
|
[97] Markl M., Shnider S., Stasheff J.: Operads in Algebra, Topology and Physics.Mathematical Surveys and Monographs, 96, American Mathematical Society, Providence, RI, 2002. Zbl 1017.18001, MR 1898414 |
| Reference:
|
[98] McCrimmon K.: A Taste of Jordan Algebras.Springer, New York, 2004. Zbl 1044.17001, MR 2014924 |
| Reference:
|
[99] Meyberg K.: Lectures on Algebras and Triple Systems.The University of Virginia, Charlottesville, 1972, available online: tt{http://www.math.uci.edu/{ asciitilde}brusso/Meyberg(Reduced2).pdf}. MR 0340353 |
| Reference:
|
[100] Mikhalev A.A., Zolotykh A.A.: Standard Gröbner-Shirshov bases of free algebras over rings. I. Free associative algebras.Internat. J. Algebra Comput. 8 (1998), no. 6, 689–726. Zbl 0923.16024, MR 1682224, 10.1142/S021819679800034X |
| Reference:
|
[101] Mora F.: Groebner bases for noncommutative polynomial rings.Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, 229, pp. 353–362, Springer, Berlin, 1986. MR 0864254, 10.1007/3-540-16776-5_740 |
| Reference:
|
[102] Mora T.: An introduction to commutative and noncommutative Gröbner bases.Theoret. Comput. Sci. 134 (1994), 131–173. Zbl 0824.68056, MR 1299371, 10.1016/0304-3975(94)90283-6 |
| Reference:
|
[103] Musson I.M.: Lie Superalgebras and Enveloping Algebras.American Mathematical Society, Providence, 2012. Zbl 1255.17001, MR 2906817 |
| Reference:
|
[104] Newman M.H.A.: On theories with a combinatorial definition of “equivalence”.Annals of Math. 43 (1942), no. 2, 223–243. Zbl 0060.12501, MR 0007372, 10.2307/1968867 |
| Reference:
|
[105] The OEIS Foundation: On-line Encyclopedia of Integer Sequences.tt{http://oeis.org/}. |
| Reference:
|
[106] Pérez-Izquierdo J.-M.: An envelope for Bol algebras.J. Algebra 284 (2005), no. 2, 480–493. MR 2114566, 10.1016/j.jalgebra.2004.09.038 |
| Reference:
|
[107] Pérez-Izquierdo J.-M.: Algebras, hyperalgebras, nonassociative bialgebras and loops.Adv. Math. 208 (2007), no. 2, 834–876. MR 2304338, 10.1016/j.aim.2006.04.001 |
| Reference:
|
[108] Pérez-Izquierdo J.-M., Shestakov I.P.: An envelope for Malcev algebras.J. Algebra 272 (2004), no. 1, 379–393. MR 2029038, 10.1016/S0021-8693(03)00389-2 |
| Reference:
|
[109] Qiu J.: Gröbner-Shirshov bases for commutative algebras with multiple operators and free commutative Rota-Baxter algebras.tt{arXiv:1301.5018}. |
| Reference:
|
[110] Rajaee S.: Non-associative Gröbner bases.J. Symbolic Comput. 41 (2006), no. 8, 887–904. Zbl 1236.17006, MR 2246715 |
| Reference:
|
[111] Rutherford D.E.: Substitutional Analysis.Edinburgh, at the University Press, 1948. Zbl 0174.31202, MR 0027272 |
| Reference:
|
[112] Shestakov I.P., Umirbaev U.U.: Free Akivis algebras, primitive elements, and hyperalgebras.J. Algebra 250 (2002), no. 2, 533–548. Zbl 0993.17002, MR 1899864, 10.1006/jabr.2001.9123 |
| Reference:
|
[113] Shirshov A.I.: Some algorithmic problems for $\epsilon$-algebras.Sibirsk. Mat. Zh. 3 (1962), 132–137. MR 0183744 |
| Reference:
|
[114] Shirshov A.I.: On a hypothesis in the theory of Lie algebras.Sibirsk. Mat. Zh. 3 (1962), 297–301. MR 0182684 |
| Reference:
|
[115] Shirshov A.I.: Selected Works of A.I. Shirshov.translated by M.R. Bremner and M.V. Kotchetov, edited by L.A. Bokut, V.N. Latyshev, I.P. Shestakov and E. Zelmanov, Birkhäuser, Basel, 2009. Zbl 1188.01028, MR 2547481 |
| Reference:
|
[116] Ufnarovski V.S.: Introduction to noncommutative Gröbner bases theory.Gröbner Bases and Applications, pp. 259–280, Cambridge Univ. Press, Cambridge, 1998. Zbl 0902.16002, MR 1708883, 10.1017/CBO9780511565847.015 |
| Reference:
|
[117] Vallette B.: Manin products, Koszul duality, Loday algebras and Deligne conjecture.J. Reine Angew. Math. 620 (2008), 105–164. Zbl 1159.18001, MR 2427978 |
| Reference:
|
[118] Young A.: The Collected Papers of Alfred Young (1873–1940)., University of Toronto Press, 1977. MR 0439548 |
| Reference:
|
[119] Zhukov A.I.: Reduced systems of defining relations in non-associative algebras.Mat. Sbornik N.S. 27 (69) (1950), 267–280. MR 0037831 |
| Reference:
|
[120] Zinbiel G.W.: Encyclopedia of types of algebras 2010.Operads and Universal Algebra, pp. 217–297, Nankai Ser. Pure Appl. Math. Theoret. Phys., 9, World Sci. Publ., Hackensack, NJ, 2012. MR 3013090 |
| . |
