Previous |  Up |  Next

Article

Keywords:
non-associative algebra; Akivis algebra; universal enveloping algebra; Pre-Lie algebra; Gröbner-Shirshov basis
Summary:
In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I. P. Shestakov's result that any Akivis algebra is linear and D. Segal's result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra ${\rm PLie}(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov's Composition-Diamond lemma for non-associative algebras.
References:
[1] Akivis, M. A.: The local algebras of a multidimensional three-web. Sibirsk. Mat. Z. 17 (1976), 5-11 Russian English translation: Siberian Math. J. 17 (1976), 3-8. MR 0405261
[2] Bergman, G. M.: The diamond lemma for ring theory. Adv. Math. 29 (1978), 178-218. DOI 10.1016/0001-8708(78)90010-5 | MR 0506890
[3] Bokut, L. A.: Embeddings into simple associative algebras. Algebra Log. 15 (1976), 73-90. DOI 10.1007/BF01877233 | MR 0506423
[4] Bokut, L. A., Fong, Y., Ke, W.-F., Kolesnikov, P. S.: Gröbner and Gröbner-Shirshov bases in algebra and conformal algebras. Fundam. Appl. Prikl. Mat. 6 (2000), 669-706. MR 1801321
[5] Bokut, L. A., Kolesnikov, P. S.: Gröbner-Shirshov bases: from their inception to the present time. J. Math. Sci. 116 (2003), 2894-2916. DOI 10.1023/A:1023490323855 | MR 1811792
[6] Bokut, L. A., Kolesnikov, P. S.: Gröbner-Shirshov bases, conformal algebras and pseudo-algebras. J. Math. Sci. 131 (2005), 5962-6003. DOI 10.1007/s10958-005-0454-y | MR 2153696
[7] Buchberger, B.: An algorithmical criteria for the solvability of algebraic systems of equations. Aequationes Math. 4 (1970), 374-383 German.
[8] Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, Vol. 150. Springer Berlin (1995). DOI 10.1007/978-1-4612-5350-1 | MR 1322960
[9] Hironaka, H.: Resolution of singularities of an algebraic variety over a feild of characteristic zero. I, II. Ann. Math. 79 (1964), 109-203, 205-326. DOI 10.2307/1970486 | MR 0199184
[10] Knuth, D. E., Bendix, P. B.: Simple word problems in universal algebras. Comput. Probl. Abstract Algebra. Proc. Conf. Oxford 1967 (1970), 263-297. MR 0255472 | Zbl 0188.04902
[11] Kurosh, A. G.: Nonassociative free algebras and free products of algebras. Mat. Sb. N. Ser. 20 (1947), 239-262 Russian. MR 0020986 | Zbl 0041.16803
[12] Reutenauer, C.: Free Lie Algebras. Clarendon Press Oxford (1993). MR 1231799 | Zbl 0798.17001
[13] Segal, D.: Free left-symmetric algebras and an analogue of the Poincaré-Birkhoff-Witt Theorem. J. Algebra 164 (1994), 750-772. DOI 10.1006/jabr.1994.1088 | MR 1272113 | Zbl 0831.17001
[14] Shestakov, I. P.: Every Akivis algebra is linear. Geom. Dedicata 77 (1999), 215-223. DOI 10.1023/A:1005157524168 | MR 1713296 | Zbl 1043.17002
[15] Shestakov, I. P., Umirbaev, U.: Free Akivis algebras, primitive elements and hyperalgebras. J. Algebra 250 (2002), 533-548. DOI 10.1006/jabr.2001.9123 | MR 1899864 | Zbl 0993.17002
[16] Shirshov, A. I.: Subalgebras of free Lie algebras. Mat. Sb., N. Ser. 33 (1953), 441-452 Russian. MR 0059892
[17] Shirshov, A. I.: Subalgebras of free commutative and free anti-commutative algebras. Mat. Sbornik 34 (1954), 81-88 Russian. MR 0062112
[18] Shirshov, A. I.: Certain algorithmic problems for $\epsilon$-algebras. Sib. Mat. Zh. 3 (1962), 132-137.
[19] Shirshov, A. I.: Certain algorithmic problems for Lie algebras. Sib. Mat. Zh. 3 (1962), 292-296 Russian.
[20] Bokut, L. A., Latyshev, V., Shestakov, I., Zelmanov, E.: Selected Works of A. I. Shirshov Series. Contemporary Mathematicians. Basel, Boston, Berlin (2009).
[21] Witt, E.: Subrings of free Lie rings. Math. Z. 64 (1956), 195-216 German.
[22] Zhukov, A. I.: Reduced systems of defining relations in nonassociative algebras. Mat. Sb., N. Ser. 27 (1950), 267-280 Russian. MR 0037831 | Zbl 0038.17001
Partner of
EuDML logo