Relative Gröbner–Shirshov bases for algebras and groups

Bokut, L.; Shum, K.

doi:10.1090/S1061-0022-08-01025-X

Relative Gröbner–Shirshov bases for algebras and groups
HTML articles powered by AMS MathViewer

by L. A. Bokut and K. P. Shum
Translated by: the authors

St. Petersburg Math. J. 19 (2008), 867-881

DOI: https://doi.org/10.1090/S1061-0022-08-01025-X

Published electronically: August 21, 2008

PDF | Request permission

Abstract:

The notion of a relative Gröbner–Shirshov basis for algebras and groups is introduced. The relative composition lemma and relative (composition-)diamond lemma are established. In particular, it is shown that the relative normal forms of certain groups arising from Malcev’s embedding problem are the irreducible normal forms of these groups with respect to their relative Gröbner–Shirshov bases. Other examples of such groups are given by showing that any group $G$ in a Tits system $(G, B, N,S)$ has a relative ($B$-)Gröbner–Shirshov basis such that the irreducible words are the Bruhat words of $G$.

References

S. I. Adjan, Defining relations and algorithmic problems for groups and semigroups, Trudy Mat. Inst. Steklov. 85 (1966), 123 (Russian). MR 0204501
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
A. J. Bowtell, On a question of Mal′cev, J. Algebra 7 (1967), 126–139. MR 230750, DOI 10.1016/0021-8693(67)90071-3
L. A. Bokut′, Some examples of rings without zero divisors, Algebra i Logika Sem. 3 (1964), no. 5-6, 5–28 (Russian). MR 0174586
L. A. Bokut′, Factorization theorems for certain classes of rings without divisors of zero. I, Algebra i Logika Sem. 4 (1965), no. 4, 25–52 (Russian). MR 0197501
L. A. Bokut′, The imbedding of rings in sfields, Dokl. Akad. Nauk SSSR 175 (1967), 755–758 (Russian). MR 0222115
L. A. Bokut′, Groups with a relative standard basis, Sibirsk. Mat. Ž. 9 (1968), 499–521 (Russian). MR 0237671
L. A. Bokut′, Groups of fractions of multiplicative semigroups of certain rings. I, II, III, Sibirsk. Mat. Ž. 10 (1969), 246–286; ibid. 10 (1969), 744–799; ibid. 800–819 (Russian). MR 0255606
L. A. Bokut′, The problem of Mal′cev, Sibirsk. Mat. Ž. 10 (1969), 965–1005 (Russian). MR 0255607
L. A. Bokut′, Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1173–1219 (Russian). MR 0330250
L. A. Bokut′, Imbeddings into simple associative algebras, Algebra i Logika 15 (1976), no. 2, 117–142, 245 (Russian). MR 0506423
L. A. Bokut′, Mal′cev’s problem and groups with a normal form, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Studies in Logic and the Foundations of Mathematics, vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 29–53. With the collaboration of D. J. Collins. MR 579936
L. A. Bokut′, Imbedding of rings, Uspekhi Mat. Nauk 42 (1987), no. 4(256), 87–111 (Russian). MR 912062
L. A. Bokut′, Abstract semigroups and groups of formal series of dependent variables, Siberian Adv. Math. 6 (1996), no. 3, 1–26. Siberian Advances in Mathematics. MR 1469036
L. A. Bokut′ and P. S. Kolesnikov, Gröbner-Shirshov bases: from inception to the present time, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), no. Vopr. Teor. Predst. Algebr i Grupp. 7, 26–67, 345 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 116 (2003), no. 1, 2894–2916. MR 1811792, DOI 10.1023/A:1023490323855
Roger W. Carter, Simple groups of Lie type, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Reprint of the 1972 original; A Wiley-Interscience Publication. MR 1013112
P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
V. N. Gerasimov, Inverting homomorphisms of rings, Algebra i Logika 18 (1979), no. 6, 648–663, 754 (Russian). MR 596422
Irving Kaplansky, Problems in the theory of rings, Report of a conference on linear algebras, June, 1956, National Academy of Sciences-National Research Council, Washington, D.C., 1957, pp. 1-3. Publ. 502. MR 0096696
Abraham A. Klein, Rings nonembeddable in fields with multiplicative semi-groups embeddable in groups, J. Algebra 7 (1967), 100–125. MR 230749, DOI 10.1016/0021-8693(67)90070-1
Serge Lang, $\textrm {SL}_2(\textbf {R})$, Graduate Texts in Mathematics, vol. 105, Springer-Verlag, New York, 1985. Reprint of the 1975 edition. MR 803508
D. Lazard, Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, Computer algebra (London, 1983) Lecture Notes in Comput. Sci., vol. 162, Springer, Berlin, 1983, pp. 146–156. MR 774807, DOI 10.1007/3-540-12868-9_{9}9
A. Malcev, On the immersion of an algebraic ring into a field, Math. Ann. 113 (1937), no. 1, 686–691. MR 1513116, DOI 10.1007/BF01571659
A. I. Mal′tsev, Izbrannye trudy. Tom I. Klassicheskaya algebra, Izdat. “Nauka”, Moscow, 1976 (Russian). Edited by M. A. Lavrent′ev and A. I. Širšov. MR 0424502
A. I. Širšov, Some algorithm problems for Lie algebras, Sibirsk. Mat. Ž. 3 (1962), 292–296 (Russian). MR 0183753
B. L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar Publishing Co., New York, N. Y., 1949. Translated from the second revised German edition by Fred Blum; With revisions and additions by the author. MR 0029363