Generic, almost primitive and test elements of free Lie algebras

Mikhalev, Alexander; Umirbaev, Ualbai; Yu, Jie-Tai

doi:10.1090/S0002-9939-01-06203-7

Generic, almost primitive and test elements of free Lie algebras
HTML articles powered by AMS MathViewer

by Alexander A. Mikhalev, Ualbai Umirbaev and Jie-Tai Yu PDF

Proc. Amer. Math. Soc. 130 (2002), 1303-1310 Request permission

Abstract:

We construct a series of generic elements of free Lie algebras. New almost primitive and test elements were found. We present an example of an almost primitive element which is not generic.

References

Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR 886063
L. A. Bokut′ and G. P. Kukin, Algorithmic and combinatorial algebra, Mathematics and its Applications, vol. 255, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1292459, DOI 10.1007/978-94-011-2002-9
A. M. Brunner, R. G. Burns, and Sheila Oates-Williams, On almost primitive elements of free groups with an application to Fuchsian groups, Canad. J. Math. 45 (1993), no. 2, 225–254. MR 1208114, DOI 10.4153/CJM-1993-011-9
P. M. Cohn, Subalgebras of free associative algebras, Proc. London Math. Soc. (3) 14 (1964), 618–632. MR 167504, DOI 10.1112/plms/s3-14.4.618
Leo P. Comerford Jr., Generic elements of free groups, Arch. Math. (Basel) 65 (1995), no. 3, 185–195. MR 1344016, DOI 10.1007/BF01195088
Albrecht Dold, Nullhomologous words in free groups which are not nullhomologous in any proper subgroup, Arch. Math. (Basel) 50 (1988), no. 6, 564–569. MR 948271, DOI 10.1007/BF01193627
Benjamin Fine, Gerhard Rosenberger, Dennis Spellman, and Michael Stille, Test, generic and almost primitive elements in free groups, Mat. Contemp. 14 (1998), 45–59. Algebra Meeting (Rio de Janeiro, 1996). MR 1663638
B. Fine, G. Rosenberger, D. Spellman, and M. Stille, Test words, generic elements and almost primitivity, Pacific J. Math. 190 (1999), no. 2, 277–297. MR 1722895, DOI 10.2140/pjm.1999.190.277
R. C. Lyndon, The equation $a^{2}b^{2}=c^{2}$ in free groups, Michigan Math. J. 6 (1959), 89–95. MR 103218, DOI 10.1307/mmj/1028998143
Alexander A. Mikhalev and Jie-Tai Yu, Test elements and retracts of free Lie algebras, Comm. Algebra 25 (1997), no. 10, 3283–3289. MR 1465116, DOI 10.1080/00927879708826054
Alexander A. Mikhalev and Jie-Tai Yu, Test elements, retracts and automorphic orbits of free algebras, Internat. J. Algebra Comput. 8 (1998), no. 3, 295–310. MR 1627824, DOI 10.1142/S0218196798000144
Alexander A. Mikhalev and Jie-Tai Yu, Primitive, almost primitive, test, and $\Delta$-primitive elements of free algebras with the Nielsen-Schreier property, J. Algebra 228 (2000), no. 2, 603–623. MR 1764582, DOI 10.1006/jabr.2000.8288
Alexander A. Mikhalev and Andrej A. Zolotykh, Rank and primitivity of elements of free colour Lie ($p$-)superalgebras, Internat. J. Algebra Comput. 4 (1994), no. 4, 617–655. MR 1313130, DOI 10.1142/S021819679400018X
Alexander A. Mikhalev and Andrej A. Zolotykh, Test elements for monomorphisms of free Lie algebras and Lie superalgebras, Comm. Algebra 23 (1995), no. 13, 4995–5001. MR 1356115, DOI 10.1080/00927879508825513
Alexander A. Mikhalev and Andrej A. Zolotykh, Combinatorial aspects of Lie superalgebras, CRC Press, Boca Raton, FL, 1995. With 1 IBM-PC floppy disk (3.5 inch; HD). MR 1397499
Christophe Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1231799
Gerhard Rosenberger, Alternierende Produkte in freien Gruppen, Pacific J. Math. 78 (1978), no. 1, 243–250 (German). MR 513297, DOI 10.2140/pjm.1978.78.243
Gerhard Rosenberger, Über Darstellungen von Elementen und Untergruppen in freien Produkten, Groups—Korea 1983 (Kyoungju, 1983) Lecture Notes in Math., vol. 1098, Springer, Berlin, 1984, pp. 142–160 (German). MR 781367, DOI 10.1007/BFb0099671
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Vladimir Shpilrain, Recognizing automorphisms of the free groups, Arch. Math. (Basel) 62 (1994), no. 5, 385–392. MR 1274742, DOI 10.1007/BF01196426
Vladimir Shpilrain, Test elements for endomorphisms of free groups and algebras, Israel J. Math. 92 (1995), no. 1-3, 307–316. MR 1357760, DOI 10.1007/BF02762085
John R. Stallings, Problems about free quotients of groups, Geometric group theory (Columbus, OH, 1992) Ohio State Univ. Math. Res. Inst. Publ., vol. 3, de Gruyter, Berlin, 1995, pp. 165–182. MR 1355111
John R. Stallings, Generic elements in certain groups, Groups—Korea ’94 (Pusan), de Gruyter, Berlin, 1995, pp. 289–294. MR 1476973
Edward C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc. 28 (1996), no. 3, 255–263. MR 1374403, DOI 10.1112/blms/28.3.255
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Heiner Zieschang, Alternierende Produkte in freien Gruppen, Abh. Math. Sem. Univ. Hamburg 27 (1964), 13–31 (German). MR 161901, DOI 10.1007/BF02993053