Fixed algebras of residually nilpotent Lie algebras
Author:
Vesselin Drensky
Journal:
Proc. Amer. Math. Soc. 120 (1994), 10211028
MSC:
Primary 17B40
MathSciNet review:
1181161
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Abstract: Let be the free Lie algebra of rank over a field , and let be an ideal of such that and the algebra is residually nilpotent. Let be a finite group of automorphisms of and the order of be invertible in . We establish that the algebra of fixed points is not finitely generated. The class of algebras under consideration contains the free Lie algebra over an arbitrary field and the relatively free algebras in nonnilpotent varieties of Lie algebras over infinite fields of characteristic different from and .
 [1]
A. I. Belov, Periodic automorphisms of free Lie algebras and their fixed points, preprint.
 [2]
Warren
Dicks and Edward
Formanek, Poincaré series and a problem of S.
Montgomery, Linear and Multilinear Algebra 12
(1982/83), no. 1, 21–30. MR 672913
(84c:15032), http://dx.doi.org/10.1080/03081088208817467
 [3]
Vesselin
Drensky and Azniv
Kasparian, Polynomial identities of eighth degree for 3×3
matrices, Annuaire Univ. Sofia Fac. Math. Méc.
77 (1983), no. 1, 175–195 (1988) (English, with
Russian summary). MR 960570
(89h:16019)
 [4]
Joan
L. Dyer and G.
Peter Scott, Periodic automorphisms of free groups, Comm.
Algebra 3 (1975), 195–201. MR 0369529
(51 #5762)
 [5]
Edward
Formanek, Noncommutative invariant theory, Group actions on
rings (Brunswick, Maine, 1984) Contemp. Math., vol. 43, Amer. Math.
Soc., Providence, RI, 1985, pp. 87–119. MR 810646
(87d:16046), http://dx.doi.org/10.1090/conm/043/810646
 [6]
V.
K. Kharchenko, Noncommutative invariants of finite groups and
Noetherian varieties, J. Pure Appl. Algebra 31
(1984), no. 13, 83–90. MR 738208
(85j:16052), http://dx.doi.org/10.1016/00224049(84)900793
 [7]
I.
G. Macdonald, Symmetric functions and Hall polynomials, The
Clarendon Press Oxford University Press, New York, 1979. Oxford
Mathematical Monographs. MR 553598
(84g:05003)
 [8]
A.
L. Šmel′kin, Free polynilpotent groups, Izv.
Akad. Nauk SSSR Ser. Mat. 28 (1964), 91–122
(Russian). MR
0162857 (29 #161)
 [9]
E.
I. Zel′manov, On the restricted Burnside problem,
Sibirsk. Mat. Zh. 30 (1989), no. 6, 68–74
(Russian); English transl., Siberian Math. J. 30 (1989),
no. 6, 885–891 (1990). MR 1043434
(92a:20041), http://dx.doi.org/10.1007/BF00970910
 [1]
 A. I. Belov, Periodic automorphisms of free Lie algebras and their fixed points, preprint.
 [2]
 W. Dicks and E. Formanek, Poincaré series and a problem of S. Montgomery, Linear and Multilinear Algebra 12 (1982), 2130. MR 672913 (84c:15032)
 [3]
 V. Drensky and A. Kasparian, Polynomial identities of eighth degree for matrices, Ann. Univ. Sofia Fac. Math. Méc. 77 (1983), 175195. MR 960570 (89h:16019)
 [4]
 J. L. Dyer and G. P. Scott, Periodic automorphisms of free groups, Comm. Algebra 3 (1975), 195201. MR 0369529 (51:5762)
 [5]
 E. Formanek, Noncommutative invariant theory, Group Actions on Rings, Contemp. Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 87119. MR 810646 (87d:16046)
 [6]
 V. K. Kharchenko, Noncommutative invariants of finite groups and Noetherian varieties, J. Pure Appl. Algebra 31 (1984), 8390. MR 738208 (85j:16052)
 [7]
 I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Univ. Press, Clarendon, Oxford, 1979. MR 553598 (84g:05003)
 [8]
 A. L. Shmel'kin, Free polynilpotent groups. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 91122; English transl. Amer. Math. Soc. Transl. Ser. 2, vol. 55, Amer. Math. Soc., Providence, RI, 1966, pp. 270304. MR 0162857 (29:161)
 [9]
 E. I. Zelmanov, Weakened Burnside problem, Sibirsk. Math. Zh. 30 (1989), 6874; English transl. Siberian Math. J. 30 (1989), 885891. MR 1043434 (92a:20041)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199411811613
PII:
S 00029939(1994)11811613
Keywords:
Fixed points of automorphisms of Lie algebras,
residually nilpotent Lie algebras,
free Lie algebras
Article copyright:
© Copyright 1994 American Mathematical Society
