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x = ±iei + u, y = ²iei + v, where ±i, ²i " K; u, v " I. Therefore the
i=1 i=1
subalgebra generated by x and y is contained in the subalgebra
(11) Lu,v = e1, . . . , en, u, v .
Let h = h(n +2, N), where h is the Zelmanov s function mentioned in the for-
mulation of Theorem 4. It is clear that degrees of all commutators on the set
{e1, . . . , en, u, v} of length d" h will be bounded from above by a number M which
does not depend on u and v. Therefore
dimKLu,v d" D(n +2, N, M)
also does not depend on u and v. It implies that L is weakly algebraic and hence
algebraic.
4. Proof of Theorem 1
We can combine now all algebraic and topological bits and pieces together. Let
L be an algebraic Lie algebra over an infinite field K. Then the sets
An(L) ={x "L| deg(x) d" n}
by Lemma 4 are closed and since L is algebraic it is a countable union of these sets.
As linearly compact spaces are Baire spaces [11] they cannot be represented by a
countable number of closed subsets without interior points. This means that some
An(L) contains an interior point, say a. This point belongs to An(L) together with
an open neighborhood W . But in [9] it was shown that L has a neighborhood base
consisting of ideals of finite codimension. Thus we can consider that W = a + I †"
An(L), where I is an ideal of finite codimension. Now by Lemma 6 the ideal I
consists of elements of bounded degree and by Lemma 7 the same can be said for
the whole algebra L. The theorem is proved.
References
[1] Y. Bakhturin, A. Mikhalev, V. Petrogradsky and M. Zaicev, Infinite dimensional Lie Super-
algebras. Walter de Gruyter and Co, Berlin-New York, 1992.
[2] N. Bourbaki, Espaces Vectoriels Topologiques (Chap. 1 et 2). Hermann, Paris, 1966. MR
34:3277
[3] B. Cuartero and J.E. Galé, Locally PI-algebras over valued fields, In: Aportaciones Matem-
aticas en Memoria del Profesor V.M.Onieva , Santander, Universidad de Cantabria, 1991,
pp. 137 145. MR 92h:46112
[4] B. Cuartero and J.E. Galé, Bounded Degree of Algebraic Topological Algebras, Communica-
tions in Algebra, 22(1) (1994), 329 337. MR 94m:17002
[5] B. Cuartero, J.E. Galé, A. Rodriguez Palacios and A. Slinko, Bounded Degree of Weakly
Algebraic Topological Lie Algebras, Manuscripta Math. 81 (1993), 129 139. MR 94k:17008
[6] I. Kaplansky, Topological Methods in Valuation Theory, Duke Math. J. 14 (1947), 527 541.
MR 9:172f
[7] I. Kaplansky, Infinite abelian groups. University of Michigan Press, 1954. MR 16:444g
[8] D.E. Radford, Coalgebraic Coalgebras, Proc. Amer. Math. Soc. 45 (1974), 11 18. MR
50:9942
[9] A. Slinko, Local Finiteness of Coalgebraic Lie Coalgebras, Communications in Algebra, 23(3)
(1995), 1165 1170. CMP 95:08
1952 B. CUARTERO, J. E. GALÉ, AND A. M. SLINKO
[10] A. Slinko, Linearly Compact Algebras and Coalgebras, New Zealand Math. J., to appear.
CMP 96:15
[11] S. Warner, Linearly Compact Rings and Modules, Math Ann. 197 (1972), 29 43. MR 45:6874
[12] E.I. Zelmanov, Nil Rings and Periodic Groups. KMS Lecture Notes in Mathematics, Korean
Math. Soc., Seoul, 1992. MR 94c:16027
[13] E.I. Zelmanov, On Periodic Compact Groups, Israel J. Math., 77 (1992), 83 95. MR
94e:20055
[14] E.I. Zelmanov, On the Restricted Burnside Problem. In: Proceedings of the International
Congress of Mathematicians, Kyoto, Japan, 1990, The Mathematical Society of Japan, 1991,
pp. 395 402. MR 93d:20076
[15] K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov, A.I. Shirshov, Rings that are nearly associative,
Academic Press, New York - London, 1982, pp. 371. MR 83i:17001
Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
E-mail address: cuartero@cc.unizar.es
Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
E-mail address: gale@cc.unizar.es
Department of Mathematics, University of Auckland, Private Bag 92019 Auckland,
New Zealand
E-mail address: a.slinko@auckland.ac.nz [ Pobierz całość w formacie PDF ]