7 research outputs found
Subexponential estimations in Shirshov's height theorem (in English)
In 1993 E. I. Zelmanov asked the following question in Dniester Notebook:
"Suppose that F_{2, m} is a 2-generated associative ring with the identity
x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential
growth?" We show that the nilpotency degree of l-generated associative algebra
with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l
(nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by
this result. It is the consequence of one fact, which is based on combinatorics
of words. Let l, n and d>n be positive integers. Then all the words over
alphabet of cardinality l which length is greater than Psi(n,d,l) are either
n-divided or contain d-th power of subword, where a word W is n-divided, if it
can be represented in the following form W=W_0 W_1...W_n such that W_1 >'
W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov
proved that the set of non n-divided words over alphabet of cardinality l has
bounded height h over the set Y consisting of all the words of degree <n.
Original Shirshov's estimation was just recursive, in 1982 double exponent was
obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation.
We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our
proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link
arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55
Anti-commutative Groebner-Shirshov basis of a free Lie algebra
One of the natural ways to prove that the Hall words (Philip Hall, 1933)
consist of a basis of a free Lie algebra is a direct construction: to start
with a linear space spanned by Hall words, to define the Lie product of Hall
words, and then to check that the product yields the Lie identities (Marshall
Hall, 1950). Here we suggest another way using the Composition-Diamond lemma
for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962)