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Length computation of matrix subalgebras of special type. (English. Russian original) Zbl 1163.16017
J. Math. Sci., New York 155, No. 6, 908-931 (2008); translation from Fundam. Prikl. Mat. 13, No. 4, 165-197 (2007).
Let \(A\) be a finite-dimensional algebra over a field \(F\). For each set \(S\subseteq A\) which generates \(A\) as an \(F\)-algebra, define \(l(S)\) to be the least integer \(h\geq 0\) such that there is an \(F\)-basis of \(A\) consisting of words of length \(\leq h\) in \(S\). The ‘length’ \(l(A)\) of \(A\) is the maximum of \(l(S)\) over all finite generating sets \(S\).
In general the length of a subalgebra of \(A\) can be larger than the length of \(A\). If \(A\) is the full matrix algebra \(M_n(F)\), then it is known that \(l(M_n(F))=O(n^{3/2})\) and it is conjectured that \(l(M_n(F))=2n-2\) [see C. J. Pappacena, J. Algebra 197, No. 2, 535-545 (1997; Zbl 0888.16008) and A. Paz, Linear Multilinear Algebra 15, 161-170 (1984; Zbl 0536.15007)].
In the present paper the author gives short proofs that the conjecture holds for \(n=2\) and \(3\), and computes the lengths of various classes of subalgebras of \(M_n(F)\).

16S50 Endomorphism rings; matrix rings
15A30 Algebraic systems of matrices
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16P10 Finite rings and finite-dimensional associative algebras
Full Text: DOI
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