Geodesic
Upper bound for the length of commutative algebras
Sbornik. Mathematics, Tome 200 (2009) no. 12, pp. 1767-1787 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field one means the least positive integer $k$ such that the words of length not exceeding $k$ span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, an upper bound for the length of a commutative algebra in terms of a function of two invariants of the algebra, the dimension and the maximal degree of the minimal polynomial for the elements of the algebra, is obtained. As a corollary, a formula for the length of the algebra of diagonal matrices over an arbitrary field is obtained. Bibliography: 8 titles.
Keywords: length of an algebra, matrix theory, commutative algebra
Mots-clés : algebra of diagonal matrices. 
@article{SM_2009_200_12_a1,
     author = {O. V. Markova},
     title = {Upper bound for the length of commutative algebras},
     journal = {Sbornik. Mathematics},
     pages = {1767--1787},
     year = {2009},
     volume = {200},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2009_200_12_a1/}
}
TY  - JOUR
AU  - O. V. Markova
TI  - Upper bound for the length of commutative algebras
JO  - Sbornik. Mathematics
PY  - 2009
SP  - 1767
EP  - 1787
VL  - 200
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2009_200_12_a1/
LA  - en
ID  - SM_2009_200_12_a1
ER  - 
%0 Journal Article
%A O. V. Markova
%T Upper bound for the length of commutative algebras
%J Sbornik. Mathematics
%D 2009
%P 1767-1787
%V 200
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2009_200_12_a1/
%G en
%F SM_2009_200_12_a1
O. V. Markova. Upper bound for the length of commutative algebras. Sbornik. Mathematics, Tome 200 (2009) no. 12, pp. 1767-1787. http://geodesic.mathdoc.fr/item/SM_2009_200_12_a1/

[1] A. Paz, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables”, Linear and Multilinear Algebra, 15:2 (1984), 161–170 | DOI | MR | Zbl

[2] A. J. M. Spencer, R. S. Rivlin, “The theory of matrix polynomials and its application to the mechanics of isotropic continua”, Arch. Rational Mech. Anal., 2:1 (1958), 309–336 | DOI | MR | Zbl

[3] A. J. M. Spencer, R. S. Rivlin, “Further results in the theory of matrix polynomials”, Arch. Rational Mech. Anal., 4:1 (1959), 214–230 | DOI | MR | Zbl

[4] Ch. J. Pappacena, “An upper bound for the length of a finite-dimensional algebra”, J. Algebra, 197:2 (1997), 535–545 | DOI | MR | Zbl

[5] D. Constantine, M. Darnall, “Lengths of finite dimensional representations of PBW algebras”, Linear Algebra Appl., 395 (2005), 175–181 | DOI | MR | Zbl

[6] O. V. Markova, “On the length of the algebra of upper-triangular matrices”, Russian Math. Surveys, 60:5 (2005), 984–985 | DOI | MR | Zbl

[7] W. C. Brown, F. W. Call, “Maximal commutative subalgebras of $n\times n$ matrices”, Comm. Algebra, 21:12 (1993), 4439–4460 | DOI | MR | Zbl

[8] O. V. Markova, “Length computation of matrix subalgebras of special type”, J. Math. Sci. (N. Y.), 155:6 (2008), 908–931 | DOI | MR | Zbl