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Fundam. Prikl. Mat., 2012, Volume 17, Issue 6, Pages 65–173 (Mi fpm1450)  

This article is cited in 9 scientific papers (total in 9 papers)

The length function and matrix algebras

O. V. Markova

M. V. Lomonosov Moscow State University

Abstract: By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field we mean 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, we study the main ring-theoretical properties of the length function: the behavior of the length under unity adjunction, direct sum of algebras, passing to subalgebras and homomorphic images. We give an upper bound for the length of the algebra as a function of the nilpotency index of its Jacobson radical and the length of the quotient algebra. We also provide examples of the length computation for certain algebras, in particular, for the following classical matrix subalgebras: the algebra of upper triangular matrices, the algebra of diagonal matrices, the Schur algebra, Courter's algebra, and for the classes of local and commutative algebras.

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English version:
Journal of Mathematical Sciences (New York), 2013, 193:5, 687–768

UDC: 512.552+512.643

Citation: O. V. Markova, “The length function and matrix algebras”, Fundam. Prikl. Mat., 17:6 (2012), 65–173; J. Math. Sci., 193:5 (2013), 687–768

Citation in format AMSBIB
\by O.~V.~Markova
\paper The length function and matrix algebras
\jour Fundam. Prikl. Mat.
\yr 2012
\vol 17
\issue 6
\pages 65--173
\jour J. Math. Sci.
\yr 2013
\vol 193
\issue 5
\pages 687--768

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    This publication is cited in the following articles:
    1. A. E. Guterman, O. V. Markova, S. D. Sochnev, “Algebra of semimagic matrices and its length”, J. Math. Sci. (N. Y.), 199:4 (2014), 400–413  mathnet  crossref
    2. A. E. Guterman, D. K. Kudryavtsev, “The lengths of the quaternion and octotion algebras”, J. Math. Sci. (N. Y.), 224:6 (2017), 826–832  mathnet  crossref  mathscinet
    3. A. E. Guterman, O. V. Markova, “Dlina gruppovykh algebr grupp nebolshogo razmera”, Chislennye metody i voprosy organizatsii vychislenii. XXXI, Zap. nauchn. sem. POMI, 472, POMI, SPb., 2018, 76–87  mathnet
    4. Guterman A., Laffey T., Markova O., Smigoc H., “A Resolution of Paz'S Conjecture in the Presence of a Nonderogatory Matrix”, Linear Alg. Appl., 543 (2018), 234–250  crossref  mathscinet  zmath  isi  scopus
    5. A. E. Guterman, D. K. Kudryavtsev, O. V. Markova, “Dlina pryamoi summy neassotsiativnykh algebr”, Chislennye metody i voprosy organizatsii vychislenii. XXXII, Zap. nauchn. sem. POMI, 482, POMI, SPb., 2019, 73–86  mathnet
    6. O. V. Markova, M. A. Khrystik, “Dlina gruppovoi algebry gruppy diedra poryadka $2^k$”, Chislennye metody i voprosy organizatsii vychislenii. XXXIII, Zap. nauchn. sem. POMI, 496, POMI, SPb., 2020, 169–181  mathnet
    7. O. V. Markova, “Primer vychisleniya dliny gruppovoi algebry netsiklicheskoi abelevoi gruppy v modulyarnom sluchae”, Fundament. i prikl. matem., 23:2 (2020), 217–229  mathnet
    8. N. A. Kolegov, O. V. Markova, “Dlina matrichnykh algebr intsidentnosti nad malenkimi konechnymi polyami”, Chislennye metody i voprosy organizatsii vychislenii. XXXIV, Zap. nauchn. sem. POMI, 504, POMI, SPb., 2021, 102–135  mathnet
    9. O. V. Markova, D. Yu. Novochadov, “Sistemy porozhdayuschikh polnoi matrichnoi algebry, soderzhaschie tsiklicheskie matritsy”, Chislennye metody i voprosy organizatsii vychislenii. XXXIV, Zap. nauchn. sem. POMI, 504, POMI, SPb., 2021, 157–171  mathnet
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