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 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, passage 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 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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References
Yu. A. Al’pin and Kh. D. Ikramov, “Reducibility theorems for pairs of matrices as rational criteria,” Linear Algebra Appl., 313, 155–161 (2000).
Yu. A. Al’pin and Kh. D. Ikramov, “On the unitary similarity of matrix families,” Math. Notes, 74, No. 6, 772–782 (2003).
W. C. Brown and F. W. Call, “Maximal commutative subalgebras of n × n matrices,” Commun. Algebra, 21, No. 12, 4439–4460 (1993).
D. Constantine and M. Darnall, “Lengths of finite dimensional representations of PWB algebras,” Linear Algebra Appl., 395, 175–181 (2005).
R. C. Courter, “The dimension of maximal commutative subalgebras of K n ,” Duke Math. J., 32, 225–232 (1965).
M. Gerstenhaber, “On dominance and varieties of commuting matrices,” Ann. Math., 73, No. 2, 324–348 (1961).
A. E. Guterman and O. V. Markova, “Commutative matrix subalgebras and length function,” Linear Algebra Appl., 430, 1790–1805 (2009).
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge (1985).
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge (1991).
N. Jacobson, “Schur’s theorems on commutative matrices,” Bull. Am. Math. Soc., 50, 431–436 (1944).
T. J. Laffey, “The minimal dimension of maximal commutative subalgebras of full matrix algebras,” Linear Algebra Appl., 71, 199–212 (1985).
T. J. Laffey, “Simultaneous reduction of sets of matrices under similarity,” Linear Algebra Appl., 84, 123–138 (1986).
T. J. Laffey and S. Lazarus, “Two-generated commutative matrix subalgebras,” Linear Algebra Appl., 147, 249–273 (1991).
J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham (1966).
R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Reading (1983).
W. E. Longstaff, “Burnside’s theorem: irreducible pairs of transformations,” Linear Algebra Appl., 382, 247–269 (2004).
W. E. Longstaff and P. Rosenthal, “On the lengths of irreducible pairs of complex matrices,” Proc. Am. Math. Soc., 139, No. 11, 3769–3777 (2011).
A. I. Mal’cev, Basics of Linear Algebra [in Russian], Nauka, Moscow (1975).
O. V. Markova, “On the length of the algebra of upper-triangular matrices,” Russ. Math. Surv., 60, No. 5, 984–985 (2005).
O. V. Markova, “Length computation of matrix subalgebras of special type,” J. Math. Sci., 155, No. 6, 908–931 (2008).
O. V. Markova, “Characterization of commutative matrix subalgebras of maximal length over an arbitrary field,” Moscow Univ. Math. Bull., 64, No. 5, 214–215 (2009).
O. V. Markova, “Upper bound for the length of commutative algebras,” Sb. Math., 200, No. 12, 1767–1787 (2009).
O. V. Markova, “Matrix algebras and their length,” in: Matrix Methods: Theory, Algorithms, Applications, World Sci. Publ. (2010), pp. 116–139.
O. V. Markova, “On some properties of the length function,” Math. Notes, 87, No. 1, 71–78 (2010).
C. J. Pappacena, “An upper bound for the length of a finite-dimensional algebra,” J. Algebra, 197, 535–545 (1997).
A. Paz, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables,” Linear and Multilinear Algebra, 15, 161–170 (1984).
Pham Viet Hung, “An upper bound for the dimension of commutative nilpotent subalgebras of a matrix algebra,” Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, 3, 110–111 (1987).
R. S. Pierce, Associative Algebras, Springer, New York (1982).
H. Radjavi and P. Rosenthal, Simultaneous Triangularization, Springer, New York (2000).
I. Schur, “Zur Theorie der vertauschbären Matrizen,” J. Reine Angew. Math., 130, 66–76 (1905).
Y. Song, “A construction of maximal commutative subalgebra of matrix algebras,” J. Korean Math. Soc., 40, No. 2, 241–250 (2003).
A. J. M. Spencer and R. S. Rivlin, “The theory of matrix polynomials and its applications to the mechanics of isotropic continua,” Arch. Ration. Mech. Anal., 2, 309–336 (1959).
A. J. M. Spencer and R. S. Rivlin, “Further results in the theory of matrix polynomials,” Arch. Ration. Mech. Anal., 4, 214–230 (1960).
D. A. Suprunenko and R. I. Tyshkevich, Commutative Matrices, Academic Press, New York (1968); 2nd edition, URSS, Moscow (2003).
V. V. Voevodin and E. E. Tyrtyshnikov, Numerical Processes with Toeplitz Matrices [in Russian], Nauka, Moscow (1987).
A. Wadsworth, “The algebra generated by two commuting matrices,” Linear and Multilinear Algebra, 27, 159–162 (1990).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 6, pp. 65–173, 2011/12.
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Markova, O.V. The length function and matrix algebras. J Math Sci 193, 687–768 (2013). https://doi.org/10.1007/s10958-013-1495-2
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DOI: https://doi.org/10.1007/s10958-013-1495-2