ИСТИНА

Let F be a field and let R be a finite-dimensional associative F-algebra. Following [1,2], we define the length of a finite system of generators S of a given algebra R as the smallest number k such that words in S of length not greater than k generate R as a vector space, and the length of the algebra is the maximum of the lengths of its systems of generators. The length function has applications in computing methods of the mechanics of isotropic continua [3]. For the full matrix algebra M_n(F) the problem of computing the length as a function of the matrix size n is studied since 1984 and is still open. However, there exist some good bounds for the lengths of matrix sets satisfying some additional conditions. In this talk we discuss the length evaluation problem for quasi-commuting pairs of matrices (we say that A, B in M_n(F) quasi-commute if AB and BA are linearly dependent). The quasi-commutativity is an important relation in quantum physics [4,5]. Among all quasi-commuting pairs of matrices there are two special classes that can be singled out and where the analysis is easier than for general pairs. These are (I) commuting pairs and (II) quasi-commutative, non-commuting pairs with a nilpotent product. We show that in each of these cases l(S)≤n-1 and, moreover, for any l=1,...,n-1, each these two classes contains a pair of matrices with length l. If a quasi-commuting pair S={A,B}⊂M_n(F) does not belong to the class (I)∪(II), then AB=εBA, where the commutativity factor ε is a primitive k-th root of unity for some k≤n. We will show that in this case the situation is very different from the commutative and nilpotent case. We obtain sharp bounds 2k-2≤l(S)≤2n-2. Moreover, we show that the length 2n-2 is attainable if and only if the quasi-commutativity factor is a primitive n-th root of unity. For roots of unity with other degrees we provide a sharp upper bound l({A,B})≤max{2(n-k)-r; n+k}-2, where r denotes the algebraic multiplicity of 0 as an eigenvalue of AB. For the length realizability problem of quasi-commuting matrix pairs not belonging to the class (I)∪(II) we have the following partial results: 1) All even numbers between 2 and 2n-2 are realizable; 2) The numbers 1 and 2n-3 are not realizable; 3) The number 2n-5 is realizable for n=4,6 and is not realizable for n=5 or n>6; 4) For all natural numbers k,m,n, k≥ 2, n≥ (m+1)k there exist matrices An,Bn∈M_n(F) such that An and Bn quasi-commute with the factor being a primitive k-th root of unity and l({An,Bn})=(m+1)k-1. The talk is based on our paper [6]. The work was partially supported by grants MD-962.2014.1, RFBR 13-01-00234-a and 15-31-20329. References 1. A. Paz, An application of the Cayley-Hamilton theorem to matrix polynomials in several variables, Linear Multilinear Algebra, 15(1984), 161-170. 2. C.J. Pappacena, An upper bound for the length of a finite-dimensional algebra , J. Algebra, 197(1997), 535-545. 3. A.J.M. Spencer, R.S. Rivlin, The theory of matrix polynomials and its applications to the mechanics of isotropic continua, Arch. Ration. Mech. Anal., 2(1959), 309-336. 4. C. Kassel, Quantum Groups. Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995. 5. Yu.I. Manin, Quantum groups and non-commutative geometry, CRM, Montréal, 1988. 6. A.E. Guterman, O.V. Markova, V. Mehrmann, Lengths of quasi-commutative pairs of matrices, Preprint 9-2015, Institute of Mathematics, Technische Universität Berlin, 2015; submitted for publication.