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On the Mesyan conjecture. (English) Zbl 1525.15015

Let \(K\) be a field and let \(M_n(K)\) denote the algebra of \(n \times n\) matrices over \(K\) where \(n \geq 2.\) Let \(f\) be a nonzero polynomial \(f(x_1,\cdots, x_m)\) in \(m \geq 1\) variables that are noncommutative. Consider the evaluation map \begin{align*} f: \ \ M_n(K) \times \cdots \times M_n(K) \rightarrow & \quad M_n(K) \\ (a_1,a_2,\cdots, a_m) \quad \mapsto & \ \ f(a_1,\cdots, a_m) \end{align*} A well-known open question known as Lvov-Kaplansky’s conjecture asserts that if \(f\) is multilinear then the image of this map is a vector space. It can be shown that this conjecture is equivalent to asserting that the image is \(\{0\}, K\) (the space of scalar matrices), \(sl_n(K)\) (the special linear Lie algebra of order \(n\)) or \(M_n(K).\)
A weaker statement is Mesyan’s conjecture: If \(f\) is multilinear and \(m \leq n+1\) then the image of \(f\) on \(M_n(K)\) contains \(sl_n(K).\)
The motivation for both these conjectures comes from a celebrated result of K. Shoda [Japan J. Math 13, 361–365 (1937; JFM 63.0842.03)], which states that if \(M \in sl_n(K)\) then there are matrices \(A,B \in M_n(K)\) such that \(M = AB - BA\), i.e., \(M\) is in the image of the multilinear polynomial \(f(x,y) = xy - yx.\) An example due to M. Rosset and S. Rosset [Commun. Algebra 28, No. 6, 3059–3072 (2000; Zbl 0954.16021)] shows that this may not hold if we work over an arbitrary ring instead of a field.
The case \(m = 2\) of Mesyan’s conjecture follows from a result of K. Shoda [loc. cit.], while the case \(m = 3\) was proved by Z. Mesyan [Linear Multilinear Algebra 61, No. 11, 1487–1495 (2013; Zbl 1290.15011)]. The case \(m = 4\) was proved by D. Buzinski and R. Winstanley [Linear Algebra Appl. 439, No. 9, 2712–2719 (2013; Zbl 1283.15053)], but there was an error in one of the lemmas used in the proof.
In this paper, the authors provide a correct proof of this lemma to settle the case \(m= 4\) of Mesyan’s conjecture.
In addition, they prove in Theorem 3.1 that if \(m \in \{2, \cdots, 2n-1\},\) then the \(K\)-subspace generated by the image of \(f\) contains \(sl_n(K).\) This motivates them to conjecture the following: If \(m \in \{2, \cdots, 2n-1\},\) then the image of \(f\) on \(M_n(K)\) contains \(sl_n(K).\)
The paper exploits the connection between the theory of images of polynomials on algebras and the theory of polynomial identities to prove these results.

MSC:

15A54 Matrices over function rings in one or more variables
15A03 Vector spaces, linear dependence, rank, lineability
15A30 Algebraic systems of matrices
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16S50 Endomorphism rings; matrix rings
12E05 Polynomials in general fields (irreducibility, etc.)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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References:

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