The images of non-commutative polynomials evaluated on $2\times 2$ matrices

Kanel–Belov, A.; Malev, S.; Rowen, L.H.

Авторы: Kanel–Belov A., Malev S., Rowen Louis H.
Журнал: Proceedings of the American Mathematical Society. American Mathematical Society
Том: 140
Год издания: 2012
Издательство: American Mathematical Society.
Местоположение издательства: [Providence, R.I., etc.], United States
Первая страница: 465
Последняя страница: 478
DOI: 10.1090/S0002-9939-2011-10963-8
Аннотация: The authors study the following important problem, reputedly raised by Kaplansky: Let p be a polynomial in the free associative algebra K⟨x1,…,xm⟩ over an arbitrary field K. What is the image Im(p) of p evaluated on the algebra Mn(K) of n×n matrices? An important case of this problem was formulated by L’vov: If p is multilinear, is the set of values of p on Mn(K) a vector space? The answer into affirmative is equivalent to the following conjecture: If p is multilinear, then its image on Mn(K) is either {0}, the set K of scalar matrices, the set sln(K) of matrices of trace zero, or the whole algebra Mn(K). The results in the paper under review handle the case of 2×2 matrices over a quadratically closed field K of any characteristic. The latter means that the field K contains all zeros of non-constant polynomials f(x)∈K[x] of degree ≤2degp. The main result is that over a quadratically closed field K the image of the multilinear polynomial p on M2(K) is either {0}, K, sl2(K), or M2(K). This result is a consequence of a stronger result which is of independent interest: For an m-tuple of integers (w1,…,wm), the polynomial p(x1,…,xm) is semi-homogeneous of weighted degree d if for each monomial h in p, taking dj to be the degree of xj in h, it holds d1w1+⋯+dmwm=d. If the semi-homogeneous polynomial p(x1,…,xm) is evaluated on the algebra M2(K) over a quadratically closed field K, then Im(p) is either {0}, K, the set of all non-nilpotent matrices having trace 0, sl2(K), or a dense subset of M2(K) with respect to the Zariski topology.Reviewer: Vesselin Drensky (Sofia
Добавил в систему: Белов Алексей Яковлевич

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