| Résume | Elements of the free associative algebra and of the free skew field (polynomials and rational functions, as defined by Amitsur and P.M. Cohn, in noncommuting variables) can be fruitfully viewed as functions on tuples of square matrices of all sizes, in the spirit of noncommutative function theory that originated in the work of J.L. Taylor in the early 1970s and developed extensively in the last two decades. This leads naturally to certain noncommutative power series around a matrix centre, called Taylor--Taylor series after both J.L. Taylor and Brook Taylor of the calculus fame. After setting the stage, I will describe two results: one shows that the ring of these series is exactly the completion of the free algebra with respect to the (two sided) ideal of all elements vanishing at the centre (provided the centre is semisimple); the other one identifies the subring of rational series, generalizing the results of Kronecker in the single variable case and of Fliess in the case of a scalar (rather than matrix) centre.
The talk is based on joint works with Igor Klep and Jurij Volcic and with Motke Porat.
References:
S. A. Amitsur. Rational identities and applications to algebra and geometry, J. Algebra 3:304–359, 1966.
P. M. Cohn. Skew Fields. Theory of general division rings, Encyclopedia of Mathematics and its Applications 57, Cambridge University Press, Cambridge, 1995;
Free ideal rings and localization in general rings, New Mathematical Monographs 3, Cambridge University Press, Cambridge, 2006.
M. Fliess, Matrices de Hankel, J. de Math. Pures et Appliquees, 53, pp. 197–222, 1974.
Joseph L. Taylor. A general framework for a multi-operator functional calculus, Advances in Math., 9:183–252, 1972;
Functions of several noncommuting variables, Bull. Amer. Math. Soc., 79:1–34, 1973.
Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov. Foundations of free noncommutative function theory, volume 199 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2014.
I. Klep, V. Vinnikov, and Ju. Volcic. Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation, Trans. Amer. Math. Soc. 373 (2020), 5587–5625.
M. Porat, V. Vinnikov. Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras, J. London Math. Soc. 104 (2021), 1250–1299;
Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions, Int. Equat. Oper. Theory 95 (2023), article 1.
|
