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Formal proofs of operator identities by a single formal computation
Authors:Clemens G. Raab  Georg Regensburger  Jamal Hossein Poor
Affiliation:1. Graduate School of Mathematical Sciences, The University of Tokyo, Meguro-ku, Tokyo 153-8914, Japan;2. Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany;3. Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany;1. Faculty of Science, University of Toyama, 3190 Gofuku, Toyama-shi 930-8555, Japan;2. Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India
Abstract:A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.
Keywords:Matrix identities  Algebraic operator identities  Noncommutative polynomials  Linear categories  Quiver representations
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