Locally complex algebras, introduced by M. Bresar, P. Šemrl, and Š. Špenko, provide a generalization of Cayley–Dickson algebras to the case of arbitrary dimensions. The paper considers the isomorphic classes of locally complex algebras and their automorphism groups. As a characterization of the isomorphism classes, a system of specific matrix equations is used. This system allows one to derive a few necessary conditions for locally complex algebras to be isomorphic. Also classifications of locally complex algebras of dimension three and of their automorphism groups are presented. Bibliography: 5 titles.
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M. Bresar, P. Šemrl, and Š. Špenko, “On locally complex algebras and low-dimensional Cayley–Dickson algebras,” J. Algebra, 327, No. 1, 107–125, (2011).
M. Mitrouli and C. Koukouvinos, “On the computation of the Smith normal form of compound matrices,” Numer. Algorithms, 16, 95–105, (1997).
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M. Marcus, Finite Dimensional Multilinear Algebra, Marcel Dekker, New York (1973–1975).
F. R. Gantmacher, The Theory of Matrices [in Russian], Nauka, Moskow (1967).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 419, 2013, pp. 168–185.
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Smirnov, A.S. Isomorphism Classes and Automorphisms of Locally Complex Algebras. J Math Sci 199, 463–472 (2014). https://doi.org/10.1007/s10958-014-1874-3
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DOI: https://doi.org/10.1007/s10958-014-1874-3