Homomorphisms of matrix semigroups over division rings from dimension two to four
Let \(R\) be a ring and let \(M_n (R)\) denote the set of \(n \times n\) matrices over \(R\). A map \(\varphi : M_n (R)\rightarrow M_m (R)\) is called multiplicative if it satisfies \(\varphi(AB) =\varphi(A)\varphi(B)\) for all \(A, B\) in \(M_n (R)\). We say \(\varphi\) is non-degenerate if there exists a singular matrix \(M\) such that \(\varphi(M ) = 0\). The study of characterizing multiplicative maps between matrix groups over a field was started in the 1950's in the work of \textit{E. A. Khalezov} [Dokl. Akad. Nauk SSSR, n. Ser. 96, 245--248 (1954; Zbl 0055.01405); Ivanov. Gos. Ped. Inst., Uchenye Zap., Fiz.-Mat. Nauki 5, 42--56 (1954; Zbl 0059.01903)] and of \textit{L. M. Gluskin} [Usp. Mat. Nauk 11, No.~1(67), 199--206 (1956; Zbl 0070.01904)]. \textit{M. Jodiet jun.} and \textit{T.-Y. Lam} [Arch. Math. 20, 10--16 (1969; Zbl 0176.29502)] gave a classification of non-degenerate multiplicative self maps of \(M_n (R)\), where \(R\) is a principal ideal domain. \textit{S. Pierce} [ibid. 24, 25--29 (1973; Zbl 0253.20089)] generalized their result for Dedekind domains for the case \(n = 2\). The more explicit results for non-degenerate self maps can be found in the work of P. Šemrl over division rings. The degenerate case can be transformed in such a way that the known results about multiplicative maps to lower dimensional matrix spaces can be used. Recently, for a field \(\mathbb F\) and for \(n < m\), the non-degenerate multiplicative maps from \(M_n (\mathbb F)\) to \(M_m (\mathbb F)\) have been classified in several papers by \textit{D. Kokol-Bukovek}. In [Linear Algebra Appl. 346, No.~1--3, 73--95 (2002; Zbl 0998.15017)], he has studied non-degenerate, irreducible multiplicative map \(\varphi: M_2 (\mathbb C) \rightarrow M_4 (\mathbb C)\), where \(\mathbb C\) stands for the field of complex numbers. This paper gives a classification of non-degenerate multiplicative maps from \(M_2 (\mathbb D)\) to \(M_4 (\mathbb D)\), where \(\mathbb D\) is a non-commutative division ring.
scientific article; zbMATH DE number 5667889
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