Burnside type theorems in real and quaternion settings

2017/10/10 by Yahaghi, Bamdad R.
#15A30 #20M20 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1710.03849

Abstract

In this note, we consider irreducible semigroups of real, complex, and quaternionic matrices with real spectra. We prove Burnside type theorems in the settings of reals and quaternions. First, we prove that an irreducible semigroup of triangularizable matrices in M_n(ℝ) contains a vector space basis for M_n(ℝ). In other words, M_n(ℝ) is the only irreducible subalgebra of itself that is spanned by an irreducible semigroup of triangularizable matrices in M_n(ℝ). Next, we use this result to show that, up to similarity, M_n(ℝ) is the only irreducible ℝ-algebra in M_n(ℍ) that is spanned by an irreducible semigroup of matrices in M_n(ℍ) with real spectra. Some consequences of our mains results are presented.

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