ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 16, 17, 22 Michaelis, Walter J., University of New Orleans, New Orleans, LA 70148, U. S. A. The exponential matrices of non-commuting matrices may commute We show how to construct entire families of non-commuting 2-by-2 real matrices A and B for which e^{A} · e^{B} ≠ e^{A+B} and yet e^{A} · e^{B} = e^{B} · e^{A} . Though it is known that e^{sA} · e^{tB} = e^{tB} · e^{sA} , ∀ s, t ∈ ℝ ⟺ A · B = B · A and that e^{At} · e^{Bt} = e^{(A+B)t} , ∀ t ∈ ℝ ⟺ A · B = B · A, our examples provide non-commuting 2-by-2 real matrices A and B for which e^{At} · e^{B} = e^{B} · e^{At} , ∀ t ∈ ℝ. We also give examples of non-commuting 4-by-4 real matrices (and, equivalently, of 2-by-2 complex matrices) A and B such that e^{A} = e^{B} = e^{A+B} = I = the identity matrix and thus e^{A} · e^{B} = e^{A+B} = e^{B} · e^{A}.