It is proved that normalised Stiefel manifolds admit a rational parametrisation which generalises Cayley's parametrisation of the unitary groups. Applying (the quaternionic case of) this parametrisation to the Atiyah-Drinfeld-Hitchin-Manin (ADHM) instanton matrix equations, large families of new explicit rational solutions emerge. In particular, new explicit non-'t Hooft solutions are presented.
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I am grateful to Professor P.M. Cohn for showing me a private letter (dated October 1978) to him from R. Wood, in which Wood shows that, for any square quaternionic matrix A, A−αl is singular for some α∈H. He does this in analogy to the corresponding theorem for complex matrices (which uses π 1(GL(n,C))≈Z), and uses the fact that π 3(GL(n,H))≈Z, where Z = integers.
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McCarthy, P.J. Rational parametrisation of normalised stiefel manifolds, and explicit non-'t hooft solutions of the Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations for Sp(n). Lett Math Phys 5, 255–261 (1981). https://doi.org/10.1007/BF00401473
- Statistical Physic
- Group Theory
- Large Family
- Matrix Equation