nLab Cayley transform

Idea

The classical Cayley transform is an involutive bijection between the vector space of skew-symmetric matrices to the group of special orthogonal matrices, given by the formula

$$X \mapsto \frac{I + X}{I - X} \,.$$

There are many generalizations (some non-involutive), say for J-orthogonal matrices, to shifted center, for operators, in complex analysis and so on.

Literature

• M M Postnikov, Lectures on geometry, Semester V, Lie groups and Lie algebras
• wikipedia/Cayley transform
• https://encyclopediaofmath.org/wiki/Cayley_transform (for operators)
• D. Quillen, Superconnection character forms and the Cayley transform, Topology 27 (1988), no. 2, 211–238
• A. Gómez-Tato, E. Macías-Virgós, M. J. Pereira-Sáez, Trace map, Cayley transform and LS category of Lie groups, Ann Glob Anal Geom 39, 325–335 (2011) doi
• E. Macías-Virgós, M. José Pereira-Sáez, D. Tanré, Cayley transform on Stiefel manifolds, J. Geom. Phys. 123, 53-60 (2018) doi
• Michael Jauch, Peter D. Hoff, David B. Dunson, Random orthogonal matrices and the Cayley transform, Bernoulli 26:2 (2020) 1560-1586 doi

The general case related to symmetric spaces:

• F Astengo, M Cowling, B Di Blasio, The Cayley transform and uniformly bounded representations, J. Funct. Analysis 213:2 (2004) 241-269 doi