Stably Cayley Groups in Characteristic Zero.

Mikhail Borovoi, Boris Kunyavskiĭ, Nicole Lemire, Zinovy Reichstein

Research output: Contribution to journalArticlepeer-review


A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, that is, a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analog of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SOn defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if is Cayley for some r≥0. Here denotes the split r-dimensional k-torus. These notions were introduced in 2006 by Lemire, Popov, and Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero. In this paper, we study reductive Cayley groups over an arbitrary field k of characteristic zero. Our main results are a criterion for a reductive group G to be stably Cayley, formulated in terms of its character lattice, and a classification of stably Cayley simple groups.
Original languageEnglish
Pages (from-to)5340-5397
Number of pages58
JournalInternational Mathematics Research Notices
Issue number19
StatePublished - 28 Jun 2013


  • CAYLEY graphs
  • LINEAR algebraic groups
  • LIE algebras


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