Stably Cayley Semisimple Groups

A linear algebraic group $G$ over a field $k$ is called a Cayley group if it admits a Cayley map, i.e., a $G$-equivariant birational isomorphism over $k$ between the group variety $G$ and its Lie algebra ${Lie}(G)$. A prototypical example is the classical «Cayley transform" for the special orthogonal group ${\bf SO}_n$ defined by Arthur Cayley in 1846. A linear algebraic group $G$ is called stably Cayley if $G \times S$ is Cayley for some split $k$-torus $S$. We classify stably Cayley semisimple groups over an arbitrary field $k$ of characteristic 0.

2010 Mathematics Subject Classification: 20G15, 20C10.

Keywords and Phrases: Linear algebraic group, stably Cayley group, quasi-permutation lattice.

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