Stably Cayley Groups in Characteristic Zero

Abstract

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. A Cayley map can be thought of as a partial algebraic analogue 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 G × G r m is Cayley for some r ≥ 0. Here G r m 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.2010 Mathematics Subject Classification. Primary 20G15, 20C10.

Real homogeneous spaces, Galois cohomology, and Reeder puzzles

“…If they are connected by a double edge and the root α k is longer than α i , then α i , α ∨ k = −1 [9, VI.1.3, possibility (5) ], and again vertex k gives a k to the sum. However, if the vertices i and k are connected by a double edge and the root α k is shorter than α i , then α i , α ∨ k = −2 ≡ 0 (mod 2) [9, VI.1.3, possibility (5) ], hence vertex k gives nothing to the sum in (7). We conclude that formula (7) can be written as (5).…”

“…Now we consider cases. If vertices i and k are connected by a single edge, then α i , α ∨ k = −1 [9, VI.1.3, possibility (3) ], hence vertex k gives a k to the sum in (7). If they are connected by a triple edge, then either α i , α ∨ k = −1 or α i , α ∨ k = −3 ≡ −1 (mod 2) [9, VI.1.3, possibility (7) ], and again vertex k gives a k to the sum.…”

Some New Parallels Between Groups and Lie Algebras, or What Can Be Simpler than Multiplication Table?

“…Note that like the work of Poisson-Jacobi, Cayley's invention served as a tool in theoretical mechanics (this time the Lagrangian variant). Note also that the limits up to which the Cayley transform can be generalised have recently been established; see the paper by Nicole Lemire, Vladimir Popov and Zinovy Reichstein [58] where this problem was posed and settled in the case of algebraically closed ground field and the subsequent papers [12,13] for the treatment of the general case.…”

Real reductive Cayley groups of rank 1 and 2

“…For other stably Cayley K-groups, it is a difficult problem to determine whether they are Cayley or not. By [BKLR,Lemma 5.4(c)] the answer to the question whether a K-group is Cayley depends only on the equivalence class of G up to inner twisting.…”

“…By [BKLR,Corollary 7.1] all the reductive K-groups of rank ≤ 2 over a field K of characteristic 0 are stably Cayley (by the rank we always mean the absolute rank). We would like to know, which of those stably Cayley K-groups of rank ≤ 2 are Cayley.…”