The classical Cayley map,
X
↦
(
I
n
−
X
)
(
I
n
+
X
)
−
1
X \mapsto (I_n-X)(I_n+X)^{-1}
, is a birational isomorphism between the special orthogonal group SO
n
_n
and its Lie algebra
s
o
n
{\mathfrak so}_n
, which is SO
n
_n
-equivariant with respect to the conjugating and adjoint actions, respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the answer is usually “no", with a few exceptions. In particular, we show that a Cayley map for the group SL
n
_n
exists if and only if
n
⩽
3
n \leqslant 3
, answering an old question of Luna.
Upper bounds on the essential dimension of algebraic groups can be found by examining related questions about the integral representation theory of lattices for their Weyl groups. We examine these questions in detail for all simple affine algebraic groups, expanding on work of Lorenz and Reichstein for PGLn. This results in upper bounds on the essential dimensions of these simple affine algebraic groups which match or improve on the previously known upper bounds.
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.
Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G F of F . Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H n (U, F p ) as an F p [G F /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H 1 (U, F p ) was determined only for F a local field, a case settled by Borevič and Faddeev in the 1960s. For the case when the maximal pro-p quotient T of G F is finitely generated, we apply these results to study the partial Euler-Poincaré characteristics of χ n (N ) of open subgroups N of T . We show in particular that the nth partial Euler-Poincaré characteristic χ n (N ) is determined by only χ n (T ) and the conorm in H n (T, F p ).
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