Galois cohomology of real quasi-connected reductive groups

Computing Galois cohomology of a real linear algebraic group

“…Proof See [2, section 6, Corollary 1 of Proposition 8], or [15, Lemma 3.2.1], or [18, Lemma 1.3].$$\Box$$…”

“…The result of [8] has been used in a few articles, in particular, in [27, 60] and [57]. Using this result, Borovoi, Gornitskii, and Rosengarten [14] described the Galois cohomology of quasi‐connected reductive $${\mathbb {R}}$$‐groups (normal subgroups of connected reductive $${\mathbb {R}}$$‐groups). In [13], Borovoi and Evenor used the result of [8] to describe explicitly the Galois cohomology of simply connected semisimple $${\mathbb {R}}$$‐groups.…”

“…However, as said above, for our purposes the size is not enough: We need explicit cocycles representing the elements of $${\rm H}^1{\bf G}$$. Moreover, the explicit results of [17] and [18] for connected reductive $${\mathbb {R}}$$‐groups, and the less explicit results of [14] for quasi‐connected reductive $${\mathbb {R}}$$‐groups, are not enough either: we need to be able to compute $${\rm H}^1{\bf G}$$ for linear algebraic groups that are not necessarily reductive, or connected, or quasi‐connected. In this paper, we give algorithms, implemented on computer, to determine the Galois cohomology set of an arbitrary real linear algebraic group.…”

Galois cohomology of reductive algebraic groups over the field of real numbers

“…Since it was announced in [3], our Theorem 3.1 has been used in a few articles, in particular, in [17], [10], and [15]. In [4], Gornitskii, Rosengarten, and the author described, using Theorem 3.1, the Galois cohomology of quasi-connected reductive R-groups (normal subgroups of connected reductive R-groups). Our description in [4] is similar to that of Theorem 3.1.…”

“…In [4], Gornitskii, Rosengarten, and the author described, using Theorem 3.1, the Galois cohomology of quasi-connected reductive R-groups (normal subgroups of connected reductive R-groups). Our description in [4] is similar to that of Theorem 3.1. In [5] Evenor and the author used Theorem 3.1 to describe explicitly the…”