Pseudo-reductive algebraic group
In mathematics, a pseudo-reductive group over a field k (sometimes called a k-reductive group) is a smooth connected affine algebraic group defined over k whose k-unipotent radical (i.e., largest smooth connected unipotent normal k-subgroup) is trivial. Over perfect fields these are the same as (connected) reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. A pseudo-reductive k-group need not be reductive (since the formation of the k-unipotent radical does not generally commute with non-separable scalar extension on k, such as scalar extension to an algebraic closure of k). Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic (even over a perfect field of constants).
Springer (1998) gives an exposition of Tits' results on pseudo-reductive groups, while Conrad, Gabber & Prasad (2010) builds on Tits' work to develop a general structure theory, including more advanced topics such as construction techniques, root systems and root groups and open cells, classification theorems, and applications to rational conjugacy theorems for smooth connected affine groups over arbitrary fields. The general theory (with applications) as of 2010 is summarized in Rémy (2011), and later work in the second edition Conrad, Gabber & Prasad (2015) and in Conrad & Prasad (2016) provides further refinements.
Suppose that k is a non-perfect field of characteristic 2, and a is an element of k that is not a square. Let G be the group of nonzero elements x + y√a in k[√a]. There is a morphism from G to the multiplicative group Gm taking x + y√a to its norm x2 – ay2, and the kernel is the subgroup of elements of norm 1. The underlying reduced scheme of the geometric kernel is isomorphic to the additive group Ga and is the unipotent radical of the geometric fiber of G, but this reduced subgroup scheme of the geometric fiber is not defined over k (i.e., it does not arise from a closed subscheme of G over the ground field k) and the k-unipotent radical of G is trivial. So G is a pseudo-reductive k-group but is not a reductive k-group. A similar construction works using a primitive nontrivial purely inseparable finite extension of any imperfect field in any positive characteristic, the only difference being that the formula for the norm map is a bit more complicated than in the preceding quadratic examples.
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