In mathematics, a pseudo-reductive group over a fieldk is a smooth connected affine algebraic group defined over k whose k-unipotent radical is trivial. Over perfect fields these are the same as 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. Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic. gives an exposition of Tits' results on pseudo-reductive groups, while 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 as of 2010 is summarized in , and later work in the second edition and in provides further refinements.
Examples of pseudo reductive groups that are not reductive
Suppose that k is a non-perfect field of characteristic 2, and a is an element ofk that is not a square. Let G be the group of nonzero elements x + y in k. There is a morphism from G to the multiplicative groupGm taking x + y 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 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. More generally, if K is a non-trivial purely inseparable finite extension of k and G is any non-trivial connected reductive K-group defined then the Weil restrictionH=RK/k is a smooth connected affine k-group for which there is a homomorphism from HK onto G. The kernel of this K-homomorphism descends the unipotent radical of the geometric fiber of H and is not defined over k, so RK/k is pseudo-reductive but not reductive. The previous example is the special case using the multiplicative group and the extension K=k.
Classification and exotic phenomena
Over fields of characteristic greater than 3, all pseudo-reductive groups can be obtained from reductive groups by the "standard construction", a generalization of the construction above. The standard construction involves an auxiliary choice of a commutative pseudo-reductive group, which turns out to be a Cartan subgroup of the output of the construction, and the main complication for a general pseudo-reductive group is that the structure of Cartan subgroups is mysterious. The commutative pseudo-reductive groups admit no useful classification, but modulo this one has a useful description of the situation away from characteristics 2 and 3 in terms of reductive groups over some finite extensions of the ground field. Over imperfect fields of characteristics 2 and 3 there are some extra pseudo-reductive groups coming from the existence of exceptional isogenies between groups of types B and C in characteristic 2, between groups of type F₄ in characteristic 2, and between groups of type G₂ in characteristic 3, using a construction analogous to that of the Ree groups. Moreover, in characteristic 2 there are additional possibilities arising not from exceptional isogenies but rather from the fact that for simply connected type C there are roots that are divisible in the weight lattice; this gives rise to examples whose root system is non-reduced; such examples exist with a split maximal torus and an irreducible non-reduced root system of any positive rank over every imperfect field of characteristic 2. The classification in characteristic 3 is as complete as in larger characteristics, but in characteristic 2 the classification is most complete when =2. Subsequent work of, building on additional material included in the second edition , completes the classification in characteristic 2 up to a controlled central extension by providing an exhaustive array of additional constructions that only exist when >2, ultimately resting on a notion of special orthogonal group attached to regular but degenerate and not fully defective quadratic spaces in characteristic 2.
