349:. Moreover, in characteristic 2 there are additional possibilities arising not from exceptional isogenies but rather from the fact that for simply connected type C (I.e., symplectic groups) there are roots that are divisible (by 2) in the weight lattice; this gives rise to examples whose root system (over a separable closure of the ground field) 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
333:
is mysterious. The commutative pseudo-reductive groups admit no useful classification (in contrast with the connected reductive case, for which they are tori and hence are accessible via Galois lattices), 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 (possibly inseparable) extensions of the ground field.
332:
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 (which are always commutative and pseudo-reductive)
230:-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.
336:
Over imperfect fields of characteristics 2 and 3 there are some extra pseudo-reductive groups (called exotic) coming from the existence of exceptional isogenies between groups of types B and C in characteristic 2, between groups of type
78:). 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).
328:
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
353:(due to complications caused by the examples with a non-reduced root system, as well as phenomena related to certain regular degenerate quadratic forms that can only exist when
92:, 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
365:, completes the classification in characteristic 2 up to a controlled central extension by providing an exhaustive array of additional constructions that only exist when
88:
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,
369:, ultimately resting on a notion of special orthogonal group attached to regular but degenerate and not fully defective quadratic spaces in characteristic 2.
185:, 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
541:
507:
466:
438:
400:
312:) is pseudo-reductive but not reductive. The previous example is the special case using the multiplicative group and the extension
422:
384:
567:
493:"Groupes algébriques pseudo-réductifs et applications (d'après J. Tits et B. Conrad--O. Gabber--G. Prasad)"
89:
35:
21:
472:
537:
513:
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462:
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388:
551:
525:
484:
461:, Annals of Mathematics Studies, vol. 191, Princeton, NJ: Princeton University Press,
448:
410:
547:
521:
480:
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406:
62:
found some examples of pseudo-reductive groups that are not reductive. A pseudo-reductive
55:
536:, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston,
561:
70:-unipotent radical does not generally commute with non-separable scalar extension on
51:
492:
59:
456:
198:, but this reduced subgroup scheme of the geometric fiber is not defined over
517:
430:
392:
476:
346:
286:-homomorphism descends the unipotent radical of the geometric fiber of
84:
gives an exposition of Tits' results on pseudo-reductive groups, while
46:-unipotent radical (i.e., largest smooth connected unipotent normal
345:
in characteristic 3, using a construction analogous to that of the
361:, building on additional material included in the second edition
271:-group for which there is a (surjective) homomorphism from
421:, New Mathematical Monographs, vol. 26 (2 ed.),
383:, New Mathematical Monographs, vol. 17 (1 ed.),
108:
Examples of pseudo reductive groups that are not reductive
66:-group need not be reductive (since the formation of the
237:
is a non-trivial purely inseparable finite extension of
294:(i.e., does not arise from a closed subgroup scheme of
194:
and is the unipotent radical of the geometric fiber of
74:, such as scalar extension to an algebraic closure of
202:(i.e., it does not arise from a closed subscheme of
417:Conrad, Brian; Gabber, Ofer; Prasad, Gopal (2015),
379:Conrad, Brian; Gabber, Ofer; Prasad, Gopal (2010),
362:
97:
85:
341:in characteristic 2, and between groups of type G
116:is a non-perfect field of characteristic 2, and
8:
358:
101:
458:Classification of pseudo-reductive groups.
249:-group defined then the Weil restriction
81:
245:is any non-trivial connected reductive
96:, and later work in the second edition
455:Conrad, Brian; Prasad, Gopal (2016),
7:
93:
324:Classification and exotic phenomena
363:Conrad, Gabber & Prasad (2015)
98:Conrad, Gabber & Prasad (2015)
86:Conrad, Gabber & Prasad (2010)
54:these are the same as (connected)
14:
128:be the group of nonzero elements
267:) is a smooth connected affine
226:-group but is not a reductive
104:provides further refinements.
58:, but over non-perfect fields
1:
152:to the multiplicative group
50:-subgroup) is trivial. Over
532:Springer, Tonny A. (1998),
148:. There is a morphism from
584:
423:Cambridge University Press
385:Cambridge University Press
359:Conrad & Prasad (2016)
124:that is not a square. Let
102:Conrad & Prasad (2016)
431:10.1017/CBO9781316092439
393:10.1017/CBO9780511661143
290:and is not defined over
34:) is a smooth connected
534:Linear algebraic groups
491:RĂ©my, Bertrand (2011),
419:Pseudo-reductive groups
381:Pseudo-reductive groups
357:). Subsequent work of
90:classification theorems
222:is a pseudo-reductive
214:-unipotent radical of
206:over the ground field
36:affine algebraic group
18:pseudo-reductive group
330:standard construction
282:. The kernel of this
27:(sometimes called a
233:More generally, if
16:In mathematics, a
543:978-0-8176-4021-7
509:978-2-85629-326-3
468:978-0-691-16793-0
440:978-1-107-08723-1
402:978-0-521-19560-7
120:is an element of
575:
568:Algebraic groups
554:
528:
502:(339): 259–304,
497:
487:
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413:
218:is trivial. So
176:
175:
143:
142:
56:reductive groups
32:-reductive group
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60:Jacques Tits
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94:RĂ©my (2011)
500:Astérisque
373:References
347:Ree groups
210:) and the
518:0303-1179
562:Category
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