By Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen

This milestone paintings at the mathematics concept of linear algebraic teams is now on hand in English for the 1st time. **Algebraic teams and quantity thought offers the 1st systematic exposition in mathematical literature of the junction of staff thought, algebraic geometry, and quantity conception. The exposition of the subject is outfitted on a synthesis of equipment from algebraic geometry, quantity thought, research, and topology, and the result's a scientific assessment ofalmost the entire significant result of the mathematics idea of algebraic teams got thus far.
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Algebraic Groups 51 for any L-group G. Now let K = Q and let wl, . . ,wd be a base of O/Z where O = OL, the ring of integers of L. Taking the regular representation Q with respect to this base, we obtain RLIK(G)z 2 Go, and RLIK(G)zpe! Go, for any prime number p. n VIP in the n2d2 variables y ~ ' , where a , @ = 1 , .. ,d and i, j = 1 , . . ,j . Then the image of G i in Mnd(K) under Q is defined by the equations (where 0 in the last equation denotes the zero matrix in Md(K)). 3) in GLn(R). Then GI is the desired algebraic K-group.

Then any order B c A is contained in some maximal order. PROOF: As above, this reduces to the case of a central simple K-algebra A. It suffices to show that the set {Bi) of orders in A containing B is finite. First this assertion is proved for a matrix algebra A = Mn(K). Clearly here A has a maximal order C = Mn(0). 15 it follows that B, = C, is a maximal order in AKv = Mn(Kv) for almost all v in v ~ K . Moreover, for the remaining v the number of orders in AK. containing B, is finite. 15, yields the required result.

In particular, G is always C GLn(R) be a unipotent K-group. Then G is Now let G connected. , there exists a matrix g in GLn(K) such that gGg-i is contained in the group U n of upper unitriangular matrices. It follows, in particular, that G is nilpotent. - (6, for i = 0, . . , n - 1. Note that most of the above statements do not carry over for positive characteristic. We shall require a technical assertion about unipotent groups. such that the factors Gi/Gi+1 are isomorphic to G, or (6,. 5) of K-subgroups such that the Gi/Gi+l are K-isomorphic to 6 , or G,, then G is said to be K-split.