By Alfonso Di Bartolo, Giovanni Falcone, Peter Plaumann, Karl Strambach

ISBN-10: 3540785833

ISBN-13: 9783540785835

ISBN-10: 3540785841

ISBN-13: 9783540785842

Algebraic teams are handled during this quantity from a bunch theoretical perspective and the bought effects are in comparison with the analogous concerns within the conception of Lie teams. the most physique of the textual content is dedicated to a class of algebraic teams and Lie teams having simply few subgroups or few issue teams of other kind. particularly, the variety of the character of algebraic teams over fields of optimistic attribute and over fields of attribute 0 is emphasised. this can be printed through the plethora of three-d unipotent algebraic teams over an ideal box of optimistic attribute, in addition to, through many concrete examples which conceal a space systematically. within the ultimate part, algebraic teams and Lie teams having many closed basic subgroups are determined.

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**Example text**

Hence can ﬁnd two isogenies hi : Ai −→ A such that h1 f1 = h2 f2 and we obtain h2 [φ2 ]g2 = h1 [φ1 ]g1 . We have already mentioned that the groups SL2 and P SL2 show that the existence of an isogeny is not a symmetric relation. However, if there exists an isogeny from a connected commutative unipotent group G1 onto G2 then an isogeny from G2 onto G1 exists as well (see [89], Proposition 10, p. 176). Already for unipotent connected non-commutative algebraic groups this is not any more the case as the following example shows.

We recall that the ring Endk (Ga ) of k-endomorphisms of the additive group Ga is isomorphic to the non-commutative ring k[F] of p-polynomials, where F is the Frobenius homomorphism and i αi Fi : x → αi xp . i i The following proposition shows that in odd characteristic any factor system can be derived by Φ1 and η1 only. 9 Proposition. If the characteristic of the ground ﬁeld is greater than 2, then in the free left End(Ga )-module H2 (Ga , Ga ) we have k−1 2k−1 Fi [η1 (1 + F2(k−i)−1 )] − [η2k ] = i=0 k−1 i 2(k−i) F [η1 (1 + F i=0 [η1 ] i=0 k [η2k+1 ] = Fi )] + F [η1 ] − k F [η1 ] − k i i=1 Fk+i [η1 ].

Let X1 , X2 be connected commutative complex Lie groups of maximal rank n1 , n2 and let P1 , P2 be the corresponding period matrices. The period matrix P = P1 Σ ∈ Mn,n+q1 +q2 (C) 0 P2 (n = n1 + n2 ) deﬁnes an extension of X1 by X2 which is isogenous to a split one, via an lIn1 0 lIn1 +q1 0 isogeny f such that ρa (f ) = and ρr (f ) = , if and 0 In2 0 In2 +q2 only if Σ = P1 M − AP2 with A ∈ Mn1 ,n2 (C) and M ∈ Mn1 +q1 ,n2 +q2 (Q), where l ∈ Z is such that lM has integral entries. Extensions of complex tori X1 and X2 which are not isogenous to a split analytic extension X1 ⊕ X2 are called Shafarevich extensions in [7], Ch.

### Algebraic groups and lie groups with few factors by Alfonso Di Bartolo, Giovanni Falcone, Peter Plaumann, Karl Strambach

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