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 taken care of during this quantity from a gaggle theoretical perspective and the got effects are in comparison with the analogous matters within the thought of Lie teams. the most physique of the textual content is dedicated to a category of algebraic teams and Lie teams having purely few subgroups or few issue teams of alternative kind. specifically, the range of the character of algebraic teams over fields of confident attribute and over fields of attribute 0 is emphasised. this can be printed by means of the plethora of third-dimensional unipotent algebraic teams over an ideal box of confident attribute, in addition to, by way of 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**

Let A (respectively A1 , A2 ) be either the Witt group Wm or the vector group (Ga )m . Let Gψ (respectively Gφ1 , Gφ2 ) be a central extension of A (respectively A1 , A2 ) by the (not necessarily commutative) algebraic group B (respectively B1 , B2 ). If ηi : Gψ −→ Gφi are isogenies with ηi (A) = Ai , then there exist hi : Ai −→ A and gi : B −→ Bi such that h2 [φ2 ]g2 = h1 [φ1 ]g1 . Proof. By (i) there exist isogenies fi : A −→ Ai and gi : B −→ Bi , (i = 1, 2), such that f1 [ψ] = [φ1 ]g1 and f2 [ψ] = [φ2 ]g2 .

The next proposition shows the crucial rˆ ole played by the Ore condition in the context of extensions of algebraic groups and isogenies. 13 Proposition. Let A (respectively A1 , A2 ) be either the Witt group Wm or the vector group (Ga )m . Let Gψ (respectively Gφ1 , Gφ2 ) be a central extension of A (respectively A1 , A2 ) by the (not necessarily commutative) algebraic group B (respectively B1 , B2 ). If ηi : Gψ −→ Gφi are isogenies with ηi (A) = Ai , then there exist hi : Ai −→ A and gi : B −→ Bi such that h2 [φ2 ]g2 = h1 [φ1 ]g1 .

Thus X is a C∗ -ﬁber bundle over the complex tori deﬁned by the period matrices √ 10i √ i 10i 2 √ 0 CH(2,3) = , , CH(1,3) = 010i 2 01 0 i 2 CH(1,2) = 10 √ i i 01i 20 . Let X1 = Cn1 /Λ1 , X2 = Cn2 /Λ2 be connected commutative complex Lie groups of maximal ranks n1 , n2 and let P1 , P2 be the corresponding period matrices. Let 0 −→ X1 −→ X −→ X2 −→ 0 be an exact sequence of connected commutative complex Lie groups. 2 we ﬁnd a basis such that the corresponding period matrix is P = P1 Σ ∈ Mn,n+q1 +q2 (C).

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

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