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

Algebraic teams are taken care of during this quantity from a gaggle theoretical standpoint and the bought effects are in comparison with the analogous concerns within the concept of Lie teams. the most physique of the textual content is dedicated to a category of algebraic teams and Lie teams having in basic terms few subgroups or few issue teams of other variety. particularly, 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 through the plethora of 3-dimensional unipotent algebraic teams over an ideal box of confident attribute, in addition to, via many concrete examples which conceal a space systematically. within the ultimate part, algebraic teams and Lie teams having many closed general subgroups are determined.

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**Example text**

11 Remark. We illustrate here the fact that the functor H2 (B, A) is contra-variant in B and co-variant in A in the special case of isogenies. The arguments and the notations are essentially those of [89], VII, 1. p. 164-165. 1) For any explicit central extension Gφ1 1 −→ A1 −→ Gφ1 −→ B −→ 1 of A1 by B, deﬁned by the factor system φ1 : B × B −→ A1 , and any isogeny α : A1 −→ A2 there exists a unique (up to equivalence) explicit central extension Gφ2 1 −→ A2 −→ Gφ2 −→ B −→ 1 and an isogeny α∗ : Gφ1 −→ Gφ2 , such that the following diagram commutes 1 −→ A1 −→ Gφ1 −→ B −→ 1 idB↓ α↓ α∗↓ 1 −→ A2 −→ Gφ2 −→ B −→ 1.

2, we call P the period matrix of X. The imaginary part of G has real rank q, because the columns of P are R-independent. Up to a permutation of the vectors of the basis we can assume that the imaginary part of T is invertible. For q = 0 we have Λ = Zn , hence the group X = Cn /Zn ∼ = (C∗ )n is a linear torus, whereas for q = n the group X is a complex torus by deﬁnition ( [7], p. 1). 11, p. 9, if P = (In G) is the matrix of a R-basis of the lattice Λ, the group Cn /Λ is toroidal if and only the following irrationality condition holds: where the columns of G = for any non-zero v ∈ Zn the vector vG is never contained in Zq .

Now, by the Hurwitz relations we have (0 In−k) P1 Σ 0 A = P2 (0 In+q−k−q1) from which it follows that A = P2 . Now we look for closed linear subtori of a connected commutative complex Lie group X ∼ = Cn /Λ of maximal rank n with period matrix P = (In G). Denote by H = H(l1 , · · · , ln−m ) the m-dimensional subspace of Cn deﬁned by H = (z1 , · · · , zn ) ∈ Cn : zlk = 0 for lk ∈ {1, · · · , n} and k = 1, · · · , n − m and let CH (P ) be the matrix obtained from P in the following way: we cancel in P any row with exception of those labeled by l1 , · · · , ln−m as well as any of the ﬁrst n columns with exception of those labeled by l1 , · · · , ln−m .