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

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

Algebraic teams are handled during this quantity from a gaggle theoretical perspective and the got effects are in comparison with the analogous matters within the idea of Lie teams. the most physique of the textual content is dedicated to a category of algebraic teams and Lie teams having in simple terms few subgroups or few issue teams of other style. particularly, the variety of the character of algebraic teams over fields of confident attribute and over fields of attribute 0 is emphasised. this can be published by means of the plethora of 3-dimensional unipotent algebraic teams over an ideal box of confident attribute, in addition to, by means of many concrete examples which hide a space systematically. within the ultimate part, algebraic teams and Lie teams having many closed basic subgroups are determined.

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7 Remark. 5, p. 148, that G/zG is either a one-dimensional torus or a unipotent group. Hence G is nilpotent and G/zG is a unipotent chain. This yields that G/zG has a unique d-dimensional connected algebraic subgroup for any d ≤ dim G/zG. In the first theorem of this section we relate the size of the commutator subgroup to the size of centre for these groups. 8 Theorem. Let G be a non-commutative connected algebraic k-group. If the connected component z◦ G of zG is a maximal connected subgroup of G, then z◦ G has co-dimension one in G, the characteristic of k is positive and the commutator subgroup G is a (central) vector group.

5, p. 148, that G/zG is either a one-dimensional torus or a unipotent group. Hence G is nilpotent and G/zG is a unipotent chain. This yields that G/zG has a unique d-dimensional connected algebraic subgroup for any d ≤ dim G/zG. In the first theorem of this section we relate the size of the commutator subgroup to the size of centre for these groups. 8 Theorem. Let G be a non-commutative connected algebraic k-group. If the connected component z◦ G of zG is a maximal connected subgroup of G, then z◦ G has co-dimension one in G, the characteristic of k is positive and the commutator subgroup G is a (central) vector group.

If A and B are algebraic groups, it is necessary to assume that the factor system F and all the automorphisms a → ab for all b ∈ B are rational maps, in order to have the extension of A by B as an algebraic group. ), a + a +φ(b0 , b1 , b0 , b1 )). 6) Let H = {(b0 , b1 , a) ∈ Gφ : b0 = 1}. Under the assumption that [G, H] ≤ A we want to find the factor system γ corresponding to the section τ : B2 −→ Gφ , τ (b0 ) = (b0 , 0, 0) of the non-central extension 1 −→ H −→ Gφ −→ B2 −→ 1 and we want to compare this factor system with φ.

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