By Alfonso Di Bartolo, Giovanni Falcone, Peter Plaumann, Karl Strambach
Algebraic teams are taken care of during this quantity from a gaggle theoretical perspective and the got effects are in comparison with the analogous concerns within the thought of Lie teams. the most physique of the textual content is dedicated to a class of algebraic teams and Lie teams having merely few subgroups or few issue teams of alternative sort. specifically, the variety of the character of algebraic teams over fields of optimistic attribute and over fields of attribute 0 is emphasised. this is often published through the plethora of three-d 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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Extra info for Algebraic groups and lie groups with few factors
6. In algebraic groups over ﬁelds of characteristic zero or Lie groups G the sum of the dimensions of the commutator subgroup and the center can be smaller, equal or larger than dim G. Let g be a nilpotent Lie algebra over a ﬁeld of characteristic zero such that g is a free Lie algebra of nilpotency class two. If g has n generators, then the commutator subalgebra g of g coincides with its center z and has dimension n(n−1) . One has dim g + dim z > dim g if and only if n ≥ 4. 12 Proposition. Let G be a connected algebraic k-group such that G/zG is a chain.
Let k be a ﬁeld of characteristic p > 0. On the n-dimensional aﬃne space we deﬁne a multiplication by (x0 , · · · , xn−1 )(y0 , · · · , yn−1 ) = (z0 , · · · , zn−1 ) where z0 = x0 + y0 and, for i = 1, · · · , n − 1, i−1 j+1 p xj yi−j−1 . 2) Mn−1 (x0 , · · · , xn−1 ) = ⎜ ⎟. ⎜ ⎟ . ⎝ ⎠ 1 This representation is faithful of smallest possible degree, because the nilpotency class of this group is n: in fact, the centre zJn of Jn is the onedimensional connected subgroup deﬁned by x0 = · · · = xn−2 = 0, and the factor group Jn /zJn is isomorphic to Jn−1 , hence Jn is a group of maximal possible nilpotency class.
This implies that its kernel ΠTu is deﬁned over k, which is a contradiction. Before leaving the questions of rationality and turning to the connections between factor systems and isogenies, we want to remark once more that the existence of a k-rational regular factor system is guaranteed if A is a unipotent group deﬁned over a perfect ﬁeld k and B is aﬃne. For any factor system φ ∈ C2 (B, A) there is precisely one factor system φ0 , equivalent to φ, satisfying φ0 (1, 1) = 0. In the group Gφ0 , equivalent to Gφ , we have the useful identity (b, a) = (b, 0)(1, a).
Algebraic groups and lie groups with few factors by Alfonso Di Bartolo, Giovanni Falcone, Peter Plaumann, Karl Strambach