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

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 gaggle theoretical standpoint and the acquired effects are in comparison with the analogous concerns within the idea of Lie teams. the most physique of the textual content is dedicated to a type of algebraic teams and Lie teams having in simple terms few subgroups or few issue teams of other kind. 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 published via the plethora of three-d unipotent algebraic teams over an ideal box of optimistic 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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Extra resources for Algebraic groups and lie groups with few factors

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1 Maximal Nilpotency Class 33 a1,j = xj−2 and i ai+1,i+1+t = aα i,i+t , i where we write xα for αi (x). With this notation we can introduce the following family of subgroups of Jn : S0 = Jn = Jn (F), S1 = {(0, x1 , 0, x3 , · · · ) : xi ∈ ¯k} ∼ = Jm1 (F2 ), S2 = {(0, 0, x2 , 0, 0, x5 , · · · ) : xi ∈ ¯k} ∼ = Jm2 (F3 ), .. n . 9 Proposition. For n ≥ 3, the group Jn (α) is the only group G of linear transformations on an (n + 1)-dimensional vector space V containing a vector 0 = v ∈ V , fixed by any element of Jn (α), such that: i) on the factor space V / v the group Jn (α) induces the group of linear transformations represented by a matrix M (α) = (aα i,j ), with M = (ai,j ) representing an element of Jn−1 (α); ii) Jn (α) induces the group of linear transformations Jn−1 (α) on the factor space V ∗ / v∗ , where v∗ , V ∗ are the canonical dual objects of v, V .

The assertion follows from the fact that any normal connected algebraic k-subgroup of G contains z◦ G. Closing this section we show that the descending central series Ci+1 (G) = [G, Ci (G)] (with C0 (G) = G) of an affine algebraic group of dimension n and 36 3 Groups of Extreme Nilpotency Class nilpotency class n has particular properties. Furthermore, we show that the non-commutative two-dimensional unipotent group never occurs as a commutator subgroup of a nilpotent group. To reach our goal we need two propositions, the first of which holds for abstract groups.

Let E be a purely inseparable extension of k of degree [E : k] = n = pt . 9. The group ΠT has dimension n and contains T up to a birational k-isomorphism. There exists a surjective E-homomorphism ρ : ΠT −→ T , the kernel ker ρ of which is the unipotent radical of ΠT , is connected, has dimension n − 1 and does not contain non-trivial algebraic k-subgroups of ΠT . In particular, ker π is not defined over k. 6, p. 137). Assume by contradiction that the exact sequence π 1 −→ T −→ ΠT −→ ΠT /T −→ 1 has a k-rational regular cross section.

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

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