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

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 bunch theoretical standpoint and the got effects are in comparison with the analogous matters 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 basically few subgroups or few issue teams of other style. 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 published by means of the plethora of third-dimensional unipotent algebraic teams over an ideal box of optimistic attribute, in addition to, via many concrete examples which hide a space systematically. within the ultimate part, algebraic teams and Lie teams having many closed general subgroups are determined.

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Proof. Let X2 ∼ = Cn−m /Λ2 be the connected commutative complex Lie group of maximal rank n − m having CH (P ) as a period matrix and let fˆ : Cn −→ Cn−m be the homomorphism defined by fˆ(z1 , · · · , zn ) = (zl1 , · · · , zln−m ). Since fˆ(Λ) ≤ Λ2 , a homomorphism f : X −→ X2 is induced such that X1 is the kernel. This proves that X1 is a closed subgroup. In order to prove that X1 is a linear torus we show that H ∩ Λ has real rank m. This follows from the fact that the columns of CH (P ) are R-independent, hence no non-trivial linear combination of the columns l1 , · · · , ln−m of the matrix P with integral (or even real) coefficients enters in H.

0 In2 +q2 P1 Σ defines therefore a split extension of X1 by X2 0 P2 if and only if Σ = P1 M −AP2 with A ∈ Mn1 ,n2 (C) and M ∈ Mn1 +q1 ,n2 +q2 (Z). P1 Σ is such that Σ = P1 M − AP2 Moreover, if the period matrix P = 0 P2 with M ∈ Mn1 +q1 ,n2 +q2 (Q), then P defines an extension of X1 by X2 which is isogenous to a split one. An isogeny f : X1 → X2 is given by ρa (f ) = lIn1 +q1 0 lIn1 0 and ρr (f ) = , where l ∈ Z is such that lM has 0 In2 0 In2 +q2 integral entries. 5 Proposition. Let X1 , X2 be connected commutative complex Lie groups of maximal rank n1 , n2 and let P1 , P2 be the corresponding period matrices.

For instance, in the three-dimensional toroidal group X having ⎛ ⎞ 100 √ i i 0 ⎠ P = ⎝0 1 0 i 2 √ 001 0 i 2 as a period matrix, the three subgroups H(2, 3), H(1, 3) and H(1, 2) are onedimensional maximal closed linear subtori. 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.

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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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