Algebraic Groups and Lie Groups with Few Factors

Our Friend and Mathematician Karl Strambach

“…This was a short survey of what was just one half of the contribution given by Karl Strambach to the academic training of G. Falcone, who cannot help but mention in what follows one of the many results contained in the monograph [7]. While climbing the stairs of the Erlangen Mathematics Institute in Bismarckstraße 1 1/2, Karl asked G. Falcone whether connected algebraic groups where every connected subgroup was normal always have onedimensional commutator subgroups.…”

“…which J. Dieudonné found of some interest. The authors of the paper [7] generalized this group to the n-dimensional unipotent algebraic group J n , of maximal nilpotent class n, having the following linear representation…”

On liftings of projective indecomposable G(1)-modules

“…Indeed, the extension is determined by a 2-cocycle β : U α × U α → U i /U i+1 , which is a morphism G a × G a → G a , and as such is given by a polynomial in two variables. The cohomology class representatives of H 2 (G a , G a ) are given in [4,Remark 2.1.6], where these polynomials have a k-basis consisting of the polynomials: , i ≥ 0, and xy p j p i , j > 0, i ≥ 0.…”

The periods of the generalized Jacobian of a complex elliptic curve

“…When one considers an -dimensional toroidal group G = ℂ / (sometimes called quasi-torus) of real rank = + 1 (see, e.g., [1]), it is easy to see that the toroidal group is an extension of an ( − 1)-dimensional maximal linear sub-torus by an elliptic curve C with period matrix (1,̂ ), witĥ ∉ ℝ (cf. [4,Proposition 2.2.3,p. 26]) and, because the functor Ext(C, (ℂ * ) −1 ) is additive, one can restrict to the case where = 2 and G is de ned by the lattice generated by (1, 0), (0, 1) and (̂ , ), for a given ∈ ℂ such that the only pair ( , ) ∈ ℤ 2 such that ̂ + is an integer is the pair (0, 0).…”