Group varieties and group structures

Since group operations of algebraic groups agree with the structure of their underlying varieties, there must be a dependence between them. A striking illustration of it is the classical theorem about commutativity of every connected algebraic group whose group variety is complete. In an explicit or implicit form, this problem was considered in the classical papers of A. Weil, C. Chevalley, A. Borel, A. Grothendieck, M. Rosenlicht, M. Lazard. This talk is aimed to discuss to what extent the group variety of a connected algebraic group or the group manifold of a connected real Lie group determines its group structure.