[For the entire collection see Zbl 0563.00007.]
Let k be a field, R a k-algebra. Using as examples groups of automorphisms of R, and group-gradings of R (or rather the projections onto the graded pieces), the author considers families of linear operators on R with specified laws of composition, and specified behavior on products of two elements of R. Passing to the k-vector space H spanned by these operators, he thus motivates the notion of a bialgebra \((algebra+coalgebra+compatibility\) conditions). Hopf algebras (bialgebras with antipode) are motivated by group actions on modules and their dual modules. Another motivating example is Lie algebras L of derivations of R (or rather U(L)). Although this point of view is not new (see, for example, the books of Hochschild on Lie groups, on affine algebraic groups, and on algebraic groups and Lie algebras), it seems useful to point out the common features of the motivating situations. It also leads to natural discussions of actions and coactions, identities and co- identities, and duality notions. The author notes the variations on the purely algebraic ideas motivated by algebraic topology and algebraic geometry. His remarks on duality are motivated by the latter (formal group schemes and completed tensor products). While this is useful, he might also have mentioned the passage from algebras to coalgebras via linear representative functions, which keeps the discussion algebraic (i.e. ordinary tensor products). The article is written in the author’s usual rambling style, so that various types of mathematicians can find sections relating to their own domains. However, the reviewer would recommend that such a reader take the trouble to read the whole article for general cultural purposes and to realize the generality of the Hopf algebra concept.