This is the first of two physically-oriented chapters in which we take a break from R-matrices and such topics and explore instead a completely different setting in which Hopf algebras arise naturally. This is the setting of logic, quantum mechanics and probability theory, in all of which contexts the coproduct Δ of the Hopf algebra plays the role of ‘sharing out’ possibilities. The theory has potential applications in computer science as well. In Boolean algebra and quantum mechanics, the product in the algebra corresponds in some sense to deduction of facts (‘putting two and two together’), and supplementing this structure with a coproduct provides a kind of reverse of this process by sharing out possible explanations of a fact. This provides the general theme of the present chapter. The chapter can also be viewed as background material for the next chapter, where we present a number of concrete models of quantum-mechanical Hopf algebras of observables.

The simplest example of a combinatorial Hopf algebra is the shuffle algebra associated to an alphabet of symbols. Here the coproduct of a word is the formal sum of all pairs of words which could be shuffled to give the original word. Here, a shuffle of two words takes symbols from each of the words and interleaves them randomly while preserving the order of each word (this is exactly what you do when shuffling a pack of cards).