Quantum Groups

the long version

Algebra has moved well beyond the topics discussed in standard undergraduate texts on ¬emodern algebra¬i. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn the latest algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an ¬ealgebra¬i. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a ¬ecoalgebra¬i. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term ¬equantum group¬i, along with revolutionary new examples, was launched by Drinfel'd in 1986.