Introduction

Summary

Algebra has moved well beyond the topics discussed in standard undergraduate texts on “modern algebra”. 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 “algebra”. 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 “coalgebra”. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term “quantum group”, along with revolutionary new examples, was unleashed on the mathematical community by Drinfel'd [Dri87] at the International Congress in 1986. Before launching into an explanation of the duality required, I should mention here that an ordinary group gives rise to a quantum group by taking the vector space with the group as basis.

When pushed to provide formal proofs of our claims, mathematicians generally resort to set theory. We build our structures on sets and feel satisfied when we can be explicit about the elements of our constructed objects. Up to the mid twentieth century, algebraic objects were sets with selected operations which assigned elements to lists of elements. Typically, we would have binary operations which might be called addition, multiplication or Lie bracket respectively assigning a sum, product or formal commutator to each pair of elements.