“According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple Lie algebra. The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof.” With these sentences George Lusztig lays out the large scale structure of the discussion that follows in the 348 pages of his Introduction to Quantum Groups. Manifestly this is not an introduction aimed at raw beginners, and accordingly the book is pitched at quite a high level. The engaged reader should be well-versed in algebra proper, the theory of Lie groups and Lie algebras (in the broadest sense, where the phrases “Cartan datum,” “Serre relations,” and “complete reducibility” ring a loud bell), and in the category theory dominated field of (what to call it?) geometric Lie theory (?), geometric algebra (?), geometric representation theory (?) — it’s really a weighted sum of all of these, and other things besides.
A good idea of what is involved can be gleaned from the material in the book’s second part, where we start with perverse sheaves, quivers, and the Fourier-Deligne transform, hit Verdier duality not too long afterwards, and then return to Cartan data, this time in connection with graphs with automorphisms. All this is in the service of providing a geometric realization of the algebra f, a quotient algebra (by a special ideal) of a certain finitely generated free associative unital Q(v)-algebra, v being a symmetric bilinear form on the free Z-module on the stipulated generating set. It is this f that launches the whole adventure (on p. 2 of the book) in the sense that the centrally important Drinfeld-Jimbo algebra U arises later in the first part of the book as a homomorphic image of f.
Whew! All this cannot help but bring to mind Weil’s famous descriptive phrase in his review essay on Chevalley’s book on Lie groups: “This is algebra with a vengeance.”
So let’s bring things down to earth a bit, courtesy of the world wide web, or, more specifically, the internet version of the Notices of the AMS. Vol. 53, No. 1 (Jan. 2006) contains the illuminating article, “What is a Quantum Group,” by Shahn Majid, on pp. 30–31. There we learn that, to be sure, first of all a quantum group is a Hopf algebra, and Majid accommodates us with four axioms defining such: we’re dealing with unital k-algebras (for a field k) that are also counital coalgebras of a nice sort, and there is an antipode map. Then we learn that such objects occur naturally in algebraic geometry, e.g. for algebraic varieties over k that possess group structure; we learn that the group algebra kG of any group G and the enveloping algebra U(g) of a Lie algebra g are Hopf algebras with commutative coalgebra. Then we learn that “Hopf algebras are the next simplest category after abelian groups admitting Fourier transform.” So, in any event, these objects certainly do occur with some frequency in nature and we have a reasonable context for what Lusztig presents us with. Certainly now the presence of Fourier-Deligne transforms in the context of a geometric realization begins to resonate with a more familiar frequency, so to speak.
This said, what does Lusztig himself have to say about his offering, given that “Part I contains an elementary treatment of the algebras of Drinfeld and Jimbo,” and that “Part II contains the construction of canonical bases using perverse sheaves” which he says can be omitted if the reader is willing to “accept the theorems in Chapter 14 without proof”? Well, Part III is concerned with work of Masaki Kashiwara (specifically his “operators”), Part IV talks about canonical bases for “a modified form… of the quantized enveloping algebra,” Part V deals with roots on unity, and Part VI deals with the action of the braid group.
It’s unfortunate that the book’s index is not only very spare but also refers not to page numbers but to sections (in the Bourbaki style). But this is very minor and possibly petty criticism of what is otherwise obviously a significant and important work. Again, it’s not pitched at raw beginners: be prepared for hard-core algebra and a lot of work. But it’s terrific stuff, elegant and deep, and Lusztig presents it very well indeed, of course.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.