There is no rigorous, universally accepted definition of the term quantum group.
However, it is generally agreed that this term includes certain deformations of
classical objects associated to algebraic groups. Before I go on, let me briefly
illustrate by an example what the word deformation means in this context.
Take the polynomial algebra k[X, Y] in two variables X and Y over a field k.
We can think of k[X, Y] as the (associative) fc-algebra with generators X and Y
and relation YX = XY. Consider now for each q € k the (associative) A;-algebra
kq[X,Y] with generators X and Y and relation YX = qXY. One checks easily
that all monomials XmYn are a basis of fcg[X, Y] over k (for any q) and that the
multiplication of two basis elements is given by
ym~yn yryrs nr ym+r-yn+s
So the kq [X, Y] are a family of algebras (parametrized by q G k) where the multi-
plication depends in a "nice" way on the parameter q and where we get for q = 1
our old algebra k[X, Y], This family and its members are then called deformations
This example should give you an intuitive feeling of what we mean by a defor-
mation of a given algebra A: It should be a family of algebras depending "nicely"
on a parameter q such that we get back A for some special value of q. (Of course,
this is not a rigorous definition of a deformation.)
In the theory of quantum groups one deals with deformations Uq(g) of the
enveloping algebra U(Q) of a semisimple Lie algebra g and with deformations kq[G]
of the ring k[G] of regular functions on a semisimple algebraic group G. In both
cases the original algebras have an additional structure (a comultiplication), and it
is essential that the deformations are compatible with this additional structure.
The algebras Uq(g) are often called quantized enveloping algebras. They are
— strictly speaking — deformations not of U(g) but of a certain covering of U(g).
These algebras (or minor modifications thereof) were introduced independently by
DrinfePd and Jimbo around 1985. They were first used to construct solutions to the
quantum Yang-Baxter equations, cf. 3.17, 7.6. Since then they have found numerous
applications in areas ranging from theoretical physics via symplectic geometry and
knot theory to modular representations of reductive algebraic groups.
By now, this subject has become so large that any textbook will have to restrict
itself to parts of the theory. The choices made here can be explained in part by my
personal interests, in part by the circumstances of its genesis. This book has its
origin in notes that I wrote for a course on quantum groups in the spring quarter
of 1994 at the University of Oregon. (Since then the notes have been revised and
the Chapters 5A and 9-11 have been added.)