Introduction to Ringel-Hall Algebras
Lecture Notes
Course Summary
The representation theory of quivers began with the study by Gabriel,
where he proved that a quiver has finitely many isomorphism classes of
indecomposable representations if and only if it is an oriented Dynkin
diagram. In this case the isomorphism classes are in bijection with the
set of positive roots of the semisimple Lie algebra sharing the same
Dynkin diagram, this bijection being given by the dimension vectors of
the indecomposable representations. Special cases of
representations of quivers had
been considered long before this, for example by Kronecker, though not
using this modern language.
This theorem of Gabriel was extended to the extended Dynkin diagrams by
Donovan and Frieslich, and independently by Nazarova. In this case, the
associated Lie algebra is an affine Kac-Moody Lie algebra, and has
infinitely many roots. These roots belong to two classes: the real
roots, obtained from the simple roots by the Weyl algebra, and the
imaginary roots, in this case non-zero integer multiples of a positive
root, commonly denoted δ. The theorem then states that the set of
dimension vectors of the indecomposable representations again coincides
with the set of positive roots, and that, up to isomorphism, there is a
unique indecomposable representation corresponding to each positive
real root. Moreover, for each positive multiple of δ, the isomorphism
classes of indecomposable representations are parametrised by the
projective line together with finitely many points.
The full generality of this theorem was finally proved by Kac, where he
showed that for an arbitrary quiver, the set of dimension vectors of
indecomposable representations coincides precisely with the set of
positive roots of the associated symmetric Kac-Moody Lie algebra.
Moreover, there is a unique indecomposable representation (up to
isomorphism) corresponding to a given positive root if and only if this
root is real.
In his study of quiver representations, Ringel generalised the
construction of the Hall algebra, originally constructed from the
category of finite dimensional k[[T]]-modules, to an arbitrary module
category, satisfying some finiteness conditions. As a vector space,
this has as basis the set of isomorphism classes of modules, and the
multiplication is determined by the possible extensions of modules. In
particular, for any representation directed algebra, he proved that the
structure constants are given by evaluations of poynomials, and thus
constructed a generic Ringel-Hall algebra. He then proved that this
algebra is isomorphic to the quantum group (the positive part of the
quantised enveloping algebra) of the semisimple Lie algebra, which had
recently been
introduced by Drinfeld and Jimbo. This deepened the understanding
between quiver representations and Lie algebras first observed by
Gabriel.
Green later proved that the Ringel-Hall algebra is naturally a
self-dual Hopf algebra (after extending by a quantum torus). In
particular, there is a natural comultiplication as well as a positive
definite bilinear form for which multiplication and comultiplication
are adjoint to one another. This enabled him to show that for an
arbitrary quiver, the subalgebra generated by the simple modules is
isomorphic to the quantum group of the associated Kac-Moody Lie algebra.
This result was later extended by Sevenhant and Van den Bergh to show
that the full Ringel-Hall algebra is isomorphic to the quantum group of
a Borcherds algebra, or generalised Kac-Moody Lie algebra. By
considering the character of this algebra (that is, the generating
function for the dimensions of graded pieces), Deng and Xiao recovered
Kac's theorem on the dimension vectors of the indecomposable
representations.
In a different direction, the representation theory approach to quantum
groups inspired Lusztig in the development of the canonical basis in
the semisimple Lie algebra case. By considering natural lifts to the
quantum group of the Poincaré-Birkhoff-Witt bases of the
universal enveloping algebra, he proved that these all generate the
same lattice in the quantum group, and that there is a unique basis
generating this same lattice and invariant under the so-called bar
involution. The lifts of the Poincaré-Birkhoff-Witt bases are
the natural bases in the Ringel-Hall algebras corresponding to the
various orientations of the Dynkin diagrams.
Course Summary
In this course I intend to introduce the basic results in the
representation theory of quivers. I will then study the Ringel-Hall
algebra and prove Green's theorem on the Hopf algebra structure. I will
also outline the necessary results from Lie algebra theory to prove the
results by Sevenhant - Van den Bergh and Deng - Xiao.
If time permits, I would then like to show how Green's comultiplication
gives an easy proof of the existence of Hall polynomials in the case of
Dynkin quivers, as well as studying Lusztig's construction of the
canonical basis, again in the Dynkin case.
Exercise sheets will be distributed during the course, and I will try
to motivate the results with concrete examples.