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.