Ringel-Hall Algebras

The representation theory of quivers lies at the interface of various areas of mathematics, including algebraic and combinatorial representation theory, algebraic geometry, and Lie theory and quantum groups. The theory of Ringel-Hall algebras provides one approach to understanding and deepening many of these connections.

Although the subject can appear very technical and demanding, many of the calculations involved are actually fairly amenable, despite their initially intimidating appearance. On the other hand, once these computations have been made, they can have powerful ramifications when combined with, say, results from algebraic geometry.

In this seminar we aim to study several of these important results, and thus reveal the interplay between different mathematical disciplines in the setting of quiver representations and Ringel-Hall algebras.

Overview of the talks

Section one: Basic structure

The first lecture Ringel-Hall algebras introduces the basic definitions of the Ringel-Hall algebra of an abelian category over a finite field. In particular, we can define an algebra structure and a coalgebra structure, and both the associativity and coassociativity follow from analysing pull-backs and push-outs in the category. We will also show that the multiplication and comultiplication are in general incompatible, even for the category of vector spaces.

The second lecture The algebra of partitions is an extended example, analysing the structure of the Ringel-Hall algebra arising from the module category of a discrete valuation ring. The two main instances are of course the finite dimensional modules over the power series ring $k[[t]]$, as well as the category of finite abelian $p$-groups, so the finite length modules over the ring of $p$-adic integers. In either case the Ringel-Hall algebra is in fact a self-dual Hopf algebra, isomorphic to a polynomial ring on countably many variables, each of which is a primitive element for the comultiplication. This is the classical algebra of partitions, and is isomorphic to Macdonald's ring of symmetric functions, which itself plays a central role in the theory of representations of the symmetric groups or the general linear groups.

The next two lectures Green's formula, I and II address the issue of the compatibility of the multiplication and comultiplication. We will begin by examining the situation for the category of vector spaces. In this case the Ringel-Hall algebra is isomorphic as an algebra to the polynomial ring on a single variable $u$, and this is a primitive element. If this were an ordinary Hopf algebra, then the comultiplication of $u^r$ would be a sum of terms $u^a\otimes u^{r-a}$ occuring with the usual binomial coefficient, but the natural comultiplication for the Ringel-Hall algebra actually has $u^a\otimes u^{r-a}$ occuring with the quantum binomial coefficient (evaluated at the reciprocal of the size of the finite field).

It follows that, in general, the Ringel-Hall algebra is not a bialgebra. J.A. Green showed how to rectify this situation when the abelian category is hereditary (so submodules of projective modules are again projective), by introducing a twist corresponding to the Euler characteristic of the category. Thus the Ringel-Hall algebra of an hereditary abelian category is a twisted bialgebra. In fact the structure is even richer: there is an invertible antipode, so we have a twisted Hopf algebra, and a non-degenerate symmetric Hopf pairing, so the Ringel-Hall algebra is self dual.

Section two: Quiver representations

We then move onto the particular case of quiver representations Quiver representations and quantum Serre relations. After a (brief) introduction to quiver representations, we show that quantum Serre relations hold in the Ringel-Hall algebra. These are deformations of the usual Serre relations for Kac-Moody Lie algebras, and occur in the presentation of quantum groups, which themselves are deformations of the universal enveloping algebras of the Lie algebras.

The sixth talk The Kronecker quiver is again an extended example, this time for the representations of the Kronecker quiver. We will describe the category, including classifying all possible indecomposable representations, and then use this information to describe the corresponding Ringel-Hall algebra. In particular we will see that the composition algebra, that is the subalgebra generated by the two simple representations, has an explicit PBW-type basis, and we will give the relevant straightening rules for the multiplication. In this case it is even possible to show directly that the composition algebra has a presentation by the two generators corresponding to the two simple modules, and the two quantum Serre relations. We can also compute the characters of both the composition algebra and the whole Ringel-Hall algebra, and give factorisations of these as infinite products. For the composition algebra we get one factor for each positive root of the affine Lie algebra of type $\mathfrak{sl}_2$, and for the whole Ringel-Hall algebra we get one factor for each indecomposable module.

The seventh talk Coherent sheaves on the projective line is similar, this time for the category of coherent sheaves on the projective line. We will see that the algebra structure of the Ringel-Hall algebra is very similar to that for the category of Kronecker representations, and in fact they have naturally isomorphic subalgebras. On the other hand, the comultiplication in the Ringel-Hall algebra for coherent sheaves on the projective line only lies in a completed tensor product, so there is a significant difference between the coalgebra structures in the two settings. We observe that the isomorphism between the subalgebras is a shadow of the derived equivalence between the two categories. One can also regard the distinction between the two Ringel-Hall algebras as coming from the two possible constructions of the affine Lie algebra: either as a Kac-Moody Lie algebra given by its usual Chevalley generators and Serre relations, or as a loop algebra of the simple Lie algebra of type $\mathfrak{sl}_2$.

Section three: Quantum groups

The next talk Quantum groups introduces the quantum group associated to a symmetric generalised Cartan matrix, also called the quantised enveloping algebra of the corresponding Kac-Moody Lie algebra. We will sketch the various tools needed to prove that the quantum group has a presentation given by the Chevalley generators and quantum Serre relations, and that this holds over any field $k$ of characteristic zero together with a chosen invertible element $v\in k$ which is not a root of unity. In particular, we will deduce that we can define the (positive part of the) quantum group over a suitable subring of the function field $\mathbb Q(v)$, which will allow us to specialise $v$ to any prime power. One of the important tools we will need is that of the braid group action on the quantum group, and using this we will also be able to describe the character of the (positive part of the) quantum group.

The second talk in this section Green's main theorem pulls together many of the results discussed so far, and hence gives our first major result. Using that the quantum Serre relations hold in the Ringel-Hall algebra and that the bilinear form is non-degenerate (even positive definite), we can prove that there is a map from the (positive part of the) quantum group to the Ringel-Hall algebra, sending the Chevalley generators to the simple modules (so has image the composition subalgebra) and sending the parameter $v$ to (a square root of) the size of the finite field.

This allows us to compute the character of the composition algebra in terms of roots for the corresponding Kac-Moody Lie algebra, which has an entirely combinatorial description. With a little bit more work we can then prove that the set of dimension vectors of the indecomposable representations for the quiver is precisely the set of positive roots, and that there is a unique indecomposable representation for each positive real root. This is Kac's Theorem for finite fields.

We will also observe that the braid group action on the quantum group has a natural interpretation in the Ringel-Hall algebra via the concept of reflection functors.

Section four: Quiver Grassmannians

The next two talks aim to showcase the interplay between Ringel-Hall algebras and geometry. We begin in Exceptional modules and root vectors by discussing exceptional representations, so those indecomposables having no self-extensions. Using reflection functors/the braid group action, we will see that the basis element in the Ringel-Hall algebra corresponding to an exceptional representation actually lies in the composition subalgebra. We can then extend this to a basis of the composition algebra, and lift this back to the quantum group. It follows that for each product of the generators in the quantum group, there is a polynomial giving the coefficient of the basis element corresponding to our chosen exceptional representation. Evaluating this polynomial at a particular prime power, we can interpret this in the Ringel-Hall algebra and thus show that this polynomial counts the number of composition series of the exceptional representation such that the composition factors are in a prescribed order (the same order as the product of the generators). In particular, choosing the order of the composition factors appropriately, we can count the number of subrepresentations of a specific dimension vector in this way.

We next analyse the same situation geometrically in the talk Quiver Grassmannians. For each exceptional representation and each dimension vector we can construct a smooth projective variety by taking those subrepresentations of the exceptional of this given dimension vector, a so-called quiver Grassmannian. Moreover, this projective variety can be defined over the integers such that specialising to any field yields the corresponding quiver Grassmannian. It follows that the number of rational points of this smooth projective variety over a finite field is given by the polynomial we constructed using the Ringel-Hall algebra. Combining this information with the Weil conjectures for smooth projective varieties. we obtain some strong results. For example, the polynomial has non-negative coefficients and is palindromic. On the other hand, the quiver Grassmannians have no odd cohomology and strictly positive Euler characteristic (provided they are non-empty).