BIREP – Representations of finite dimensional algebras at Bielefeld

# Seminar on Ringel-Hall Algebras

Time and place: Wednesdays 10-12 in room V3-204

# Ringel-Hall Algebras

In [5], P. Hall showed how one could construct an algebra by taking as basis the isomorphism classes of finite $p$-groups, and where the multiplication counts all possible extensions of the two groups. This is the classical Hall algebra, or algebra of partitions, so named since the isomorphism classes of finite $p$-groups are in bijection with the set of partitions. As an algebra, this is isomorphic to a polynomial ring on countably many variables, but it has a much richer structure, that of a self-dual Hopf algebra. Nowadays this algebra is often referred to as Macdonald's ring of symmetric functions, and encodes a lot of interesting combinatorics [10]. For example, the Kostka numbers occur as entries of the base-change matrix between the bases of Schur functions and monomial functions. Note that this algebra has a history going back even further; for example it was discussed by Steinitz already in 1901 [13].

This idea was picked up by Ringel in [11], who showed that one could attach in a similar way an algebra to any finitary abelian category, the so-called Ringel-Hall algebra; here finitary means that the homomorphism and first extension groups are all finite sets. In particular, this works for the category of finite-dimensional modules over a $k$-algebra whenever $k$ is a finite field, but also for the category of coherent sheaves on projective space, again over a finite field. There is also a natural coalgebra structure, but in general these two structures are not compatible.

In fact, analysing the Ringel-Hall algebra coming from the representations of a quiver, Green showed in [4] that the multiplication and comultiplication are compatible precisely when the category is hereditary (so the second and higher extension groups all vanish), and in this case we again obtain a self-dual Hopf algebra. Moreover, he proved that a certain natural subalgebra (that generated by the simple objects) is isomorphic, again as a self-dual Hopf algebra, to a quantum group, so the (positive part of the) quantised enveloping algebra of a symmetrisable Kac-Moody Lie algebra [9]. For the case of $p$-groups, or similarly the finite-dimensional modules over the Laurent polynomial ring $k[[t]]$, the Lie algebra is the affinisation (or loop algebra) of $\mathfrak{sl}_2$.

As an application, one can compare the Weyl denominator formula for the quantum group with the natural factorisation of the Ringel-Hall algebra derived from the Krull-Remak-Schmidt Theorem on the module category to ascertain that the dimension vectors of the indecomposable modules are precisely the positive roots of the Kac-Moody Lie algebra [2]. In some sense, this completed the circle, since Kac had already shown that this result for dimension vectors holds [7], and Ringel was interested in obtaining a deeper structural relationship between the module category and the Kac-Moody Lie algebra suggested by this combinatorial result.

On the other hand, using the natural basis for the Ringel-Hall algebra (more precisely, comparing these bases for the different orientations of the quiver), Lusztig was inspired to construct his canonical basis of the quantum group [8], which has particularly nice properties with respect to a certain class of modules over the quantum group.

More recently there has been work developing Ringel-Hall algebras for elliptic curves [1], weighted projective lines [12], configurations of abelian categories and stacks [6], as well as for Frobenius categories [3] and triangulated categories [14].

A more detailed overview of Ringel-Hall algebras is also available.

# Schedule of Talks

• 13 April: Ringel-Hall algebras (Baolin Xiong)
• 20 April: The algebra of partitions (Andrew Hubery): lecture notes
• 27 April: Green's formula, I (Andre Beineke)
• 04 May: Green's formula, II (Ögmundur Eiriksson)
• 11 May: Quiver representations and quantum Serre relations (Rolf Farnsteiner): lecture notes
• 18 May: The Kronecker quiver (Fajar Yuliawan): Kronecker representations
• 25 May: Coherent sheaves on the projective line (Thomas Cauwbergs)
• 01 June: Quantum groups (Florian Gellert)
• 08 June: Green's main theorem (Philipp Lampe): lecture notes
• 15 June: Exceptional modules and root vectors (Paula Lins)
• 22 June: Quiver Grassmannians (Christopher Voll)

# References

 [1] I. Burban and O. Schiffmann, On the Hall algebra of an elliptic curve, I, Duke Math. J. 161 (2012), 1171-1231 [2] B. Deng and J. Xiao, A new approach to Kac's theorem on representations of valued quivers, Math. Z. 245 (2003), 183-199 [3] M. Gorsky, Semi-derived and derived Hall algebras for stable categories, preprint arXiv:1409.679 [4] J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), 361-377 [5] P. Hall, The algebra of partitions, Proceedings 4th Canadian Mathematical Congress (1959), 147-159 [6] D. Joyce, Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), 635-706 [7] V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57-92 [8] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498 [9] G. Lusztig, Introduction to quantum groups, Progress in Math. 110 (Birkhäuser Boston, Boston, MA, 1993) [10] I. G. Macdonald, Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky, Oxford Mathematical Monographs. Oxford Science Publications. (The Clarendon Press, Oxford University Press, New York, 1995) [11] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-591 [12] O. Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math. J. 121 (2004), 113-168 [13] E. Steinitz, Zur Theorie der Abelschen Gruppen, Jahresberichts der DMV 9 (1901), 80-85 [14] B. Toën, Derived Hall algebras, Duke Math. J. 135 (2006), 587-615