The composition Hall algebra of a weighted projective line (Q2838630)

Let \(H({\mathbb X})=H(\text{Coh}({\mathbb X}))\) be the Hall algebra of the abelian category of coherent sheaves on the weighted projective line \(\mathbb X\) over a finite field. In the paper under review, the authors study the composition algebra \(U({\mathbb X})\) of \(H({\mathbb X})\) and its reduced Drinfeld double \(DU({\mathbb X})\).NEWLINENEWLINEThe motivation comes from several places:NEWLINENEWLINE (i) The importance of Hall algebras of the category of nilpotent representations of finite quivers and their role in the theory of quantized Kac-Moody algebras;NEWLINENEWLINE (ii) The relation of the Hall algebra of the category of coherent sheaves on the classical projective line \({\mathbb P}^1\) with Drinfeld's new realization of the quantized enveloping algebra \(U_q(\widehat{{\mathfrak s}{\mathfrak l}}_2)\);NEWLINENEWLINE (iii) An attempt to generalize the results of the previous work by the authors [Glasg. Math. J. 54, No. 2, 283--307 (2012; Zbl 1247.17010)] to other quantized enveloping algebras.NEWLINENEWLINEThe first result of the paper presents a new definition of the composition algebra \(U({\mathbb X})\) and gives that \(U({\mathbb X})\) is a topological bialgebra, i.e., a subalgebra of \(H({\mathbb X})\) closed under the Green comultiplication. Then the authors characterize the subalgebra \(\bar{U}({\mathbb X})_{\text{tor}}\) of \(U({\mathbb X})\) generated by the skyscraper sheaves as a tensor product of the Macdonald ring of symmetric functions and algebras \(U_q^+(\widehat{{\mathfrak s}{\mathfrak l}}_p)\). The authors also obtain other important properties of \(U({\mathbb X})\). In particular, for any weighted projective line \(\mathbb X\) the Drinfeld double \(DU({\mathbb X})\) of \(U({\mathbb X})\) contains a subalgebra isomorphic to \(U_q(\widehat{{\mathfrak s}{\mathfrak l}}_2)\). The same technique leads to the construction of new embeddings, e.g., for the quantized enveloping algebras \(U_q(\hat{A}_3)\to U_q(\hat{D}_4)\). Moreover, the approach of the paper allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types \(D^{(1,1)}_4,E^{(1,1)}_6,E^{(1,1)}_7\) and \(E^{(1,1)}_8\). In particular, the method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.