Drinfeld double and Ringel-Green theory of Hall algebras
As a continuation of the work of \textit{C. M. Ringel} [Invent. Math. 101, 583-592 (1990; Zbl 0735.16009)] and \textit{J. A. Green} [Invent. Math. 120, 361-377 (1995; Zbl 0836.16021)] on Hall algebras, the Hopf algebra structure of an extended twisted Hall algebra is presented. By introducing the Ringel pairing of Hall algebras, the Drinfeld double of a Hall algebra is constructed. The reduced form of this Drinfeld double is applicable to the generic composition algebra, which gives a complete realization of the quantum group. This Ringel-Green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of Hall algebras. This paper is organized as follows. In Section 2, the definitions of Cartan datum, skew-Hopf pairing and Drinfeld double are given. The skew-Hopf pairing belonging to a Cartan datum \(C\) is considered; this class is denoted by \({\mathcal L}(C)\). Section 3 is a review of Green's result with a slight modification only. For \((A^+,A^-,\varphi)\in {\mathcal L}(C)\) an algorithm is given to calculate \(\varphi\) that only depends on the Cartan datum. It follows that the constructions of Lusztig and Joseph are canonically isomorphic to each other. In Section 4, the author obtains Theorem 4.5, one of the main results, which gives the Hopf algebra structure of the extended twisted Hall algebra. In Section 5, the positive and negative Hall algebras and their Ringel pairing are formulated precisely, then the reduced Drinfeld double of Hall algebras is given, which on restriction to the generic composition algebras gives a realization of quantum groups (Theorem 5.8). A formula for \(u_{\lambda'}^-u_\lambda^+-u_\lambda^+u_{\lambda'}^-\) is given at last; which has some simplified form in some cases.
scientific article; zbMATH DE number 1013967