Root categories and simple Lie algebras
Let \(\Delta\) be a Dynkin diagram and \(\vec\Delta\) an (oriented) quiver with underlying graph \(\Delta\). Let \(U\) be the (Drinfeld-Jimbo) quantized enveloping algebra of the finite-dimensional semisimple complex Lie algebra associated with \(\Delta\). Then the positive part \(U^+\) of \(U\) is isomorphic to the Hall algebra of \(\vec\Delta\) (by results of \textit{C. M. Ringel} [see for example Invent. Math. 101, No. 3, 583-591 (1990; Zbl 0735.16009)]). This can be seen as a far-reaching generalization of a theorem of Gabriel which yields a bijection between the positive roots of \(\Delta\) and the indecomposable representations of \(\vec\Delta\). In order to get the full quantum group \(U\) one first has to find a representation theoretic version of the set of all roots, not just the positive ones. A natural solution for this problem is to pass to the derived category of representations of \(\vec\Delta\), the indecomposables of which give countably many copies of the positive roots, and then to pass to a quotient modulo the square of the shift automorphism. For the resulting category one then has to define something like a Hall algebra. In the present paper, this approach (which comes with several technical difficulties) is used to describe at least the underlying Lie algebra itself and all its data like structure constants in terms of Hall algebras. This approach has been followed further by the authors [see \textit{L. G. Peng} and \textit{J. Xiao}, CMS Conf. Proc. 18, 539-554 (1996; Zbl 0870.16009) and \textit{J. Xiao}, Hall algebras in a root category (Bielefeld, Preprint 95-070)]\ and independently by \textit{M. Kapranov} in other situations [see e.g. J. Algebra 202, No. 2, 712-744 (1998)]\ and they obtained a (quite complicated) Hall algebra description of the full quantum group as well. Another way to construct the quantum group \(U\) out of \(U^+\) is by first applying a general construction, the so-called Drinfeld double, and then passing to a quotient (modulo a trivial Hopf ideal). This has been done by \textit{J. Xiao} [J. Algebra 190, No. 1, 100-144 (1997; Zbl 0874.16026)].
scientific article; zbMATH DE number 1106632
scientific article