On double Ringel-Hall algebras
Due to the work of \textit{J. A. Green} [Invent. Math. 120, No. 2, 361-377 (1995; Zbl 0836.16021)] the Ringel-Hall algebra of a finite-dimensional hereditary algebra \(\Lambda\) augmented by a torus algebra can be endowed with a Hopf algebra structure. Therefore one can construct its Drinfeld double \({\mathcal D}(\Lambda)\). Analogous to a result of \textit{J. A. Green} [op. cit.] and \textit{C. M. Ringel} [in: Representation theory of algebras, CMS Conf. Proc. 19, 185-245 (1996; Zbl 0860.16026)] which states that the Drinfeld double of the composition subalgebra of a Ringel-Hall algebra augmented by a torus algebra is isomorphic to Lusztig's form of the quantized enveloping algebra of the corresponding Kac-Moody algebra, the authors of the paper under review show that \({\mathcal D}(\Lambda)\) is isomorphic to the quantized enveloping algebra of a generalized Kac-Moody algebra in the sense of \textit{R. Borcherds} [J. Algebra 115, No. 2, 501-512 (1988; Zbl 0644.17010)]. In the quiver case it was proved before by \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Pure Appl. Algebra 160, No. 2-3, 319-332 (2001; Zbl 1025.17006)] that the Ringel-Hall algebra is isomorphic to the positive part of the quantized enveloping algebra of a generalized Kac-Moody algebra. The authors of the paper under review also observe that \({\mathcal D}(\Lambda)\) is a restricted non-degenerate member of some datum in the sense of \textit{J. A. Green} [op. cit.]. As a consequence, they obtain that the Ringel-Hall algebra is independent of the orientation of \(\Lambda\). In the quiver case this was pointed out by \textit{G. Lusztig} [J. Am. Math. Soc. 3, No. 2, 447-498 (1990; Zbl 0703.17008)] and proved by \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Algebra 221, No. 1, 135-160 (1999; Zbl 0955.16016)] by using a notion of Fourier transform. Moreover, the authors of the paper under review define in a natural way a highest weight category \(\mathcal O\) and integrable modules for \({\mathcal D}(\Lambda)\). The pairing between the positive and negative parts of \({\mathcal D}(\Lambda)\) yields an \(R\)-matrix which satisfies the quantum Yang-Baxter equation and induces a symmetry for the tensor product in \(\mathcal O\). It is shown that there exist enough irreducible integrable highest weight modules on which the action of this \(R\)-matrix provides solutions of the quantum Yang-Baxter equation. The authors also prove that the integrable modules in \(\mathcal O\) with strongly dominant highest weight are irreducible and satisfy a Kac-Weyl character formula.
irreducible integrable highest weight modules
scientific article; zbMATH DE number 1797466