The Serre relations in Ringel-Hall algebras (Q1848450)

The author first constructs the Casimir element of the Hall algebra. Applying the method of the Gabber-Kac theorem, it is proved that the Serre relations are the defining relations of the composition algebra. From this point of view, the Ringel-Green isomorphism theorem still holds for standard composition algebras. This means that the Drinfeld double of a composition algebra with an indeterminate \(v\) is naturally isomorphic to the quantum groups in the sense of Drinfeld-Jimbo.