Applications of spherical twist functors to Lie algebras associated to root categories of preprojective algebras

Abstract

Let ΛQ be the preprojective algebra of a finite acyclic quiver Q of non-Dynkin type and Db(repnΛQ) be the bounded derived category of finite dimensional nilpotent ΛQ-modules. We define spherical twist functors over the root category RΛQ of Db(repnΛQ) and then realize the Weyl group associated to Q as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra g(RΛQ) of RΛQ induced by spherical twist functors. We also present a concrete relation between certain Lie subalgebras of g(RΛQ) and g(RQ), where g(RQ) is the Ringel–Hall Lie algebra associated to the root category RQ of Q.