A Hopf Algebra from Preprojective Modules

Abstract

Let Q be a finite type quiver i.e. ADE Dynkin quiver. Denote by \Lambda its preprojective algebra. It is known that there are finitely many indecomposable \Lambda-modules if and only if Q is of type A1, A2, A3, A4. Extending Lusztig’s construction of Un, we study an algebra generated by these indecomposable submodules. It turns out that it forms the universal enveloping algebra of some nilpotent Lie algebra inside the function algebra on Lusztig’s nilpotent scheme. The defining relations of the corresponding nilpotent Lie algebra for type A1, A2, A3, A4 are given here.