Certain Examples of Deformed Preprojective Algebras and Geometry of Their *-Representations

TL;DR

This paper explores specific deformed preprojective algebras associated with affine types and their *-representations, providing realizations and constructing their C*-enveloping algebras, linking to known results in representation theory.

Contribution

It introduces a realization of certain deformed preprojective algebras and constructs their C*-enveloping algebras, connecting algebraic structures with geometric representation theory.

Findings

Realization of deformed preprojective algebras for specific lambda values

Construction of C*-enveloping algebras for these structures

Unification of known results within this algebraic framework

Abstract

We consider algebras $e_i \Pi^\lambda(Q) e_i$ obtained from deformed preprojective algebra of affine type $\Pi^\lambda(Q)$ and an idempotent $e_i$ for certain concrete value of the vector $\lambda$ which corresponds to the traces of $-1\in SU(2, C)$ in irreducible representations of finite subgroups of $SU(2, C)$. We give a certain realization of these algebras which allows us to construct the $C^*$-enveloping algebras for them. Some well-known results, including description of four projections with sum 2 happen to be a particular case of this picture.