Socle deformed preprojective algebras of generalized Dynkin type

Lecture 17: Deformed Preprojective Algebras

(1) If there is an indecomposable representation of dimension v, then v is a root. (2) If v is a real root, then there is a unique (up to an isomorphism) indecomposable representation of dimension v. (3) If v is primitive (meaning that GCD(vi) = 1) and there is an indecomposable representation of dimension v, then pv, the number of parameters for the isomorphism classes of indecomposable repres...

Generalisations of Preprojective algebras

In this thesis, we investigate two ways of generalising the preprojective algebra. First, we introduce the multiplicative preprojective algebra, Λ(Q), which is a multiplicative analogue of the deformed preprojective algebra, introduced by Crawley-Boevey and Holland. The special case q = 1 is the undeformed multiplicative preprojective algebra, which is an analogue of the ordinary (undeformed) p...

The Multisegment Duality and the Preprojective Algebras of Type A

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

We consider algebras eiΠ λ(Q)ei obtained from deformed preprojective algebra of affine type Π(Q) and an idempotent ei for certain concrete value of the vector λ which corresponds to the traces of −1 ∈ 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-k...

Preprojective Algebras and Mv Polytopes

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig’s nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.