Euler Form from Quiver Representations and Roots

We review notions from last lecture, including projective and simple modules over the path algebra \(kQ\) of a quiver. From the projective resolution of any module over \(kQ\), we compute dimensions of Hom and Ext functors. We use this to motivate the definition of the Euler form, classify when the symmetrized Euler form is positive (semi)definite, and draw connections to Lie theory. We finally state the definition and classification of finite type quivers. This talk will cover sections 1.4–1.7 and 3.1 of Quiver Representations and Quiver Varieties by Alexander Kirillov Jr.

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Date published: Wednesday, February 4, 2026