# zbMATH — the first resource for mathematics

Semisimple representations of quivers. (English) Zbl 0693.16018
Let Q be a finite quiver with vertices $$Q_ 0=\{1,...,n\}$$ and let us fix an algebraically closed field C of characteristic zero and a dimension vector $$\alpha =(\alpha (i))_{i\in Q_ 0}$$. In the sense of P. Gabriel [Manuscr. Math. 6, 71-103 (1972; Zbl 0232.08001)], the set of C-representations of Q with dimension vector $$\alpha$$, R(Q,$$\alpha)$$, is an affine variety where the linear reductive group $$GL(\alpha)=\prod_{i}GL_{\alpha (i)}(C)$$ acts by isomorphisms of the category of representations.
The question which is considered here is to study the orbit structure of GL($$\alpha)$$ acting on R(Q,$$\alpha)$$. A representation V in R(Q,$$\alpha)$$ is called semisimple (resp. nilpotent) if its orbit GL($$\alpha)$$$$\cdot V$$ is closed (resp. if 0 belongs to the Zariski closure of GL($$\alpha)$$$$\cdot V)$$. Every representation V has a Jordan decomposition $$V=V_ s+V_ n$$, where $$V_ s$$ is semisimple and $$V_ n$$ is nilpotent. One of the main objectives of the paper is to study the semisimple representations of Q by applying the étale slice machinery devised by D. Luna [in Bull. Soc. Math. Fr., Mém. 33, 81-105 (1973; Zbl 0286.14014)]. One of the byproducts is the determination of all dimension vectors which correspond to a semisimple representation of Q.
Reviewer: H.A.Merklen

##### MSC:
 16G20 Representations of quivers and partially ordered sets 14L30 Group actions on varieties or schemes (quotients) 20G05 Representation theory for linear algebraic groups
Full Text: