Semisimple representations of quivers.

*(English)*Zbl 0693.16018Let 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.

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 |