an:04134258
Zbl 0693.16018
Le Bruyn, Lieven; Procesi, Claudio
Semisimple representations of quivers
EN
Trans. Am. Math. Soc. 317, No. 2, 585-598 (1990).
00172867
1990
j
16G20 14L30 20G05
finite quiver; dimension vector; affine variety; linear reductive group; category of representations; Jordan decomposition; semisimple representations
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 \textit{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 ??tale slice machinery devised by \textit{D. Luna} [in Bull. Soc. Math. Fr., M??m. 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.
H.A.Merklen
Zbl 0232.08001; Zbl 0286.14014