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Geometry of representations of quivers. (English) Zbl 0632.16019

Representations of algebras, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser. 116, 109-145 (1986).
[For the entire collection see Zbl 0597.00009.]
In this elegant survey paper, the authors survey some recent results of V. Kac’s, that describe the dimension types of all indecomposables of arbitrary quivers. The starting points of the paper are the well-known facts due to P. Gabriel resp. P. Donovan - M. R. Freislich and L. A. Nazarova, that a connected quiver is of finite representation type resp. tame if and only if its associated graph is a Dynkin diagram resp. an extended Dynkin diagram. In the first case, the dimension type dim induces a bijection between (isomorphism classes of) indecomposable representations of the quiver Q and positive roots of Q, i.e. the vectors \(\alpha \in {\mathbb{N}}^ n\) with \(q(\alpha)=1\), where q is the (positive definite!) Tits form of Q and \(n=| Q|\). In the tame case a similar relation may be given as well. Of course, all other quivers are wild, showing the depth of Kac’s surprising results.
Although most results in the paper are interesting in their own right, let us concentrate on its main result: Kac’s theorem. From a geometric point of view, one considers the \(\alpha\)-representation space of Q, i.e. the affine varieties R(Q,\(\alpha)\) of all representations of Q of dimension type \(\alpha =(\alpha (1),...,\alpha (n))\in {\mathbb{N}}^ n\). The group \(G\ell (\alpha)=\prod G\ell (\alpha (i))\) acts linearly on R(Q,\(\alpha)\), and one wants to study the orbits of \(G\ell (\alpha)\) in R(Q,\(\alpha)\). One of the tools to study these, is by considering their number of parameters. In general, if G is an algebraic group acting on some variety V and if \(X\subset V\) is a G-stable subset, we let \(X_{(s)}\) consist of all \(x\in X\) with dim \(O_ x=s\) and we then define the number of parameters of X to be \(\mu (X)=\max (\dim X_{(s)}- s)\). In particular, we may thus consider the number of parameters \(\mu (R(Q,\alpha)_{ind})\) of the indecomposables in R(Q,\(\alpha)\). Let us call a vector \(\alpha \in {\mathbb{N}}^ n\) a root of Q if R(Q,\(\alpha)\) contains an indecomposable representation. If \(\mu (R(Q,\alpha)_{ind})\geq 1\), then we call \(\alpha\) an imaginary root, otherwise, we speak of a real root (i.e. \(R(Q,\alpha)_{ind}\) contains a finite number of orbits). So, the set of all roots \(\Delta\) (Q) is partitioned into \(\Delta (Q)=\Delta (Q)_{re}\cup \Delta (Q)_{im}.\)
On the other hand, a simple root is a standard basic vector \(e_ i\) of \({\mathbb{Q}}^ n\), where i is a vertex to which no loop is attached. We denote by \(\pi_ Q=\Delta (Q)_{re}\) the set of all simple roots. Clearly, \(e_ i\) is a simple root if and only if \(e_ i\) does not belong to \(F_ Q\), the fundamental set of Q, which consists of all non- zero \(\alpha \in {\mathbb{N}}^ n\) with connected support, such that \((\alpha,e_ i)\leq 0\). Finally, denote by \(W_ Q\subset Gl({\mathbb{Z}}^ n)\) the Weyl group of Q, generated by the reflections \(r_ i: {\mathbb{Z}}^ n\to {\mathbb{Z}}^ n:\alpha \to \alpha -(\alpha,e_ i)e_ i\), associated to the simple roots \(e_ i\). Of course, if Q is a Dynkin quiver, then these definitions are just the classical ones, so in particular \(W_ Q\pi_ Q=\Delta (Q)\cup -\Delta (Q)\), is the corresponding root system.
Kac’s theorem now states that \(\Delta (Q)_{re}=W_ Q\pi_ Q\cap {\mathbb{N}}^ n\) resp. \(\Delta (Q)_{im}=W_ QF_ Q\). Moreover, if \(\alpha \in \Delta (Q)_{re}\), then \(R(Q,\alpha)_{ind}\) is one orbit, whereas for \(\alpha \in \Delta (\Omega)_{im}\), we have \(\mu (R(Q,\alpha)_{ind})=1-q(\alpha)\), where, as before, q is the Tits form associated to Q. The proof of this result heavily relies upon the fact that for any \(\alpha \in {\mathbb{N}}^ n\) the number of isomorphism classes of indecomposables V with dim V\(=\alpha\), as well as \(\mu (R(Q,\alpha)_{ind})\) only depend upon the underlying graph of Q.
Reviewer: A.Verschoren

MSC:

16Gxx Representation theory of associative rings and algebras
16P10 Finite rings and finite-dimensional associative algebras
17B20 Simple, semisimple, reductive (super)algebras
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
14L30 Group actions on varieties or schemes (quotients)
15A72 Vector and tensor algebra, theory of invariants

Citations:

Zbl 0597.00009