Reference: Infinite Root Systems, Representations of Graphs (Quivers) and Invariant Theory by V.G. Kac.
Gabriel’s incredible theorem tells us that indecomposable representations of a quiver of finite type corresponds to positive roots of the finite dimensional simple Lie algebra associated to the quiver, proved by using Reflection functors which constructs all indecomposable from the simples, corresponds to Weyl group produces all positive roots from simple roots. (Is there a categorification theory for this?)
Difficulty appear in the general and tame case
not all roots are real, i.e. obtainable from reflections of simple roots
reflection can only be applied in the case of an admissible vertex.
In general classification in wild quivers appear to be difficult, but it is possible to describe the set of dimension vectors of indecomposable representations, which is exactly the set of positive roots of a contragredient Lie algebra of infinite Gelfand-Kirillov dimension.
\(\Gamma\) is a free abelian group with a system of generators \(\Pi=\{\alpha_1,\dots,\alpha_n\}\), and the set of positive (non-negative) vectors \(\Gamma_+ = \{\sum \mathbb{N}\alpha_i\}\).
Let \(A=(a_{ij})\in M_n(\mathbb{Z})\) be a Cartan matrix, i.e. such that \(a_{ii}=2\) and \(a_{ij}=0\Leftrightarrow a_{ji}=0\) for \(i\neq j\). This defines us linear forms \(\varphi_i = \alpha_i^{*}\circ A\), i.e. \(\varphi_i(\alpha_j)=a_{ij}\).
The positive root system \(\Delta_+(A)\subset \Gamma_+\) associated with a Cartan matrix \(A\) is a subset generated by \(\Pi\), satisfying
\(\Pi\subset \Delta_+(A)\)
\(2\alpha_i\not\in \Delta_+(A)\)
\(\alpha\in \Delta_+(A)\) and \(\alpha\neq \alpha_i\) implies
The positive roots that can be generated by Weyl group reflections from simple roots are called real roots. Otherwise, they are called imaginary roots.
\(A\in M_n(\mathbb{Z})\) is called Cartan matrix if
\(a_{ii}=2\)
\(a_{ij}\le 0\) for \(i\neq j\)
\(a_{ij}=0\Leftrightarrow a_{ji}=0\).
Two Cartan matrices \(A,B\) are called equivalent if they are conjugate under a permutation of indices.
A matrix \(A\) is decomposable if it is equivalent to direct sum of proper smaller matrices.
A matrix \(A\) is called symmetrizable if one of the equivalent conditions is satisfied
\(a_{ij}=0\Leftrightarrow a_{ji}=0\), and \(a_{i_1i_2}\cdots a_{i_ki_1} = a_{i_1i_k}\cdots a_{i_2i_1}\) for any \(i_1,\dots,i_k\).
There exists non-singular diagonal matrix \(D\) and symmetric matrix \(S\) such that \(A=DS\). (additionally, for indecomposable symmetrizable Cartan matrix, such decomposition is unique).
Let \(A\) be indecomposable Cartan matrix, then there is a disjoint union of three cases
(Positive) \(\det A\neq 0\), \(Au\ge 0\Rightarrow u>0\) or \(u=0\)
(Affine) \(\mathrm{rank}A = n-1\), there is a positive vector \(u\) in the kernel (isotropic imaginary root). \(Au\ge 0\Rightarrow Au=0\) (so will be a multiple of isotropic root).
(Negative) There is a positive vector \(u\) with \(Au<0\). \(u\ge 0\) and \(Au\ge 0\Rightarrow u=0\).
Let \(A\in M_n(\mathbb{Z})\) be Cartan, \(\Gamma\) with \(n\) generators \(\alpha_1,\dots,\alpha_n\). Associate with the matrix \(A\) a complex \(\Gamma\)-graded Lie algebra
\[\mathfrak{g}(A) = \bigoplus_{\alpha\in \Gamma}\mathfrak{g}_\alpha\]
characterised by
Every graded ideal which intersects \(\mathfrak{g}_0\) trivially is trivial (zero). I.e. every non-trivial graded ideal should have a non-trivial degree zero component.
\(\mathfrak{g}(A)\) is generated by the elements \(e_i,f_i,h_i\) for \(i=1,\dots,n\) such that for simple roots, \(\mathfrak{g}_{\alpha_i}=\mathbb{C}e_i\), \(\mathfrak{g}_{-\alpha_i}=\mathbb{C}f_i\).
The \(h_i\) form a basis of \(\mathfrak{g}_0\) with the following relations
\[ [h_i,h_j]=0 ,\quad [e_i,f_j]=\delta_{ij}h_i , \]
\[ [h_i,e_j]=a_{ij}e_j ,\quad [h_i,f_j]=-a_{ij}f_j . \]
For \(\Gamma\) where \(\mathfrak{g}_\alpha\neq 0\), \(\alpha\) is called a root. \(\Pi=\{\alpha_1,\dots,\alpha_n\}\) denotes the set of simple roots, \(\Delta\) denotes the set of roots, \(\Delta_+=\Gamma_+\cap \Delta\) denotes the set of positive roots.
The height of a vector \(\alpha\in \Gamma\) is the sum of coefficients of \(\alpha\) with respect to simple roots.
The linear forms \(\varphi_i = \alpha_i^{*}\circ A\) is actually \((,\alpha_i)\) where \((,)\) is the bilinear form defined by the Cartan matrix \(A\).
These allow us to define the reflections on the vector space \(L=\Gamma\otimes \mathbb{R}\) by
\[s_i(\alpha) = \alpha - \varphi_i(\alpha)\alpha_i = \alpha - (\alpha,\alpha_i)\alpha_i.\]
They satisfy the following properties
\(s_i^2=1\)
\((s_is_j)^{n_{ij}}=1\) where \(n_{ij}\) is defined as a function on \(a_{ij}a_{ji}\) as
\[ n_{ij} = \begin{cases} 2 & \text{if } a_{ij}a_{ji}=0\\ 3 & \text{if } a_{ij}a_{ji}=1\\ 4 & \text{if } a_{ij}a_{ji}=2\\ 6 & \text{if } a_{ij}a_{ji}=3\\ \infty & \text{if } a_{ij}a_{ji}\ge 4 \end{cases} \]
If \(A\) is symmetrizable Cartan matrix, the following formal formula can be used to compute the multiplicity of roots
\[\prod_{\alpha\in \Delta_+} (1-e^{\alpha})^{\dim \mathfrak{g}_\alpha} = \sum_{w\in W} (\det w) e^{s(w)}\]