Note: This question now has a sister :-)

The Coxeter transformation of a generalized Cartan matrix:

In the paper The spectral radius of the Coxeter transformations for a generalized Cartan matrix, Claus Ringel defines a Coxeter transformation as follows, in several steps:

A generalized Cartan matrix of size $n$ is a matrix $A \in M^{n \times n}(\mathbb{Z})$ such that for all $i \neq j$ the following properties are satisfied:

  • $A_{ii} = 2$
  • $A_{ij} \leq 0$
  • $A_{ij} \neq 0 \Leftrightarrow A_{ji} \neq 0$

He then goes on to define the reflection $R_i: \mathbb{R}^n \to \mathbb{R}^n$ as the linear map (depending on $A$) which is given on the canonical basis by $e(j) \mapsto e(j) - A_{ji}e(i)$.

Now if $\pi: \{1, \dots, n\} \to \{1, \dots, n\}$ is any permutation, he calls the product $C(A, \pi) := R_{\pi(n)} \cdots R_{\pi(1)}$ a Coxeter transformation for $A$.

The Coxeter transformation of a finite dimensional algebra:

Let $k$ be a field and $A$ be a finite-dimensional $k$-algebra. Let $P(1), \dots, P(n)$ and $I(1), \dots, I(n)$ be the indecomposable projective $A$-modules up to isomorphism and the indecomposable injective $A$-modules up to isomorphism, respectively (in such a way that the orders correspond, i.e. $P(i) = Ae_i$, $I(i) = D(e_iA)$ with the same idempotent $e_i$). Assume that the dimension vectors $\underline{\dim}(P(1)), \dots, \underline{\dim}(P(n))$ form a basis of $\mathbb{R}^n$. Then we can define the Coxeter transformation for $A$ (with respect to the ordering of the projective modules) by $\Phi_{A}: \mathbb{R}^n \to \mathbb{R}^n, \ \underline{\dim}(P(i)) \mapsto -\underline{\dim}(I(i))$,


What precisely is the relation between these two notions of Coxeter transformations? How do properties of a generalized Cartan matrix and properties of a finite-dimensional algebra translate via this relation?


Since I didn't want to think about permutations too much, here is an answer about relating Coxeter transformations of the form $C(A, \text{id})$ to Coxeter transformations of finite-dimensional path-algebras.

Let $A$ be a generalized Cartan matrix. Let $A_+$ and $A_-$ be defined by

$$\left( A_+\right)_{ij} = \begin{cases} A_{ji}, \ i > j \\ 1, \ i = j \ \ \ \ \ \ \ \ ; \\ 0, \text{else} \end{cases} \ \ \ (A_-)_{ij} = \begin{cases} A_{ij}, \ i > j \\ 1, \ i = j \\ 0, \text{else} \end{cases}$$

Theorem 1: We have $C(A, \text{id}) = -A_{+}^{-1}A_{-}^t$.

Proof: This is Theorem $2.9$ in this paper by Sefi Ladkani. Note that I changed the definition of $A_+$ and $A_-$ to match the definition of a Coxeter transformation I gave in my question. $\square$

Now let $H = kQ$ be the path algebra of the finite quiver $Q$ without oriented cycles. Let $S(1), \dots, S(n)$ be the simple modules, ordered in such a way that there is never an arrow in increasing direction, i.e. whenever $1 \leq i \leq j \leq n$ then there is no arrow $i \to j$ in $Q$. This can be achieved by labeling a sink of $Q$ with number $1$, a sink of the remaining quiver after killing $1$ with number $2$ and so on. The homological bilinear form $\left\langle - , -\right\rangle : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$ is given (since $H$ is hereditary) on dimension vectors by \begin{align*} \left\langle \underline{\dim} S(i), \underline{\dim} S(j)\right\rangle & = \dim_k \text{Hom}_H(S(i), S(j)) - \dim_k \text{Ext}_H^1(S(i), S(j)) \\ & = \delta_{ij} - \#\{\alpha: i \to j\}. \end{align*} It is well known that the Coxeter transformation $\Phi_H$ of $H$ is the unique Coxeter transformation of the homological bilinear form, in the sence of Ladkanis paper (i.e. $\left\langle x,y \right\rangle = - \left\langle y, \Phi_{H}x \right\rangle$). Therefore, if we set $D$ the matrix with entries $(D)_{ij} = \left\langle \underline{\dim}S(i), \underline{\dim}S(j)\right\rangle$ then we get $\Phi_H = -D^{-1}D^t$. Furthermore, the matrix $A = D + D^t$ is a symmetric generalized Cartan matrix, and since $D$ has no nonzero entries at the upper triangle (remember the order of the simple modules), we get $A_+ = D = A_-$. We get the following:

Theorem 2: $\Phi_H = C(A, \text{id})$.

Proof: By theorem $1$ we have $C(A, \text{id}) = -A_+^{-1}A_{-}^t = -D^{-1}D^t = \Phi_H$. Compare also with Corollary $2.11$ in Ladkanis paper. $\square$

Remark: I'm pretty sure we can generalize this to every permutation. Also note that we can in this way represent every Coxeter transformation of a symmetric generalized Cartan matrix as the Coxeter transformation of a path algebra (since the numbers in the lower triangle show us exactly how many arrows we have to put between the indices).

Added later: There is still the question how properties of the algebra $H = kQ$ and properties of the associated Cartan matrix $A$ correspond. Remember that we have $A_{ii} = 2$, $A_{ij} = - \#\{\alpha: i \to j\}$ for $i > j$ and $A_{ij} = A_{ji}$ and that al arrows in $Q$ go in increasing direction. Let $q_A$ be the quadratic form associated to $A$, i.e. $q_A(x) = x^{t}Ax$ for $x \in \mathbb{Z}^n$. Let $q_Q$ be the quadratic form of the quiver $Q$, i.e. $$q_Q(x) = \left\langle x, x \right\rangle = \sum_{i = 1}^{n}x_i^2 - \sum_{\alpha \in Q_1}x_{s(\alpha)}x_{t(\alpha)}.$$

Theorem 3: $q_A = 2q_Q$.

Proof: We have \begin{align*} q_A(x) & = x^tAx = \sum_{i,j} x_iA_{ij}x_j \\ & = \sum_{i = 1}^{n}A_{ii}x_{i}^2 + \sum_{i \neq j} A_{ij}x_ix_j \\ & = \sum_{i = 1}^{n}2x_i^2 + \sum_{i > j}2A_{ij}x_ix_j \\ & = 2 \left( \sum_{i = 1}^{n}x_i^2 - \sum_{\alpha \in Q_1} x_{s(\alpha)}x_{t(\alpha)}\right) \\ & = 2q_Q(x), \end{align*} proving the claim. $\square$

Therefore, the quiver $Q$ is Dynkin (i.e. $H$ is representation finite), Euclidean (i.e. $H$ is tame) or wild (i.e. $H$ is wild) iff $A$ is positive definite, positive semidefinite (but not positive definite) or indefinite as a matrix.

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