7
$\begingroup$

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 $\Phi_{A}: \mathbb{R}^n \to \mathbb{R}^n, \ \underline{\dim}(P(i)) \mapsto -\underline{\dim}(I(i))$,

Question:

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?

$\endgroup$
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.