Dimension of the mesh algebra of a finite Auslander-Reiten quiver

We show that the dimension of the mesh algebra of a finite Auslander-Reiten quiver over a field is a purely combinatorial invariant that does not depend on the ground field. Moreover, a combinatorial algorithm for computing this dimension is given along our proof of this result. Translation quivers appear naturally in the representation theory of finite dimensional algebras; see, for example, [3]. A translation quiver defines a mesh algebra over any field. A natural question arises as to whether or not the dimension of the mesh algebra depends on the field. The purpose of this note is to show that the dimension of the mesh algebra of a finite Auslander-Reiten quiver over a field is a purely combinatorial invariant of this quiver. Indeed, our proof yields a combinatorial algorithm for computing this dimension. As a further application, one may use then semicontinuity of Hochschild cohomology of algebras as in [6] to conclude that a finite Auslander-Reiten quiver contains no oriented cycle if its mesh algebra over some field admits no outer derivation. 1. Simply connected translation quivers Throughout this note, Γ denotes a translation quiver with translation τ ; see [3, (1.1)]. We assume that Γ contains neither loops nor multiple arrows but that each non-projective vertex is the end-point of at least one arrow. One defines the orbit graph O(Γ ) of Γ as follows: the τ -orbit of a vertex a is the set o(x) of vertices of the form τ(x) with n ∈ ZZ ; the vertices of O(Γ ) are the τ -orbits of Γ , and there exists an edge o(x) o(y) in O(Γ ) if Γ contains an arrow a → b or b → a with a ∈ o(x) and b ∈ o(y). If Γ contains no oriented cycle, then O(Γ ) is the graph GΓ defined in [3, (4.2)]. Now Γ is called simply connected if Γ contains no oriented cycle and O(Γ ) is a tree. By [3, (1.6), (4.1), (4.2)], this definition is equivalent to that given in [3, (1.6)]. Let p : x0 → x1 → · · · → xr be a path in Γ . Then p induces a walk w(p) : o(x0) o(x1) · · · o(xr) in O(Γ ), and w(p) in turn determines a unique reduced walk wred(p). Recall that the path p is called sectional if