Graph algebras and the Gelfand–Kirillov dimension

Abstract

We study some properties of the Gelfand–Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with direct limits. We then give some applications in the context of graph algebras, which embraces, among some others, path algebras and Cohn and Leavitt path algebras. In particular, we determine the GK-dimension of these algebras in full generality, so extending the main result in A. Alahmadi, H. Alsulami, S. K. Jain and E. Zelmanov, Leavitt Path algebras of finite Gelfand–Kirillov dimension, J. Algebra Appl. 11(6) (2012) 1250225–1250231.

On Leavitt path algebras of Hopf graphs

“…Otherwise, S G,r is not a submonoid of G, and so Γ G,r has no cycles, by Proposition 2.5 (6). Then, by [23,Theorem 3.21], GKdim(L K (Γ G,r )) = 0, thus finishing the proof.…”

“…In [11] Alahmadi, Alsulami, Jain and Zelmanov determined the Gelfand-Kirillov dimension of Leavitt path algebras of finite graphs. In [23] Moreno-Fernández and Siles Molina extended this to arbitrary graphs. We should mention this result here.…”

“…We say that such a chain has an exit if the cycle c k has an exit. Let d 1 be the maximal length of a chain of cycles in E, and let d 2 be the maximal length of chain of cycles with an exit in E. For every field K, by [23,Theorem 3.21], GKdim(L K (E)) is finite if and only if E satisfies Condition (EXC) and the maximal length of chains of cycles in E is finite. In this case, GKdim(L K (E)) = max{2d 1 − 1, 2d 2 }.…”

“…In particular, we extend the criteria for the simplicity and Invariant Basis Number property of Leavitt path algebras of Cayley graphs, introduced in [22,24], to Hopf graphs. We should mention that graph-theoretic characterizations on graphs E of these properties for Leavitt path algebras L K (E) have been established in literatures (see, e.g, [6], [13], [18] and [23], respectively); while based on these criteria, our characterizations are established completely on properties of both ramification data r and G.…”

“…Proof. If C∈C r C |C| = 1 and S G,r is a submonoid of G, then by Corollary 2.9 (2), Γ G,r is a disjoint union of single cycles, and so GKdim(L K (Γ G,r )) = 2, by[23, Theorem 3.21]. If C∈C r C |C| ≥ 2 and S G,r is a submonoid of G, then by Proposition 2.5(6), Γ G,r has a cycle.…”