Applications of Normal Forms for Weighted Leavitt Path Algebras: Simple Rings and Domains

Abstract

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras LK (n, n + k) constructed by Leavitt. Using Bergman's diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular.

The Injective and Projective Leavitt Complexes

“…The following notion is inspired by a basis for Leavitt path algebra (refer to [1] and [13] for the basis).…”

“…Denote by I • = (I l , ∂ l ) l∈Z and I •′ = (I l ′ , (∂ l ) ′ ) l∈Z the injective Leavitt complexes of E with S and S ′ sets of special edges, respectively. Let Λ ′ be the k-basis of L k (E) given by [1,Theorem 1] and [13,Corollary 17] with S ′ the set of special edges.…”

The V-monoid of a weighted Leavitt path algebra

“…Remark 3. In [4] and [5], E 1 was denoted by E st . The set {e i | e ∈ E 1 , 1 ≤ i ≤ w(e)} was denoted by E 1 .…”

“…Let n ≥ 1 and k ≥ 0. If (E, w) is a weighted graph with precisely one vertex and precisely n + k edges each of which has weight n, then L K (E, w) is isomorphic to the Leavitt algebra L K (n, n + k), for details see [5,Example 4].…”

“…A presentation for V (L K (E, w)) where K is a field and (E, w) is a row-finite weighted graph was given in [4,Theorem 5.21]. In [5] this presentation was used in order to show that there is a huge class C of wLpas that are domains but neither are isomorphic to an Lpa nor to a Leavitt algebra. Unfortunately, [4,Theorem 5.21] is wrong as we will show in Section 4.…”