The commutative core of a Leavitt path algebra

The cycline subalgebra of a Kumjian-Pask algebra

“…In this paper we study analogues of Leavitt path algebras associated to higher-rank graphs; these algebras are called Kumjian-Pask algebras. Concretely we extend the results given in [9] to Kumjian-Pask algebras.…”

“…At the same time we prove a more general version of the main results of [9] in the context of 1-graphs. In [9] the second and third named authors prove a uniqueness theorem for Leavitt path algebras which establishes that the injectivity of a representation depends only on its injectivity on a certain commutative subalgebra [9,Theorem 5.2].…”

“…At the same time we prove a more general version of the main results of [9] in the context of 1-graphs. In [9] the second and third named authors prove a uniqueness theorem for Leavitt path algebras which establishes that the injectivity of a representation depends only on its injectivity on a certain commutative subalgebra [9,Theorem 5.2]. Note that these results are not corollaries of the ones we obtain in this paper, since in [9] arbitrary graphs are considered and here we suppose that k-graphs are row-finite with no sources.…”

“…Let E be a 1-graph. An infinite path in E is called essentially aperiodic if it is either aperiodic or of the form µccc · · · , where c is a cycle without entry (see [9,Definition 3.5 (B)] for more details). In [6,Example 5.4], it is seen that T E is the set of essentially aperiodic paths of E.…”

“…In[6, Example 4.9], it is seen that a standard generator s α s β * is cycline if and only if α = β, α = βc or β = αc for some cycle c ∈ s(α)Es(α) without entry. Thus the cycline subalgebra M coincides with the commutative core M R (E) defined in[9, Definition 3.15].Define D ′ such that(2) D ′ := {a ∈ KP R (Λ) : ad = da for every d ∈ D}.…”