Group gradations on Leavitt path algebras

Abstract

Given a directed graph [Formula: see text] and an associative unital ring [Formula: see text] one may define the Leavitt path algebra with coefficients in [Formula: see text], denoted by [Formula: see text]. For an arbitrary group [Formula: see text], [Formula: see text] can be viewed as a [Formula: see text]-graded ring. In this paper, we show that [Formula: see text] is always nearly epsilon-strongly [Formula: see text]-graded. We also show that if [Formula: see text] is finite, then [Formula: see text] is epsilon-strongly [Formula: see text]-graded. We present a new proof of Hazrat’s characterization of strongly [Formula: see text]-graded Leavitt path algebras, when [Formula: see text] is finite. Moreover, if [Formula: see text] is row-finite and has no source, then we show that [Formula: see text] is strongly [Formula: see text]-graded if and only if [Formula: see text] has no sink. We also use a result concerning Frobenius epsilon-strongly [Formula: see text]-graded rings, where [Formula: see text] is finite, to obtain criteria which ensure that [Formula: see text] is Frobenius over its identity component.

Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

“…Crucial to our investigation, Nystedt andÖinert have shown that a Leavitt path algebra associated to a finite directed graph is epsilon-strongly Z-graded (see [23,Thm. 1.2]).…”

“…However, since an epsilon-strong Z-grading is strong if and only if ǫ i = 1 for every i ∈ Z (see [23,Prop. 3.2]), it follows that the above Z-grading of M 2 (R) is not strong.…”

“…2]), corner skew Laurent polynomial rings (see [20,Thm. 8.1]) and Leavitt path algebras of finite graphs (see [23,Thm. 1.2]).…”

Properties of the gradings on ultragraph algebras via the underlying combinatorics

“…Given the exposed above, the importance of understanding the grading of a Leavitt path algebra is clear. In particular, the integer grading of a Leavitt path algebra has been studied in [20], where the unital Leavitt path algebras which are strongly graded are completely characterized, in [10,18], where strongly graded Leavitt path algebras are characterized in terms of Condition (Y), and in [23], where it is shown that the Leavitt path algebra associated to a finite graph is epsilon-strongly Z-graded. The grading over the free group of the edges has been introduced in [13] and has been used to give alternative proofs of interesting results, such as the Reduction Theorem and the simplicity criteria for Leavitt path algebras (see [13,16,12]) and is related to branching systems, see [6].…”

The graded structure of algebraic Cuntz-Pimsner rings

“…1.2]). Seeking to extend their result, they introduced the notion of a nearly epsilonstrongly graded ring (see Definition 2.2) and proved that every Leavitt path algebra (even for infinite graphs) is nearly epsilon-strongly Z-graded (see [17,Thm. 1.3]).…”