Leavitt path algebras satisfying a polynomial identity

Abstract

Abstract. Leavitt path algebras L of an arbitrary graph E over a field K satisfying a polynomial identity are completely characterized both in graphtheoretic and algebraic terms. When E is a finite graph, L satisfying a polynomial identity is shown to be equivalent to the Gelfand-Kirillov dimension of L being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph E, the Leavitt path algebra L has Gelfand-Kirillov dimension zero if and only if E has no cycles. Likewise, L has Gelfand-Kirillov dimension one if and only if E contains at least one cycle, but no cycle in E has an exit.

Leavitt path algebras with bounded index of nilpotence

“…Examples show that the converse of these statements do not hold. Interestingly, it turns out that the graphical conditions on E that ensure L has a bounded index of nilpotence are exactly the same graphical conditions on E that were shown in [4] to imply that L satisfies a polynomial identity. When E is a finite graph, these graphical conditions also imply that L has GK-dimension ≤ 1.…”

“…For instance, the Amitsur-Levitzky theorem (see [13]) states that, for any n ≥ 1, the matrix ring M n (R) over a commutative ring R satisfies a polynomial identity of degree 2n. In [4] it is shown that the Leavitt path algebra L K (E) of an arbitrary graph E over a field K satisfies a polynomial identity if and only if no cycle in E has an exit and there is a fixed positive integer d such that the number of distinct paths that end at any given vertex v (including v, but not including the entire cycle c in case v lies on c) is less than or equal to d. When E is a finite graph, then the Leavitt path algebra L K (E) satisfying a polynomial identity is known to be equivalent to the Gelfand-Kirillov dimension of L K (E) being at most one [4]. (1) L has index of nilpotence less than or equal to n;…”

A Survey of Some of the Recent Developments in Leavitt Path Algebras

“…Theorem 4.7 ( [9], [17], [25], [27], [28], [43] ) For a finite graph E, the following conditions are equivalent for L := L K (E):…”

“…A Γ-graded ring R is said to be a graded Baer ring, if the left/right annihilator of every subset X of homogeneous elements is generated by a homogeneous idempotent. A ring R is said to have bounded index of nilpotence if there is a positive integer n which is such that a n = 0 for every nilpotent element a ∈ R. Theorem 4.7 ( [9], [17], [25], [27], [28], [43] ) For a finite graph E, the following conditions are equivalent for L := L K (E):…”

“…[17]) Let E be an arbitrary graph. Then the following properties are equivalent for L K (E):(a) L K (E) satisfies a polynomial identity; (b) No cycle in E has an exit, there is a fixed positive integer d such that the number of distinct paths that end at any vertex v is ≤ d and the only infinite paths in E are paths that are eventually of the form ggg • ••, for some cycle g;…”