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Register a new userLeavitt path algebras over a poset of fields
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Authors
Pere Ara
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Let $E$ be a finite directed graph, and let $I$ be the poset obtained as the antisymmetrization of its set of vertices with respect to a pre-order $le$ that satisfies $vle w$ whenever there exists a directed path from $w$ to $v$. Assuming that $I$ is a tree, we define a poset of fields over $I$ as a family $mathbf K = { K_i :iin I }$ of fields $K_i$ such that $K_isubseteq K_j$ if $jle i$. We define the concepts of a Leavitt path algebra $L_{mathbf K} (E)$ and a regular algebra $Q_{mathbf K}(E)$ over the poset of fields $mathbf K$, and we show that $Q_{mathbf K}(E)$ is a hereditary von Neumann regular ring, and that its monoid $mathcal V (Q_{mathbf K}(E))$ of isomorphism classes of finitely generated projective modules is canonically isomorphic to the graph monoid $M(E)$ of $E$.
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A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.
Given a graph E we define E-algebraic branching systems, show their existence and how they induce representations of the associated Leavitt path algebra. We also give sufficient conditions to guarantee faithfulness of the representations associated to E-algebraic branching systems and to guarantee equivalence of a given representation (or a restriction of it) to a representation arising from an E-algebraic branching system.
We show that if $E$ is an arbitrary acyclic graph then the Leavitt path algebra $L_K(E)$ is locally $K$-matricial; that is, $L_K(E)$ is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field $K$. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph $E$: (1) $L_K(E)$ is von Neumann regular. (2) $L_K(E)$ is $pi$-regular. (3) $E$ is acyclic. (4) $L_K(E)$ is locally $K$-matricial. (5) $L_K(E)$ is strongly $pi$-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.
We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${mathbb Z}$-graded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the $K_0$ groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in $K_0$, classifies the algebras in these sets up to isomorphism. Second, we show that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using number-theoretic methods, can be reobtained via appropriate isomorphisms between Leavitt path algebras.
We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations induce Morita equivalence of the corresponding Leavitt path algebras. As a consequence we obtain our two main results: the first gives sufficient conditions for which the Leavitt path algebras in a certain class are Morita equivalent, while the second gives sufficient conditions which yield isomorphisms. We discuss a possible approach to establishing whether or not these conditions are also in fact necessary. In the final section we present many additional operations on graphs which preserve Morita equivalence (resp., isomorphism) of the corresponding Leavitt path algebras.
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