Flow invariants in the classification of Leavitt path algebras.(English)Zbl 1263.16007
Leavitt algebras \(L_K(1,n)\) for \(n\geq 2\) and any field \(K\) were introduced byW. G. Leavitt in 1962 [in Trans. Am. Math. Soc. 103, 113-130 (1962;Zbl 0112.02701)] as universal examples of algebras not satisfying the IBN (invariant basis number) property. Leavitt path algebras \(L_K(E)\) for row-finite graphs \(E\) were independently introduced byP. Ara, M. A. Moreno andE. Pardo [in Algebr. Represent. Theory 10, No. 2, 157-178 (2007;Zbl 1123.16006)], and byG. Abrams and the reviewer [in J. Algebra 293, No. 2, 319-334 (2005;Zbl 1119.16011)]. These \(L_K(E)\) are algebras associated to directed graphs and are the algebraic analogs of the graph \(C^*\)-algebras [I. Raeburn, Graph algebras. Providence: AMS (2005;Zbl 1079.46002)].
In the study of \(C^*\)-algebras, an important role is played by the Classification Theorem of purely infinite simple unital nuclear \(C^*\)-algebras (see e.g. [E. Kirchberg, “The classification of purely infinite \(C^*\)-algebras using Kasparov theory” (preprint)] and [N. C. Phillips, Doc. Math., J. DMV 5, 49-114 (2000;Zbl 0943.46037)]). Specifically, Kirchberg and Phillips independently showed that if \(X\) and \(Y\) are purely infinite simple unital \(C^*\)-algebras (satisfying certain additional conditions), then \(X\cong Y\) as \(C^*\)-algebras if and only if (i) \(K_0(X)\cong K_0(Y)\) via an isomorphism \(\varphi\) for which \(\varphi([1_X])=[1_Y]\), and (ii) \(K_1(X)\cong K_1(Y)\).
The reformulation of the previous theorem for the context of Leavitt path algebras (for instance showing that the \(K_1\) data is irrelevant) was initially tackled byG. Abrams, P. N. Ánh, A. Louly andE. Pardo [in J. Algebra 320, No. 5, 1983-2026 (2008;Zbl 1205.16013)], where the authors also gave a partial proof of it, obtaining a Classification Theorem for purely infinite simple unital Leavitt path algebras whose graphs do not have parallel edges and have at most three vertices.
In the paper under review the authors obtain an affirmative answer for a significantly wider class of graphs. They do so by analyzing in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. Thus, in Theorem 1.25 a sufficient set of conditions on the graphs \(E\) and \(F\) which ensure that \(L_K(E)\) is Morita equivalent to \(L_K(F)\) is obtained. Concretely it is shown that if \(E\) and \(F\) are finite graphs, such that \(L_K(E)\) and \(L_K(F)\) are purely infinite simple, the \(K_0\)-groups are isomorphic and \(\det(I_n-A_E^t)=\det(I_m-A_F^t)\) where \(n\) and \(m\) are the number of vertices of \(E\) and \(F\) respectively, and \(A_E\) is the adjacency matrix of the graph \(E\), then \(L_K(E)\) is Morita equivalent to \(L_K(F)\).
Later in §2, by exploiting the Morita equivalence results in the previous section, sufficient conditions which ensure isomorphism are reached, thus obtaining the aforementioned partial affirmative answer to the Classification Question. In addition to this, the authors examine the remaining difficulty in getting a potentially affirmative answer for all germane graphs.
Finally, in the last section of the paper, several results about Morita equivalence and isomorphism to certain classes of graphs \(E\) for which \(L_K(E)\) is not necessarily purely infinite simple unital are extended, thereby giving more general results than have been previously known about isomorphism and Morita equivalence of Leavitt path algebras.
The paper contains many useful and concrete examples of the operations in graphs, which definitely helps the reader in grasping the material more quickly. The results on the other hand are interesting and relevant, with involved proofs containing several different ideas and techniques. Furthermore, a great deal of the current research in Leavitt path algebras in fact builds upon many of the results herein. Therefore, in the reviewer’s opinion, all this makes the paper under review a definitely must-read for any algebraist working in the field.
In the study of \(C^*\)-algebras, an important role is played by the Classification Theorem of purely infinite simple unital nuclear \(C^*\)-algebras (see e.g. [E. Kirchberg, “The classification of purely infinite \(C^*\)-algebras using Kasparov theory” (preprint)] and [N. C. Phillips, Doc. Math., J. DMV 5, 49-114 (2000;Zbl 0943.46037)]). Specifically, Kirchberg and Phillips independently showed that if \(X\) and \(Y\) are purely infinite simple unital \(C^*\)-algebras (satisfying certain additional conditions), then \(X\cong Y\) as \(C^*\)-algebras if and only if (i) \(K_0(X)\cong K_0(Y)\) via an isomorphism \(\varphi\) for which \(\varphi([1_X])=[1_Y]\), and (ii) \(K_1(X)\cong K_1(Y)\).
The reformulation of the previous theorem for the context of Leavitt path algebras (for instance showing that the \(K_1\) data is irrelevant) was initially tackled byG. Abrams, P. N. Ánh, A. Louly andE. Pardo [in J. Algebra 320, No. 5, 1983-2026 (2008;Zbl 1205.16013)], where the authors also gave a partial proof of it, obtaining a Classification Theorem for purely infinite simple unital Leavitt path algebras whose graphs do not have parallel edges and have at most three vertices.
In the paper under review the authors obtain an affirmative answer for a significantly wider class of graphs. They do so by analyzing in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. Thus, in Theorem 1.25 a sufficient set of conditions on the graphs \(E\) and \(F\) which ensure that \(L_K(E)\) is Morita equivalent to \(L_K(F)\) is obtained. Concretely it is shown that if \(E\) and \(F\) are finite graphs, such that \(L_K(E)\) and \(L_K(F)\) are purely infinite simple, the \(K_0\)-groups are isomorphic and \(\det(I_n-A_E^t)=\det(I_m-A_F^t)\) where \(n\) and \(m\) are the number of vertices of \(E\) and \(F\) respectively, and \(A_E\) is the adjacency matrix of the graph \(E\), then \(L_K(E)\) is Morita equivalent to \(L_K(F)\).
Later in §2, by exploiting the Morita equivalence results in the previous section, sufficient conditions which ensure isomorphism are reached, thus obtaining the aforementioned partial affirmative answer to the Classification Question. In addition to this, the authors examine the remaining difficulty in getting a potentially affirmative answer for all germane graphs.
Finally, in the last section of the paper, several results about Morita equivalence and isomorphism to certain classes of graphs \(E\) for which \(L_K(E)\) is not necessarily purely infinite simple unital are extended, thereby giving more general results than have been previously known about isomorphism and Morita equivalence of Leavitt path algebras.
The paper contains many useful and concrete examples of the operations in graphs, which definitely helps the reader in grasping the material more quickly. The results on the other hand are interesting and relevant, with involved proofs containing several different ideas and techniques. Furthermore, a great deal of the current research in Leavitt path algebras in fact builds upon many of the results herein. Therefore, in the reviewer’s opinion, all this makes the paper under review a definitely must-read for any algebraist working in the field.
Reviewer: Gonzalo Aranda Pino (Malaga)
MSC:
| 16S88 | Leavitt path algebras |
| 16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
| 16G20 | Representations of quivers and partially ordered sets |
| 46L05 | General theory of \(C^*\)-algebras |
| 16E20 | Grothendieck groups, \(K\)-theory, etc. |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
Cite
References:
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