Leavitt path algebras of hypergraphs. (English) Zbl 1445.16029
The author defines Leavitt path algebras of hypergraphs. This class of algebras unifies Leavitt path algebras of separated graphs and Leavitt path algebras of vertex-weighted graphs (a weighted graph is vertex-weighted if any two edges emitted by the same vertex have the same weight). The author shows results on Leavitt path algebras of hypergraphs related to linear bases, Gelfand-Kirillov dimension, and some ring-theoretic properties such are being simple, Artinian, Noetherian, von Neumann regular, and being a domain. The author describes the monoid of the isomorphism classes of finitely generated projective modules and the monoid of the graded isomorphism classes of finitely generated graded projective modules over the Leavitt path algebra of a hypergraph.
Reviewer: Lia Vas (Philadelphia)
MSC:
| 16S88 | Leavitt path algebras |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
References:
| [1] | Abrams, G., Ara, P., Siles, Molina, M.: Leavitt path algebras, Lecture Notes in Mathematics 2191, Springer, Berlin (2017) · Zbl 1393.16001 |
| [2] | Abrams, G.; Pino, Ga, The Leavitt path algebra of a graph, J. Algebra, 293, 2, 319-334 (2005) · Zbl 1119.16011 · doi:10.1016/j.jalgebra.2005.07.028 |
| [3] | Ara, Pere; Hazrat, Roozbeh; Li, Huanhuan; Sims, Aidan, Graded Steinberg algebras and their representations, Algebra & Number Theory, 12, 1, 131-172 (2018) · Zbl 1387.22005 · doi:10.2140/ant.2018.12.131 |
| [4] | Ara, P.; Goodearl, Kr, Leavitt path algebras of separated graphs, J. Reine angew. Math., 669, 165-224 (2012) · Zbl 1281.46050 |
| [5] | Ara, P.; Moreno, Ma; Pardo, E., Nonstable \(K\)-theory for graph algebras, Algebr. Represent. Theory, 10, 2, 157-178 (2007) · Zbl 1123.16006 · doi:10.1007/s10468-006-9044-z |
| [6] | Bergman, Gm, Coproducts and some universal ring constructions, Trans. Am. Math. Soc., 200, 33-88 (1974) · Zbl 0264.16018 · doi:10.1090/S0002-9947-1974-0357503-7 |
| [7] | Bergman, Gm, The diamond lemma for ring theory, Adv. Math., 29, 2, 178-218 (1978) · Zbl 0326.16019 · doi:10.1016/0001-8708(78)90010-5 |
| [8] | Cohn, Pm, On the free product of associative rings, III. J. Algebra, 8, 376-383 (1968) · Zbl 0157.07802 · doi:10.1016/0021-8693(68)90066-5 |
| [9] | Cuntz, J., Simple \(C^*\)-algebras generated by isometries, Comm. Math. Phys., 57, 2, 173-185 (1977) · Zbl 0399.46045 · doi:10.1007/BF01625776 |
| [10] | Hazrat, R., The graded structure of Leavitt path algebras, Israel J. Math., 195, 2, 833-895 (2013) · Zbl 1308.16005 · doi:10.1007/s11856-012-0138-5 |
| [11] | Hazrat, R.; Preusser, R., Applications of normal forms for weighted Leavitt path algebras: simple rings and domains, Algebr. Represent. Theor., 20, 1061-1083 (2017) · Zbl 1376.16026 · doi:10.1007/s10468-017-9674-3 |
| [12] | Leavitt, Wg, Modules over rings of words, Proc. Am. Math. Soc., 7, 188-193 (1956) · Zbl 0073.02401 · doi:10.1090/S0002-9939-1956-0077520-9 |
| [13] | Leavitt, Wg, Modules without invariant basis number, Proc. Am. Math. Soc., 8, 322-328 (1957) · Zbl 0073.02402 · doi:10.1090/S0002-9939-1957-0083986-1 |
| [14] | Leavitt, Wg, The module type of a ring, Trans. Am. Math. Soc., 103, 113-130 (1962) · Zbl 0112.02701 · doi:10.1090/S0002-9947-1962-0132764-X |
| [15] | Leavitt, Wg, The module type of homomorphic images, Duke Math. J., 32, 305-311 (1965) · Zbl 0158.28801 · doi:10.1215/S0012-7094-65-03231-X |
| [16] | Moreno-Fernandez, Jm; Siles-Molina, M., Graph algebras and the Gelfand-Kirillov dimension, J. Algebra Appl., 17, 5, 1850095 (2018) · Zbl 1410.16027 · doi:10.1142/S0219498818500950 |
| [17] | Phillips, Nc, A classification theorem for nuclear purely infinite simple \(C^*\)-algebras, Doc. Math., 5, 49-114 (2000) · Zbl 0943.46037 |
| [18] | Preusser, R.: The V-monoid of a weighted Leavitt path algebra. Israel J. Math. (2018, accepted) · Zbl 1478.16017 |
| [19] | Preusser, R.: The Gelfand-Kirillov dimension of a weighted Leavitt path algebra. J. Algebra Appl. (2019, accepted) · Zbl 1478.16017 |
| [20] | Raeburn, I.: Graph algebras. In: CBMS Regional Conference Series in Mathematics, 103. American Mathematical Society, Providence, RI (2005) · Zbl 1079.46002 |
| [21] | Small, Lw; Warfield, Rb Jr, Prime affine algebras of Gelfand-Kirillov dimension one, J. Algebra, 91, 386-389 (1984) · Zbl 0545.16011 · doi:10.1016/0021-8693(84)90110-8 |
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