Maximal commutative subalgebras of Leavitt path algebras. (English) Zbl 07805953
Summary: Let \(K\) be a field, and let \(E\) be a row-finite (directed) graph. We present a construction of a wealth of maximal commutative subalgebras of the Leavitt path algebra \(L_K(E)\), which is a far-reaching generalization of the construction of the commutative core as a maximal commutative subalgebra of \(L_K(E)\).
MSC:
| 16W99 | Associative rings and algebras with additional structure |
| 16U80 | Generalizations of commutativity (associative rings and algebras) |
References:
| [1] | Abrams, G., Leavitt path algebras: The first decade, Bull. Math. Sci.5(1) (2015) 59-120. · Zbl 1329.16002 |
| [2] | Abrams, G., Ara, P. and Siles Molina, M., Finite-dimensional Leavitt path algebras, J. Pure Appl. Algebra209(3) (2007) 753-762. · Zbl 1128.16008 |
| [3] | Abrams, G., Ara, P. and Siles Molina, M., Locally finite Leavitt path algebras, Israel J. Math.165 (2008) 329-348. · Zbl 1169.16008 |
| [4] | G. Abrams, P. Ara and M. Siles Molina, Leavitt path algebras, Lecture Notes in Mathematics, Vol. 2191 (Springer Verlag, London, 2017). · Zbl 1393.16001 |
| [5] | Abrams, G. and Aranda Pino, G., The Leavitt path algebra of a graph, J. Algebra293(2) (2005) 319-334. · Zbl 1119.16011 |
| [6] | Abrams, G. and Aranda Pino, G., Purely infinite simple Leavitt path algebras, J. Pure Appl. Algebra207(3) (2006) 553-563. · Zbl 1137.16028 |
| [7] | Abrams, G., Bell, J. P. and Rangaswamy, K. M., On prime nonprimitive von Neumann regular algebras, Trans. Amer. Math. Soc.366(5) (2014) 2375-2392. · Zbl 1336.16002 |
| [8] | Abrams, G. and Tomforde, M., Isomorphism and Morita equivalence of graph algebras, Trans. Amer. Math. Soc.363(7) (2011) 3733-3767. · Zbl 1230.16012 |
| [9] | Amitsur, S. A. and Levitzki, J., Minimal identities for algebras, Trans. Amer. Math. Soc.1 (1950) 449-463. · Zbl 0040.01101 |
| [10] | Ara, P., Moreno, M. A. and Pardo, E., Nonstable \(K\)-theory for graph algebras, Algebr. Represent. Theore10(2) (2007) 157-178. · Zbl 1123.16006 |
| [11] | Aranda Pino, G., Clark, J., an Huef, A. and Raeburn, I., Kumjian-Pask algebras of higher-rank graphs, Trans. Amer. Math. Soc.365(7) (2013) 3613-3641. · Zbl 1282.16006 |
| [12] | Aranda Pino, G., Pardo, E. and Siles Molina, M., Exchange Leavitt path algebras and stable rank, J. Algebra305(2) (2006) 912-936. · Zbl 1108.46038 |
| [13] | Aranda Pino, G., Pardo, E. and Siles Molina, M., Prime spectrum and primitive Leavitt path algebras, Indiana Univ. Math. J.58(2) (2009) 869-890. · Zbl 1189.16014 |
| [14] | Bajor, G., van Wyk, L. and Ziembowski, M., Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras, Forum Math.33(6) (2021) 1573-1590. · Zbl 1523.16034 |
| [15] | Bavula, V. V., Structure of maximal commutative subalgebras of the first Weyl algebra, Ann. Univ. Ferrara Sez. VII (N.S.)51 (2005) 1-14. · Zbl 1120.16022 |
| [16] | Brown, J. H. and an Huef, A., Centers of algebras associated to higher-rank graphs, Rev. Mat. Iberoam.30(4) (2014) 1387-1396. · Zbl 1315.16009 |
| [17] | Clark, L. O., Gil Canto, C. and Nasr-Isfahani, A., The cycline subalgebra of a Kumjian-Pask algebra, Proc. Amer. Math. Soc.145(5) (2017) 1969-1980. · Zbl 1359.16020 |
| [18] | Clark, L. O., Martín Barquero, D., Martín Gonzalez, C. and Siles Molina, M., Using the Steinberg algebra model to determine the center of any Leavitt path algebra, Israel J. Math.230(1) (2019) 23-44. · Zbl 1469.16063 |
| [19] | Corrales García, M. G., Martín Barquero, D., Martín Gonzalez, C., Siles Molina, M. and Solanilla Hernández, J. F., Extreme cycles. The center of a Leavitt path algebra, Publ. Mat.60(1) (2016) 235-263. · Zbl 1350.16005 |
| [20] | Deicke, K., Hong, J. H. and Szymański, W., Stable rank of graph algebras. Type I graph algebras and their limits, Indiana Univ. Math. J.52 (2003) 963-979. · Zbl 1080.46038 |
| [21] | Domokos, M., On the dimension of faithful modules over finite dimensional basic algebras, Linear Algebra Appl.365 (2003) 155-157. · Zbl 1022.16013 |
| [22] | Domokos, M. and Zubor, M., Commutative subalgebras of the Grassmann algebra, J. Algebra Appl.14 (2015) 1550125. · Zbl 1318.15011 |
| [23] | Drensky, V., Free Algebras and PI-Algebras, (Springer-Verlag, Singapore, 2000). · Zbl 0936.16001 |
| [24] | Drensky, V. and Formanek, E., Polynomial Identity Rings, (Birkhäuser-Verlag, Basel, 2004). · Zbl 1077.16025 |
| [25] | Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Mat. Sb.30(2) (1952) 349-462. (in Russian) · Zbl 0048.01701 |
| [26] | Elduque, A., On maximal subalgebras of central simple Malcev algebras, J. Algebra103(1) (1986) 216-227. · Zbl 0592.17014 |
| [27] | Elduque, A., Laliena, S. and Sacristán, S., Maximal subalgebras of associative superalgebras, J. Algebra275(1) (2004) 40-58. · Zbl 1087.16029 |
| [28] | Gil Canto, C. and Nasr-Isfahani, A., The commutative core of a Leavitt path algebra, J. Algebra511 (2018) 227-248. · Zbl 1444.16040 |
| [29] | Gustafson, W. H., On maximal commutative algebras of linear transformations, J. Algebra42(2) (1976) 557-563. · Zbl 0335.13009 |
| [30] | Hazrat, R. and Li, H., A note on the centralizer of a subalgebra of the Steinberg algebra, St. Petersburg Math. J.33 (2022) 179-184. · Zbl 1487.18002 |
| [31] | Jacobson, N., Schur’s theorems on commutative matrices, Bull. Amer. Math. Soc.50 (1944) 431-436. · Zbl 0063.03016 |
| [32] | Kajiwara, T. and Watatani, Y., Maximal abelian subalgebras of \(C^\ast \)-algebras associated with complex dynamical systems and self-similar maps, J. Math. Anal. Appl.455(2) (2017) 1383-1400. · Zbl 1373.37025 |
| [33] | Karamzadeh, N. S., Maximal abelian subalgebras of \(C^\ast \)-algebras associated with complex dynamical systems and self-similar maps, Math. Inequal. Appl.13(3) (2010) 625-628. · Zbl 1209.15004 |
| [34] | Krakowski, D. and Regev, A., The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc.181 (1973) 429-438. · Zbl 0289.16015 |
| [35] | Leavitt, W. G., The module type of a ring, Trans. Amer. Math. Soc.103 (1962) 113-130. · Zbl 0112.02701 |
| [36] | Malcev, A., Commutative subalgebras of semi-simple Lie algebras, Amer. Math. Soc. Trans.1951(40) (1951) 15. |
| [37] | Mirzakhani, M., Commutative subalgebras of semi-simple Lie algebras, Amer. Math. Mon.105(3) (1998) 260-262. · Zbl 0916.15004 |
| [38] | Nagy, G. and Reznikoff, S., Abelian core of graph algebras, J. London Math. Soc.85(3) (2012) 889-908. · Zbl 1251.46029 |
| [39] | Öinert, J., Richter, J. and Silvestrov, S. D., Maximal commutative subrings and simplicity of Ore extensions, J. Algebra Appl.12(4) (2013) 1250192. · Zbl 1277.16016 |
| [40] | Racine, M. L., On maximal subalgebras, J. Algebra30 (1974) 155-180. · Zbl 0282.17009 |
| [41] | Racine, M. L., Maximal subalgebras of central separable algebras, Proc. Amer. Math. Soc.68(1) (1978) 11-15. · Zbl 0381.16003 |
| [42] | Raeburn, I., Graph Algebras, , Vol. 103. (American Mathematical Society, Providence, RI, 2005). · Zbl 1079.46002 |
| [43] | Schur, J., Zur Theorie der vertauschbaren Matrizen, J. Reine Angew. Math.130 (1905) 66-76. (in German) · JFM 36.0140.01 |
| [44] | Szigeti, J., van den Berg, J., van Wyk, L. and Ziembowski, M., The maximum dimension of a Lie nilpotent subalgebra of \(\mathbb{M}_n(F)\) of index \(m\), Trans. Amer. Math. Soc.372(7) (2019) 4553-4583. · Zbl 1456.16027 |
| [45] | Tomforde, M., Leavitt path algebras with coefficients in a commutative ring, J. Pure Appl. Algebra215(4) (2011) 471-484. · Zbl 1213.16010 |
| [46] | van Wyk, L. and Ziembowski, M., Lie solvability and the identity \([ x_1, y_1][ x_2, y_2]\cdots[ x_q, y_q]=0\) in certain matrix algebras, Linear Algebra Appl.533 (2017) 235-257. · Zbl 1370.16022 |
| [47] | Z. Wan and G. Li, The two theorems of Schur on commutative matrices, Acta Math. Sin.14 (1964) 143-150 (in Chinese); translated as Chinese Math.5 (1964) 156-164. · Zbl 0166.03602 |
| [48] | Wang, S. and Liu, W., The abelian subalgebras of maximal dimensions for general linear Lie superalgebras, Linear Multilinear Algebra533(10) (2016) 2081-2089. · Zbl 1378.17019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
