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2017 | OriginalPaper | Buchkapitel

1. The Basics of Leavitt Path Algebras: Motivations, Definitions and Examples

verfasst von : Gene Abrams, Pere Ara, Mercedes Siles Molina

Erschienen in: Leavitt Path Algebras

Verlag: Springer London

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Abstract

We introduce the central idea, that of a Leavitt path algebra. We start by describing the classical Leavitt algebras. We then proceed to give the definition of the Leavitt path algebra L K (E) for an arbitrary directed graph E and field K. After providing some basic examples, we show how Leavitt path algebras are related to the monoid realization algebras of Bergman, as well as to graph C ∗-algebras. We then introduce the more general construction of relative Cohn path algebras C K X (E), and show how these are related to Leavitt path algebras. We finish by describing how any Cohn (specifically, Leavitt) path algebra may be constructed as a direct limit of Cohn (specifically, Leavitt) path algebras corresponding to finite graphs. We conclude the chapter with an historical overview of the subject.

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Literatur
1.
2.
Gene Abrams and Pham N. Ánh. Some ultramatricial algebras which arise as intersections of Leavitt algebras. J. Algebra Appl., 1(4):357–363, 2002.MathSciNetMATHCrossRef
3.
Gene Abrams, Pham N. Ánh, Adel Louly, and Enrique Pardo. The classification question for Leavitt path algebras. J. Algebra, 320(5):1983–2026, 2008.MathSciNetMATHCrossRef
4.
Gene Abrams, Pham N. Ánh, and Enrique Pardo. Isomorphisms between Leavitt algebras and their matrix rings. J. Reine Angew. Math., 624:103–132, 2008.MathSciNetMATH
5.
Gene Abrams and Gonzalo Aranda Pino. The Leavitt path algebra of a graph. J. Algebra, 293(2):319–334, 2005.
6.
Gene Abrams and Gonzalo Aranda Pino. Purely infinite simple Leavitt path algebras. J. Pure Appl. Algebra, 207(3):553–563, 2006.
7.
Gene Abrams, Gonzalo Aranda Pino, Francesc Perera, and Mercedes Siles Molina. Chain conditions for Leavitt path algebras. Forum Math., 22(1):95–114, 2010.
8.
Gene Abrams, Gonzalo Aranda Pino, and Mercedes Siles Molina. Locally finite Leavitt path algebras. Israel J. Math., 165:329–348, 2008.
9.
Gene Abrams, Jason P. Bell, Pinar Colak, and Kulumani M. Rangaswamy. Two-sided chain conditions in Leavitt path algebras over arbitrary graphs. J. Algebra Appl., 11(3):1250044, 23, 2012.
10.
Gene Abrams, Jason P. Bell, and Kulumani M. Rangaswamy. On prime nonprimitive von Neumann regular algebras. Trans. Amer. Math. Soc., 366(5):2375–2392, 2014.MathSciNetMATHCrossRef
11.
Gene Abrams, Adel Louly, Enrique Pardo, and Christopher Smith. Flow invariants in the classification of Leavitt path algebras. J. Algebra, 333:202–231, 2011.MathSciNetMATHCrossRef
12.
Gene Abrams and Kulumani M. Rangaswamy. Regularity conditions for arbitrary Leavitt path algebras. Algebr. Represent. Theory, 13(3):319–334, 2010.MathSciNetMATHCrossRef
13.
Gene Abrams and Mark Tomforde. Isomorphism and Morita equivalence of graph algebras. Trans. Amer. Math. Soc., 363(7):3733–3767, 2011.MathSciNetMATHCrossRef
14.
Gene Abrams and Mark Tomforde. A class of C ∗-algebras that are prime but not primitive. Münster J. Math., 7(2):489–514, 2014.MathSciNetMATH
15.
16.
17.
Pere Ara. The exchange property for purely infinite simple rings. Proc. Amer. Math. Soc., 132(9):2543–2547 (electronic), 2004.
18.
Pere Ara. Rings without identity which are Morita equivalent to regular rings. Algebra Colloq., 11(4):533–540, 2004.MathSciNetMATH
19.
Pere Ara. The realization problem for von Neumann regular rings. In Ring theory 2007, pages 21–37. World Sci. Publ., Hackensack, NJ, 2009.
20.
Pere Ara and Miquel Brustenga. \(K_{1}\) of corner skew Laurent polynomial rings and applications. Comm. Algebra, 33(7):2231–2252, 2005.
21.
22.
Pere Ara and Miquel Brustenga. Module theory over Leavitt path algebras and K-theory. J. Pure Appl. Algebra, 214(7):1131–1151, 2010.MathSciNetMATHCrossRef
23.
Pere Ara, Miquel Brustenga, and Guillermo Cortiñas. K-theory of Leavitt path algebras. Münster J. Math., 2:5–33, 2009.MathSciNetMATH
24.
Pere Ara and Alberto Facchini. Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids. Forum Math., 18(3):365–389, 2006.MathSciNetMATHCrossRef
25.
Pere Ara, María A. González-Barroso, Kenneth R. Goodearl, and Enrique Pardo. Fractional skew monoid rings. J. Algebra, 278(1):104–126, 2004.MathSciNetMATHCrossRef
26.
Pere Ara and Kenneth R. Goodearl. Stable rank of corner rings. Proc. Amer. Math. Soc., 133(2):379–386 (electronic), 2005.
27.
Pere Ara and Kenneth R. Goodearl. Leavitt path algebras of separated graphs. J. Reine Angew. Math., 669:165–224, 2012.MathSciNetMATH
28.
Pere Ara, Kenneth R. Goodearl, Kevin C. O’Meara, and Enrique Pardo. Separative cancellation for projective modules over exchange rings. Israel J. Math., 105:105–137, 1998.MathSciNetMATHCrossRef
29.
Pere Ara, Kenneth R. Goodearl, and Enrique Pardo. \(K_{0}\) of purely infinite simple regular rings. K-Theory, 26(1):69–100, 2002.
30.
Pere Ara and Martin Mathieu. Local multipliers of C ∗ -algebras. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2003.MATHCrossRef
31.
Pere Ara, María A. Moreno, and Enrique Pardo. Nonstable K-theory for graph algebras. Algebr. Represent. Theory, 10(2):157–178, 2007.MathSciNetMATHCrossRef
32.
33.
Pere Ara and Enrique Pardo. Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras. J. K-Theory, 14(2):203–245, 2014.MathSciNetMATHCrossRef
34.
Pere Ara and Enrique Pardo. Representing finitely generated refinement monoids as graph monoids. J. Algebra, 480:79–123, 2017.MathSciNetMATHCrossRef
35.
Pere Ara, Gert K. Pedersen, and Francesc Perera. An infinite analogue of rings with stable rank one. J. Algebra, 230(2):608–655, 2000.MathSciNetMATHCrossRef
36.
37.
Gonzalo Aranda Pino, John Clark, Astrid an Huef, and Iain Raeburn. Kumjian-Pask algebras of higher-rank graphs. Trans. Amer. Math. Soc., 365(7):3613–3641, 2013.
38.
Gonzalo Aranda Pino and Kathi Crow. The center of a Leavitt path algebra. Rev. Mat. Iberoam., 27(2):621–644, 2011.
39.
Gonzalo Aranda Pino, Kenneth R. Goodearl, Francesc Perera, and Mercedes Siles Molina. Non-simple purely infinite rings. Amer. J. Math., 132(3):563–610, 2010.
40.
Gonzalo Aranda Pino, Enrique Pardo, and Mercedes Siles Molina. Exchange Leavitt path algebras and stable rank. J. Algebra, 305(2):912–936, 2006.
41.
Gonzalo Aranda Pino, Enrique Pardo, and Mercedes Siles Molina. Prime spectrum and primitive Leavitt path algebras. Indiana Univ. Math. J., 58(2):869–890, 2009.
42.
Gonzalo Aranda Pino, Kulumani Rangaswamy, and Lia Vaš. ∗-regular Leavitt path algebras of arbitrary graphs. Acta Math. Sin. (Engl. Ser.), 28(5):957–968, 2012.
43.
44.
Hyman Bass, Alex Heller, and Richard G. Swan. The Whitehead group of a polynomial extension. Inst. Hautes Études Sci. Publ. Math., 22:61–79, 1964.MathSciNetMATHCrossRef
45.
Teresa Bates, Jeong Hee Hong, Iain Raeburn, and Wojciech Szymański. The ideal structure of the \(C^{{\ast}}\)-algebras of infinite graphs. Illinois J. Math., 46(4):1159–1176, 2002.
46.
47.
Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański. The \(C^{{\ast}}\)-algebras of row-finite graphs. New York J. Math., 6:307–324 (electronic), 2000.
48.
Jason P. Bell and George M. Bergman. Personal communication, 2011.
49.
Sterling K. Berberian. Baer *-rings. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 195.
50.
51.
52.
Bruce Blackadar. K-theory for operator algebras, volume 5 of Mathematical Sciences Research Institute Publications. Cambridge University Press, Cambridge, second edition, 1998.
53.
54.
Jonathan Brown, Lisa Orloff Clark, Cynthia Farthing, and Aidan Sims. Simplicity of algebras associated to étale groupoids. Semigroup Forum, 88(2):433–452, 2014.
55.
Lawrence G. Brown. Homotopy of projection in C ∗-algebras of stable rank one. Astérisque, 232:115–120, 1995. Recent advances in operator algebras (Orléans, 1992).
56.
57.
Lawrence G. Brown and Gert K. Pedersen. Non-stable K-theory and extremally rich C ∗-algebras. J. Funct. Anal., 267(1):262–298, 2014.MathSciNetMATHCrossRef
58.
Nathan Brownlowe and Adam P. W. Sørensen. \(L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}\) does not embed in \(L_{2,\mathbb{Z}}\). J. Algebra, 456:1–22, 2016.
59.
60.
Lisa Orloff Clark and Cain Edie-Michell. Uniqueness theorems for Steinberg algebras. Algebr. Represent. Theory, 18(4):907–916, 2015.
61.
Lisa Orloff Clark, Cynthia Farthing, Aidan Sims, and Mark Tomforde. A groupoid generalisation of Leavitt path algebras. Semigroup Forum, 89(3):501–517, 2014.
62.
Lisa Orloff Clark, Dolores Martin Barquero, Cándido Martin Gonzalez, and Mercedes Siles Molina. Using the Steinberg algebra model to determine the center of any Leavitt path algebra. To appear in Israel Journal of Mathematics. arXiv, 1604.01079:1–14, 2016.
63.
Lisa Orloff Clark and Aidan Sims. Equivalent groupoids have Morita equivalent Steinberg algebras. J. Pure Appl. Algebra, 219(6):2062–2075, 2015.
64.
65.
María G. Corrales García, Dolores Martín Barquero, Cándido Martín González, Mercedes Siles Molina, and José F. Solanilla Hernández. Extreme cycles. The center of a Leavitt path algebra. Publ. Mat., 60(1):235–263, 2016.
66.
Guillermo Cortiñas and Eugenia Ellis. Isomorphism conjectures with proper coefficients. J. Pure Appl. Algebra, 218(7):1224–1263, 2014.MathSciNetMATHCrossRef
67.
Peter Crawley and Bjarni Jónsson. Refinements for infinite direct decompositions of algebraic systems. Pacific J. Math., 14:797–855, 1964.MathSciNetMATHCrossRef
68.
69.
Joachim Cuntz. The structure of multiplication and addition in simple \(C^{{\ast}}\)-algebras. Math. Scand., 40(2):215–233, 1977.MathSciNetMATHCrossRef
70.
71.
Joachim Cuntz and Wolfgang Krieger. A class of \(C^{{\ast}}\)-algebras and topological Markov chains. Invent. Math., 56(3):251–268, 1980.MathSciNetMATHCrossRef
72.
Klaus Deicke, Jeong Hee Hong, and Wojciech Szymański. Stable rank of graph algebras. Type I graph algebras and their limits. Indiana Univ. Math. J., 52(4):963–979, 2003.
73.
74.
Jacques Dixmier. LesC ∗ -algèbres et leurs représentations. Deuxième édition. Cahiers Scientifiques, Fasc. XXIX. Gauthier-Villars Éditeur, Paris, 1969.
75.
O.I. Domanov. A prime but not primitive regular ring. Uspehi Mat. Nauk, 32(6(198)):219–220, 1977.
76.
Søren Eilers, Gunnar Restorff, Efren Ruiz, and Adam P.W. Sørensen. The complete classification of unital graph \(C^{{\ast}}\)-algebras: geometric and strong. arXiv, 1611.07120v1:1–73, 2016.
77.
Ruy Exel. Partial Dynamical Systems, Fell Bundles and Applications, volume 224 of Mathematical Surveys and Monographs Series. American Mathematical Society, Providence, R.I., 2017.
78.
Alberto Facchini and Franz Halter-Koch. Projective modules and divisor homomorphisms. J. Algebra Appl., 2(4):435–449, 2003.MathSciNetMATHCrossRef
79.
Carl Faith. Lectures on injective modules and quotient rings. Lecture Notes in Mathematics, No. 49. Springer-Verlag, Berlin-New York, 1967.
80.
Antonio Fernández López, Eulalia García Rus, Miguel Gómez Lozano, and Mercedes Siles Molina. Goldie theorems for associative pairs. Comm. Algebra, 26(9):2987–3020, 1998.
81.
82.
James Gabe, Efren Ruiz, Mark Tomforde, and Tristan Whalen. K-theory for Leavitt path algebras: Computation and classification. J. Algebra, 433:35–72, 2015.MathSciNetMATHCrossRef
83.
84.
Miguel Gómez Lozano and Mercedes Siles Molina. Quotient rings and Fountain-Gould left orders by the local approach. Acta Math. Hungar., 97(4):287–301, 2002.
85.
Kenneth R. Goodearl. Notes on real and complexC ∗ -algebras, volume 5 of Shiva Mathematics Series. Shiva Publishing Ltd., Nantwich, 1982.
86.
Kenneth R. Goodearl. von Neumann regular rings. Robert E. Krieger Publishing Co. Inc., Malabar, FL, second edition, 1991.
87.
Kenneth R. Goodearl. Leavitt path algebras and direct limits. In Rings, Modules and Representations, volume 480 of Contemp. Math. American Mathematical Society, Providence, R.I., 2009.
88.
Kenneth R. Goodearl and Robert B. Warfield, Jr. Algebras over zero-dimensional rings. Math. Ann., 223(2):157–168, 1976.
89.
Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones. Coxeter graphs and towers of algebras, volume 14 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1989.
90.
91.
Damon Hay, Marissa Loving, Martin Montgomery, Efren Ruiz, and Katherine Todd. Non-stable K-theory for Leavitt path algebras. Rocky Mountain J. Math., 44(6):1817–1850, 2014.MathSciNetMATHCrossRef
92.
Roozbeh Hazrat. The graded Grothendieck group and the classification of Leavitt path algebras. Math. Ann., 355(1):273–325, 2013.MathSciNetMATHCrossRef
93.
94.
Roozbeh Hazrat. Graded rings and graded Grothendieck groups, volume 435 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2016.CrossRef
95.
96.
97.
Danrun Huang. Automorphisms of Bowen–Franks groups of shifts of finite type. Ergodic Theory Dynam. Systems, 21(4):1113–1137, 2001.MathSciNetMATHCrossRef
98.
99.
Nathan Jacobson. Structure of rings. American Mathematical Society, Colloquium Publications, vol. 37. American Mathematical Society, 190 Hope Street, Prov., R. I., 1956.
100.
Ja A. Jeong and Gi Hyun Park. Graph C ∗-algebras with real rank zero. J. Funct. Anal., 188(1):216–226, 2002.
101.
Richard V. Kadison and John R. Ringrose. Fundamentals of the theory of operator algebras. Vol. I, volume 15 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Elementary theory, Reprint of the 1983 original.
102.
Irving Kaplansky. Algebraic and analytic aspects of operator algebras. American Mathematical Society, Providence, R.I., 1970. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 1.
103.
Eberhard Kirchberg. The classification of purely infinite C ∗-algebras using Kasparov theory. Unpublished, 3rd draft, pages 1–37, 1994.
104.
Alex Kumjian and David Pask. Higher rank graph C ∗-algebras. New York J. Math., 6:1–20, 2000.MathSciNetMATH
105.
Alex Kumjian, David Pask, and Iain Raeburn. Cuntz–Krieger algebras of directed graphs. Pacific J. Math., 184(1):161–174, 1998.MathSciNetMATHCrossRef
106.
Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault. Graphs, groupoids, and Cuntz–Krieger algebras. J. Funct. Anal., 144(2):505–541, 1997.MathSciNetMATHCrossRef
107.
Tsit-Yen Lam. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001.
108.
Tsit-Yuen Lam. Lectures on modules and rings, volume 189 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1999.CrossRef
109.
Joachim Lambek. Lectures on rings and modules. Chelsea Publishing Co., New York, second edition, 1976.
110.
111.
Hossein Larki and Abdolhamid Riazi. Stable rank of Leavitt path algebras of arbitrary graphs. Bull. Aust. Math. Soc., 88(2):206–217, 2013.MathSciNetMATHCrossRef
112.
113.
114.
Douglas Lind and Brian Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995.MATHCrossRef
115.
116.
John C. McConnell and J. Chris Robson. Noncommutative Noetherian rings, volume 30 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, revised edition, 2001. With the cooperation of L.W. Small.
117.
Pere Menal and Jaume Moncasi. Lifting units in self-injective rings and an index theory for Rickart C ∗-algebras. Pacific J. Math., 126(2):295–329, 1987.MathSciNetMATHCrossRef
118.
Gerard J. Murphy. C ∗ -algebras and operator theory. Academic Press, Inc., Boston, MA, 1990.MATH
119.
Constantin Năstăsescu and Freddy van Oystaeyen. Graded ring theory, volume 28 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1982.
120.
Constantin Năstăsescu and Freddy Van Oystaeyen. Methods of graded rings, volume 1836 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2004.
121.
122.
123.
124.
Andrei V. Pajitnov and Andrew A. Ranicki. The Whitehead group of the Novikov ring. K-Theory, 21(4):325–365, 2000. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part V.
125.
126.
127.
128.
N. Christopher Phillips. A classification theorem for nuclear purely infinite simple C ∗-algebras. Doc. Math., 5:49–114 (electronic), 2000.
129.
Iain Raeburn. Graph algebras, volume 103 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005.
130.
Iain Raeburn and Wojciech Szymański. Cuntz–Krieger algebras of infinite graphs and matrices. Trans. Amer. Math. Soc., 356(1):39–59 (electronic), 2004.
131.
Kulumani M. Rangaswamy. The theory of prime ideals of Leavitt path algebras over arbitrary graphs. J. Algebra, 375:73–96, 2013.MathSciNetMATHCrossRef
132.
Kulumani M. Rangaswamy. On generators of two-sided ideals of Leavitt path algebras over arbitrary graphs. Comm. Algebra, 42(7):2859–2868, 2014.MathSciNetMATHCrossRef
133.
Marc A. Rieffel. Dimension and stable rank in the K-theory of \(C^{{\ast}}\)-algebras. Proc. London Math. Soc. (3), 46(2):301–333, 1983.MathSciNetMATHCrossRef
134.
Mikael Rørdam. A short proof of Elliott’s theorem: \(\mathscr{O}_{2} \otimes \mathscr{O}_{2}\cong \mathscr{O}_{2}\). C. R. Math. Rep. Acad. Sci. Canada, 16(1):31–36, 1994.MathSciNet
135.
136.
Mikael Rørdam, Flemming Larsen, and Niels Laustsen. An introduction toK-theory forC ∗ -algebras, volume 49 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2000.
137.
Joseph J. Rotman. An introduction to homological algebra. Universitext. Springer, New York, second edition, 2009.
138.
Efren Ruiz and Mark Tomforde. Classification of unital simple Leavitt path algebras of infinite graphs. J. Algebra, 384:45–83, 2013.MathSciNetMATHCrossRef
139.
Konrad Schmüdgen. Unbounded operator algebras and representation theory, volume 37 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1990.CrossRef
140.
Konrad Schmüdgen. Noncommutative real algebraic geometry—some basic concepts and first ideas. In Emerging applications of algebraic geometry, volume 149 of IMA Vol. Math. Appl., pages 325–350. Springer, New York, 2009.
141.
Laurence C. Siebenmann. A total Whitehead torsion obstruction to fibering over the circle. Comment. Math. Helv., 45:1–48, 1970.MathSciNetMATHCrossRef
142.
S. Paul Smith. Category equivalences involving graded modules over path algebras of quivers. Adv. Math., 230(4–6):1780–1810, 2012.MathSciNetMATHCrossRef
143.
S. Paul Smith. The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring \(k\langle x,y\rangle /(y^{2})\). J. Noncommut. Geom., 8(2):541–586, 2014.
144.
145.
Benjamin Steinberg. Simplicity, primitivity and semiprimitivity of étale groupoid algebras with applications to inverse semigroup algebras. J. Pure Appl. Algebra, 220(3):1035–1054, 2016.MathSciNetMATHCrossRef
146.
Mark Tomforde. Extensions of graph C*-algebras. ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Dartmouth College.
147.
148.
149.
150.
Leonid N. Vaseršteĭn. The stable range of rings and the dimension of topological spaces. Funkcional. Anal. i Priložen., 5(2):17–27, 1971.MathSciNet
151.
Leonid N. Vaseršteĭn. Bass’s first stable range condition. In Proceedings of the Luminy conference on algebraicK-theory (Luminy, 1983), J. Pure Appl. Algebra, volume 34, pages 319–330, 1984.
152.
153.
R.B. Warfield, Jr. Cancellation of modules and groups and stable range of endomorphism rings. Pacific J. Math., 91(2):457–485, 1980.MathSciNetMATHCrossRef
154.
155.
Yasuo Watatani. Graph theory for \(C^{{\ast}}\)-algebras. In Operator algebras and applications, Part I (Kingston, Ont., 1980), volume 38 of Proc. Sympos. Pure Math., pages 195–197. Amer. Math. Soc., Providence, R.I., 1982.
156.
Friedrich Wehrung. Various remarks on separativity and stable rank in refinement monoids. Unpublished manuscript.
157.
158.
Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.CrossRef
159.
Charles A. Weibel. TheK-book, volume 145 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2013. An introduction to algebraic K-theory.
160.
Robert F. Williams. Classification of subshifts of finite type. Ann. of Math. (2), 98:120–153; errata, ibid. (2) 99 (1974), 380–381, 1973.
161.
Metadaten
Titel
The Basics of Leavitt Path Algebras: Motivations, Definitions and Examples
verfasst von
Gene Abrams
Pere Ara
Mercedes Siles Molina
Copyright-Jahr
2017
Verlag
Springer London
Buch
Leavitt Path Algebras

Print ISBN: 978-1-4471-7343-4

Electronic ISBN: 978-1-4471-7344-1

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-1-4471-7344-1_1