The Leavitt path algebra of a graph
For any row-finite graph E and any field K we construct the Leavitt path algebra L ( E ) having coefficients in K. When K is the field of complex numbers, then L ( E ) is the algebraic analog of the Cuntz–Krieger algebra C ∗ ( E ) described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Ma...
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| Published in | Journal of algebra Vol. 293; no. 2; pp. 319 - 334 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
15.11.2005
|
| Subjects | |
| Online Access | Get full text |
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| Abstract | For any row-finite graph
E and any field
K we construct the
Leavitt path algebra
L
(
E
)
having coefficients in
K. When
K is the field of complex numbers, then
L
(
E
)
is the algebraic analog of the Cuntz–Krieger algebra
C
∗
(
E
)
described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings
M
n
(
K
)
and the Leavitt algebras
L
(
1
,
n
)
appear as algebras of the form
L
(
E
)
for various graphs
E. In our main result, we give necessary and sufficient conditions on
E which imply that
L
(
E
)
is simple. |
|---|---|
| AbstractList | For any row-finite graph
E and any field
K we construct the
Leavitt path algebra
L
(
E
)
having coefficients in
K. When
K is the field of complex numbers, then
L
(
E
)
is the algebraic analog of the Cuntz–Krieger algebra
C
∗
(
E
)
described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings
M
n
(
K
)
and the Leavitt algebras
L
(
1
,
n
)
appear as algebras of the form
L
(
E
)
for various graphs
E. In our main result, we give necessary and sufficient conditions on
E which imply that
L
(
E
)
is simple. |
| Author | Aranda Pino, Gonzalo Abrams, Gene |
| Author_xml | – sequence: 1 givenname: Gene surname: Abrams fullname: Abrams, Gene email: abrams@math.uccs.edu organization: Department of Mathematics, University of Colorado, Colorado Springs, CO 80933, USA – sequence: 2 givenname: Gonzalo surname: Aranda Pino fullname: Aranda Pino, Gonzalo email: gonzalo@agt.cie.uma.es organization: Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain |
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| Cites_doi | 10.1016/j.jalgebra.2004.03.009 10.1215/S0012-7094-65-03231-X 10.2140/pjm.1998.184.161 10.1007/BF01389192 10.1090/S0002-9947-1962-0132764-X 10.1090/S0002-9947-03-03341-5 10.1007/BF01625776 |
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| Keywords | Cuntz–Krieger C ∗ -algebra Path algebra Leavitt algebra |
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| References | Cuntz (bib003) 1977; 57 Ara, González-Barroso, Goodearl, Pardo (bib001) 2004; 278 Kumjian, Pask, Raeburn (bib005) 1998; 184 Leavitt (bib006) 1962; 42 Bates, Pask, Raeburn, Szymański (bib002) 2000; 6 Leavitt (bib007) 1965; 32 Cuntz, Krieger (bib004) 1981; 63 Raeburn (bib008) 2005; vol. 103 Raeburn, Szymański (bib009) 2004; 356 Raeburn (10.1016/j.jalgebra.2005.07.028_bib009) 2004; 356 Raeburn (10.1016/j.jalgebra.2005.07.028_bib008) 2005; vol. 103 Bates (10.1016/j.jalgebra.2005.07.028_bib002) 2000; 6 Leavitt (10.1016/j.jalgebra.2005.07.028_bib007) 1965; 32 Cuntz (10.1016/j.jalgebra.2005.07.028_bib003) 1977; 57 Kumjian (10.1016/j.jalgebra.2005.07.028_bib005) 1998; 184 Ara (10.1016/j.jalgebra.2005.07.028_bib001) 2004; 278 Cuntz (10.1016/j.jalgebra.2005.07.028_bib004) 1981; 63 Leavitt (10.1016/j.jalgebra.2005.07.028_bib006) 1962; 42 |
| References_xml | – volume: 57 start-page: 173 year: 1977 end-page: 185 ident: bib003 article-title: Simple publication-title: Comm. Math. Phys. contributor: fullname: Cuntz – volume: 184 start-page: 161 year: 1998 end-page: 174 ident: bib005 article-title: Cuntz–Krieger algebras of directed graphs publication-title: Pacific J. Math. contributor: fullname: Raeburn – volume: 42 start-page: 113 year: 1962 end-page: 130 ident: bib006 article-title: The module type of a ring publication-title: Trans. Amer. Math. Soc. contributor: fullname: Leavitt – volume: 6 start-page: 307 year: 2000 end-page: 324 ident: bib002 article-title: The publication-title: New York J. Math. contributor: fullname: Szymański – volume: 32 start-page: 305 year: 1965 end-page: 311 ident: bib007 article-title: The module type of homomorphic images publication-title: Duke Math. J. contributor: fullname: Leavitt – volume: 278 start-page: 104 year: 2004 end-page: 126 ident: bib001 article-title: Fractional skew monoid rings publication-title: J. Algebra contributor: fullname: Pardo – volume: 356 start-page: 39 year: 2004 end-page: 59 ident: bib009 article-title: Cuntz–Krieger algebras of infinite graphs and matrices publication-title: Trans. Amer. Math. Soc. contributor: fullname: Szymański – volume: vol. 103 year: 2005 ident: bib008 article-title: Graph algebras publication-title: CBMS Reg. Conf. Ser. Math. contributor: fullname: Raeburn – volume: 63 start-page: 25 year: 1981 end-page: 40 ident: bib004 article-title: A class of publication-title: Invent. Math. contributor: fullname: Krieger – volume: 278 start-page: 104 year: 2004 ident: 10.1016/j.jalgebra.2005.07.028_bib001 article-title: Fractional skew monoid rings publication-title: J. Algebra doi: 10.1016/j.jalgebra.2004.03.009 contributor: fullname: Ara – volume: 32 start-page: 305 year: 1965 ident: 10.1016/j.jalgebra.2005.07.028_bib007 article-title: The module type of homomorphic images publication-title: Duke Math. J. doi: 10.1215/S0012-7094-65-03231-X contributor: fullname: Leavitt – volume: 6 start-page: 307 year: 2000 ident: 10.1016/j.jalgebra.2005.07.028_bib002 article-title: The C∗-algebras of row-finite graphs publication-title: New York J. Math. contributor: fullname: Bates – volume: 184 start-page: 161 issue: 1 year: 1998 ident: 10.1016/j.jalgebra.2005.07.028_bib005 article-title: Cuntz–Krieger algebras of directed graphs publication-title: Pacific J. Math. doi: 10.2140/pjm.1998.184.161 contributor: fullname: Kumjian – volume: vol. 103 year: 2005 ident: 10.1016/j.jalgebra.2005.07.028_bib008 article-title: Graph algebras contributor: fullname: Raeburn – volume: 63 start-page: 25 year: 1981 ident: 10.1016/j.jalgebra.2005.07.028_bib004 article-title: A class of C∗-algebras and topological Markov chains publication-title: Invent. Math. doi: 10.1007/BF01389192 contributor: fullname: Cuntz – volume: 42 start-page: 113 year: 1962 ident: 10.1016/j.jalgebra.2005.07.028_bib006 article-title: The module type of a ring publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1962-0132764-X contributor: fullname: Leavitt – volume: 356 start-page: 39 issue: 1 year: 2004 ident: 10.1016/j.jalgebra.2005.07.028_bib009 article-title: Cuntz–Krieger algebras of infinite graphs and matrices publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-03-03341-5 contributor: fullname: Raeburn – volume: 57 start-page: 173 year: 1977 ident: 10.1016/j.jalgebra.2005.07.028_bib003 article-title: Simple C∗-algebras generated by isometries publication-title: Comm. Math. Phys. doi: 10.1007/BF01625776 contributor: fullname: Cuntz |
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| Score | 2.332306 |
| Snippet | For any row-finite graph
E and any field
K we construct the
Leavitt path algebra
L
(
E
)
having coefficients in
K. When
K is the field of complex numbers, then... |
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| StartPage | 319 |
| SubjectTerms | Cuntz–Krieger [formula omitted]-algebra Leavitt algebra Path algebra |
| Title | The Leavitt path algebra of a graph |
| URI | https://dx.doi.org/10.1016/j.jalgebra.2005.07.028 |
| Volume | 293 |
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