853:
813:
642:
on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the
326:
The dream of noncommutative geometry is to generalize this duality to the duality between noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures, and geometric entities of certain kinds, and give an interaction between
322:
Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.
665:(over a base category), abstracting Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors. There is also another interesting approach via localization theory, due to
774:, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively and still functions as a test case for more complicated situations.
1999:
Noncommutative geometry, arithmetic, and related topics. Proceedings of the 21st meeting of the Japan-U.S. Mathematics
Institute (JAMI) held at Johns Hopkins University, Baltimore, MD, USA, March 23â26,
1068:
983:
635:
as the dual of the category of associative unital rings. There are certain analogues of
Zariski topology in that context so that one can glue such affine schemes to more general objects.
689:' direction in noncommutative geometry is his discovery of a new homology theory associated to noncommutative associative algebras and noncommutative operator algebras, namely the
301:
101:
78:
274:
685:
to formal duals of noncommutative (operator) algebras and other replacements and candidates for noncommutative spaces. One of the main starting points of
241:
1314:
1997:
716:
allow for effective extraction of numerical invariants from spectral triples. The fundamental characteristic class in cyclic cohomology, the
1855:
1793:
49:
that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an
158:
The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics,
1363:
796:
17:
1894:
GarcĂa-BeltrĂĄn, Dennise; a-BeltrĂĄn, Dennise; Vallejo, JosĂ© A.; Vorobjev, YuriÄ (2012). "On Lie
Algebroids and Poisson Algebras".
994:
311:, the cohomological properties of a site are invariants of the corresponding category of sheaves of sets viewed abstractly as a
315:(A. Grothendieck). In all these cases, a space is reconstructed from the algebra of functions or its categorified versionâsome
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2008:
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1705:
1664:
1636:
1610:
1535:
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1409:
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799:
on continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.
334:
to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization of
2231:
1971:
1387:
1099:
654:
Many properties of projective schemes extend to this context. For example, there exists an analog of the celebrated
611:. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.
2241:
362:. The sudden rise in interest in noncommutative geometry in physics follows after the speculations of its role in
138:", which played a key role in the early development of this field in 1980s and lead to noncommutative versions of
1371:
638:
There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a theorem of
2236:
1159:
1157:
Connes, Alain; Douglas, Michael R; Schwarz, Albert (1998-02-05). "Noncommutative geometry and Matrix theory".
864:
824:
1602:
760:
355:
2155:
Masson, Thierry (2006). "An informal introduction to the ideas and concepts of noncommutative geometry".
249:
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and J. J. Zhang, who add also some general ring-theoretic conditions (e.g. ArtinâSchelter regularity).
391:
143:
2216:
378:
to handle noncommutative geometry at a technical level has roots in older attempts, in particular in
2251:
1390:- New York: Dekker, 2000.- 287 p. - (Monographs and textbooks in pure and applied mathematics, 232)
888:
749:
441:
339:
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989:
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127:
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can be reconstructed up to isomorphism of schemes from the category of quasicoherent sheaves of
1401:
701:
414:
253:
165:
42:
2182:
Mahanta, Snigdhayan (2005). "On some approaches towards non-commutative algebraic geometry".
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1913:
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of smooth manifolds has been extended to spectral triples, employing the tools of operator
279:
135:
2018:
1645:
Khalkhali, Masoud; Marcolli, Matilde (2008). Khalkhali, Masoud; Marcolli, Matilde (eds.).
8:
1601:, American Mathematical Society Colloquium Publications, vol. 55, Providence, R.I.:
738:
482:
463:
456:
330:
Regarding that the commutative rings correspond to usual affine schemes, and commutative
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1678:, Lecture Notes in Physics. New Series m: Monographs, vol. 51, Berlin, New York:
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2004:
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788:
709:
639:
489:
395:
335:
188:
2197:
Sardanashvily, G. (2009). "Lectures on
Differential Geometry of Modules and Rings".
2147:
1431:
1402:"Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras"
1350:
A. L. Rosenberg, Noncommutative schemes, Compositio
Mathematica 112 (1998) 93--125,
1251:
2125:
2050:
2014:
1989:
1959:
1933:
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topologically. The algebraic invariant that recovers the
Riemannian structure is a
351:
169:
123:
108:
1328:
1309:
681:
Some of the motivating questions of the theory are concerned with extending known
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184:
1963:
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This suggests that one might define a noncommutative
Riemannian manifold as a
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1943:
1878:
1814:
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1440:
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1294:
1198:
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900:
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713:
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648:
624:
469:
452:
383:
203:
139:
131:
2087:
Marcolli, Matilde (2004). "Lectures on
Arithmetic Noncommutative Geometry".
1993:
1925:
1585:
1544:
1515:
1285:
1266:
1084:
896:
686:
375:
1477:
741:
717:
669:, Luc Willaert and Alain Verschoren, where the main concept is that of a
418:
413:
are often now called non-commutative spaces. This is by analogy with the
331:
54:
38:
2161:
852:
812:
492:
with a lot of extra structure. From its algebra of continuous functions
134:. Perhaps one of the typical examples of a noncommutative space is the "
2211:
1886:
1850:
1806:
1776:
1688:
1243:
1220:
Connes, Alain (2013). "On the spectral characterization of manifolds".
1173:
410:
122:
An approach giving deep insight about noncommutative spaces is through
2112:
2027:
Structural aspects of quantum field theory and noncommutative geometry
1893:
1354:; Underlying spaces of noncommutative schemes, preprint MPIM2003-111,
338:
as "non-commutative spaces". For this reason there is some talk about
2188:
2093:
2078:
2063:
1561:
781:
147:
119:, to be possibly carried by the noncommutative algebra of functions.
1869:
2176:
1627:
Gracia-Bondia, Jose M; Figueroa, Hector; Varilly, Joseph C (2000),
1382:
Freddy van
Oystaeyen, Algebraic geometry for associative algebras,
705:
643:
algebra is
Noetherian. This theorem is extended as a definition of
540:
by multiplication operators, and we consider an unbounded operator
401:
363:
327:
the algebraic and geometric description of those via this duality.
112:
2203:
1908:
1234:
1656:
661:
A. L. Rosenberg has created a rather general relative concept of
2072:
Khalkhali, Masoud (2004). "Very Basic Noncommutative Geometry".
2057:
Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry".
111:
is not commutative; one also allows additional structures, e.g.
1770:
Connes, Alain (2001). "C* algebras and differential geometry".
1126:
614:
2102:
Madore, J. (2000). "Noncommutative Geometry for Pedestrians".
1896:
Symmetry, Integrability and Geometry: Methods and Applications
1138:
1676:
An introduction to noncommutative spaces and their geometries
312:
164:, which are geometric in nature, can be related to numerical
1626:
1063:{\displaystyle \nabla _{r}(sa)=\nabla _{r}(s)a+s\otimes da}
1454:
Snyder, Hartland S. (1947-01-01). "Quantized Space-Time".
658:
for noncommutative projective schemes of Artin and Zhang.
568:
as a Riemannian manifold can be recovered from this data.
476:
603:, with compact resolvent, such that is bounded for all
2167:(An easier introduction that is still rather technical)
1721:
676:
345:
252:
are locally prime spectra of commutative unital rings (
978:{\displaystyle \nabla :E\to E\otimes _{A}\Omega ^{1}A}
516:, e.g. the exterior algebra bundle. The Hilbert space
1399:
1310:"Serre duality for noncommutative projective schemes"
997:
936:
282:
262:
86:
63:
1555:, World Sci. Publ., Hackensack, NJ, pp. 1â128,
1156:
752:
into a non-commutative phase space generated by the
727:
1745:
1595:
Noncommutative geometry, quantum fields and motives
1322:(3). American Mathematical Society (AMS): 697â708.
560:), such that the commutators are bounded whenever
432:. In general, one can associate to any C*-algebra
1519:
1062:
977:
587:), consisting of a representation of a C*-algebra
295:
268:
95:
72:
2003:, Baltimore, MD: Johns Hopkins University Press,
1644:
1144:
1132:
369:
236:topological spaces can be reconstructed from the
221:), and therefore it makes some sense to say that
2223:
1315:Proceedings of the American Mathematical Society
1308:Yekutieli, Amnon; Zhang, James J. (1997-03-01).
508:. It is constructed from a smooth vector bundle
402:Noncommutative C*-algebras, von Neumann algebras
168:on them. In general, such functions will form a
1726:, Lecture Notes in Mathematics, vol. 887,
1722:Van Oystaeyen, Fred; Verschoren, Alain (1981),
1551:(2008), "A walk in the noncommutative garden",
398:of an extended kind, has by now been subsumed.
1584:
1543:
1307:
2217:connection in noncommutative geometry in nLab
2196:
1988:
1464:(1). American Physical Society (APS): 38â41.
2212:Noncommutative geometry and particle physics
1856:Journal of the American Mathematical Society
1848:
697:(primarily via ConnesâChern character map).
615:Noncommutative affine and projective schemes
2029:. Hackensack New Jersey: World Scientific.
1400:Van Oystaeyen, Fred; Willaert, Luc (1995).
712:. Several generalizations of now-classical
342:, though the term also has other meanings.
1264:
2202:
2187:
2160:
2111:
2092:
2077:
2071:
2062:
1907:
1868:
1775:
1755:"C* algÚbres et géométrie différentielle"
1687:
1648:An Invitation to non-Commutative Geometry
1560:
1430:
1327:
1284:
1233:
1172:
843:
2086:
2056:
2024:
1849:Cuntz, Joachim; Quillen, Daniel (1995).
1553:An invitation to noncommutative geometry
887:is a noncommutative generalization of a
232:More specifically, in topology, compact
2181:
1851:"Algebra Extensions and Nonsingularity"
1789:"Non-commutative differential geometry"
477:Noncommutative differentiable manifolds
41:concerned with a geometric approach to
14:
2224:
2154:
2101:
1829:
1786:
1769:
1752:
1514:
1453:
1219:
595:, together with an unbounded operator
564:is smooth. A deep theorem states that
390:theory, with respect to which ergodic
172:. For instance, one may take the ring
1673:
780:Noncommutative algebras arising from
2171:Noncommutative geometry on arxiv.org
1941:
1794:Publications MathĂ©matiques de l'IHĂS
1629:Elements of Non-commutative geometry
1498:
847:
807:
677:Invariants for noncommutative spaces
346:Applications in mathematical physics
256:), and every quasi-separated scheme
2177:Theories of Noncommutative Geometry
1410:Journal of Pure and Applied Algebra
1267:"Noncommutative Projective Schemes"
556:) with compact resolvent (e.g. the
528:) of square integrable sections of
360:Noncommutative quantum field theory
53:in which the multiplication is not
24:
1982:
1724:Non-commutative algebraic geometry
1222:Journal of Noncommutative Geometry
1024:
999:
963:
937:
663:noncommutative quasicompact scheme
645:noncommutative projective geometry
25:
2263:
2104:Classical and Quantum Nonlocality
2044:
1368:Noncommutative schemes and spaces
1100:Noncommutative algebraic geometry
728:Examples of noncommutative spaces
2051:Introduction to Quantum Geometry
1746:References for Connes connection
851:
811:
374:Some of the theory developed by
1492:
1447:
1265:Artin, M.; Zhang, J.J. (1994).
899:, and was later generalized by
763:is a proposed extension of the
754:position and momentum operators
45:, and with the construction of
1944:"notes on quasi-free algebras"
1393:
1376:
1344:
1301:
1258:
1213:
1160:Journal of High Energy Physics
1150:
1039:
1033:
1017:
1008:
946:
370:Motivation from ergodic theory
107:in which one of the principal
13:
1:
1759:C. R. Acad. Sci. Paris SĂ©r. A
1603:American Mathematical Society
1507:
1329:10.1090/s0002-9939-97-03782-9
1191:10.1088/1126-6708/1998/02/003
1145:Khalkhali & Marcolli 2008
1133:Khalkhali & Marcolli 2008
910:
803:
761:noncommutative standard model
633:noncommutative affine schemes
356:Noncommutative standard model
354:are described in the entries
153:
1423:10.1016/0022-4049(94)00118-3
1120:
720:, generalizes the classical
607:in some dense subalgebra of
532:carries a representation of
7:
1417:(1). Elsevier BV: 109â122.
1073:
737:of quantum mechanics, the
240:of functions on the space (
10:
2268:
2130:10.1142/9789812792938_0007
2025:Grensing, Gerhard (2013).
1958:. 2020. pp. 201â228.
631:, we define a category of
2247:Mathematical quantization
1964:10.1017/9781108855846.009
923:, a Connes connection on
693:and its relations to the
2232:Connection (mathematics)
1674:Landi, Giovanni (1997),
1522:Non-commutative geometry
1432:10067/124190151162165141
442:spectrum of a C*-algebra
340:non-commutative topology
128:bounded linear operators
18:Non-commutative geometry
2242:Noncommutative geometry
1956:Topics in Cyclic Theory
1832:Noncommutative Geometry
1272:Advances in Mathematics
1110:Phase space formulation
1105:Noncommutative topology
895:. It was introduced by
844:In the sense of Connes
735:phase space formulation
309:Grothendieck topologies
187:-valued functions on a
126:, that is, algebras of
103:; or more generally an
43:noncommutative algebras
31:Noncommutative geometry
1926:10.3842/SIGMA.2012.006
1830:Connes, Alain (1995).
1787:Connes, Alain (1985).
1753:Connes, Alain (1980).
1286:10.1006/aima.1994.1087
1064:
979:
702:characteristic classes
683:topological invariants
415:Gelfand representation
406:The (formal) duals of
297:
270:
97:
80:does not always equal
74:
2237:Differential geometry
1478:10.1103/physrev.71.38
1065:
980:
893:differential geometry
350:Some applications in
298:
296:{\displaystyle O_{X}}
271:
98:
75:
57:, that is, for which
27:Branch of mathematics
1651:. World Scientific.
995:
934:
787:Examples related to
772:noncommutative torus
767:of particle physics.
464:von Neumann algebras
457:von Neumann algebras
451:between localizable
436:a topological space
307:âA. Rosenberg). For
280:
260:
227:commutative topology
136:noncommutative torus
84:
61:
2122:2000cqnl.conf..111M
1918:2012SIGMA...8..006G
1698:1997hep.th....1078L
1571:2006math......1054C
1470:1947PhRv...71...38S
1183:1998JHEP...02..003C
988:that satisfies the
746:classical mechanics
591:on a Hilbert space
500:), we only recover
483:Riemannian manifold
417:, which shows that
317:category of sheaves
105:algebraic structure
51:associative algebra
2053:by Micho ÄurÄevich
1834:. Academic Press.
1807:10.1007/BF02698807
1115:Quasi-free algebra
1060:
975:
863:. You can help by
823:. You can help by
695:algebraic K-theory
667:Fred Van Oystaeyen
619:In analogy to the
558:signature operator
396:homogeneous spaces
382:. The proposal of
336:topological spaces
293:
266:
246:algebraic geometry
244:). In commutative
209:), we can recover
96:{\displaystyle yx}
93:
73:{\displaystyle xy}
70:
2139:978-981-02-4296-1
2036:978-981-4472-69-2
2010:978-1-4214-0352-6
1990:Consani, Caterina
1942:Vale, R. (2009).
1841:978-0-08-057175-1
1737:978-3-540-11153-5
1707:978-3-540-63509-3
1666:978-981-270-616-4
1638:978-0-8176-4124-5
1612:978-0-8218-4210-2
1590:Marcolli, Matilde
1549:Marcolli, Matilde
1537:978-0-12-185860-5
1501:, Definition 8.1.
1090:Koszul connection
885:Connes connection
881:
880:
841:
840:
789:dynamical systems
710:cyclic cohomology
671:schematic algebra
629:commutative rings
490:topological space
269:{\displaystyle X}
250:algebraic schemes
194:. In many cases (
189:topological space
124:operator algebras
109:binary operations
37:) is a branch of
16:(Redirected from
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927:is a linear map
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468:non-commutative
455:and commutative
430:Hausdorff spaces
421:C*-algebras are
388:virtual subgroup
352:particle physics
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170:commutative ring
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2140:
2047:
2037:
2011:
1996:, eds. (2011),
1985:
1983:Further reading
1980:
1974:
1954:"Connections".
1953:
1946:
1870:10.2307/2152819
1842:
1748:
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1728:Springer-Verlag
1708:
1680:Springer-Verlag
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1457:Physical Review
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877:
871:
868:
861:needs expansion
846:
837:
831:
828:
821:needs expansion
806:
730:
722:Chern character
691:cyclic homology
679:
617:
573:spectral triple
506:spectral triple
479:
427:locally compact
408:non-commutative
404:
372:
348:
319:on that space.
287:
283:
281:
278:
277:
261:
258:
257:
254:A. Grothendieck
242:GelfandâNaimark
207:Hausdorff space
156:
85:
82:
81:
62:
59:
58:
28:
23:
22:
15:
12:
11:
5:
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2220:
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2209:
2194:
2179:
2175:MathOverflow,
2173:
2168:
2152:
2138:
2099:
2084:
2069:
2054:
2046:
2045:External links
2043:
2042:
2041:
2035:
2022:
2009:
1984:
1981:
1979:
1978:
1972:
1951:
1938:
1891:
1863:(2): 251â289.
1846:
1840:
1827:
1784:
1783:
1782:
1777:hep-th/0101093
1765:(13): 599â604.
1749:
1747:
1744:
1742:
1741:
1736:
1719:
1706:
1689:hep-th/9701078
1671:
1665:
1642:
1637:
1631:, Birkhauser,
1624:
1611:
1582:
1541:
1536:
1528:Academic Press
1526:, Boston, MA:
1511:
1509:
1506:
1504:
1503:
1491:
1446:
1392:
1375:
1343:
1300:
1279:(2): 228â287.
1257:
1212:
1174:hep-th/9711162
1149:
1137:
1135:, p. 171.
1124:
1122:
1119:
1118:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1075:
1072:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
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1026:
1022:
1019:
1016:
1013:
1010:
1005:
1001:
986:
985:
974:
969:
965:
959:
955:
951:
948:
945:
942:
939:
915:Given a right
912:
909:
905:Daniel Quillen
879:
878:
858:
856:
845:
842:
839:
838:
818:
816:
805:
802:
801:
800:
795:, such as the
785:
778:
775:
768:
765:standard model
757:
729:
726:
714:index theorems
700:The theory of
678:
675:
625:affine schemes
616:
613:
478:
475:
470:measure spaces
461:noncommutative
453:measure spaces
403:
400:
380:ergodic theory
371:
368:
366:made in 1997.
347:
344:
290:
286:
265:
238:Banach algebra
155:
152:
140:vector bundles
92:
89:
69:
66:
26:
9:
6:
4:
3:
2:
2264:
2253:
2250:
2248:
2245:
2243:
2240:
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2218:
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2210:
2205:
2200:
2195:
2190:
2185:
2180:
2178:
2174:
2172:
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2163:
2158:
2153:
2149:
2145:
2141:
2135:
2131:
2127:
2123:
2119:
2114:
2113:gr-qc/9906059
2109:
2105:
2100:
2095:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2055:
2052:
2049:
2048:
2038:
2032:
2028:
2023:
2020:
2016:
2012:
2006:
2002:
2001:
1995:
1994:Connes, Alain
1991:
1987:
1986:
1975:
1973:9781108855846
1969:
1965:
1961:
1957:
1952:
1945:
1939:
1935:
1931:
1927:
1923:
1919:
1915:
1910:
1905:
1901:
1897:
1892:
1888:
1884:
1880:
1876:
1871:
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1862:
1858:
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1837:
1833:
1828:
1824:
1820:
1816:
1812:
1808:
1804:
1800:
1796:
1795:
1790:
1785:
1778:
1773:
1768:
1767:
1764:
1761:(in French).
1760:
1756:
1751:
1750:
1739:
1733:
1729:
1725:
1720:
1717:
1713:
1709:
1703:
1699:
1695:
1690:
1685:
1681:
1677:
1672:
1668:
1662:
1658:
1654:
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1640:
1634:
1630:
1625:
1622:
1618:
1614:
1608:
1604:
1597:
1596:
1591:
1587:
1586:Connes, Alain
1583:
1580:
1576:
1572:
1568:
1563:
1558:
1554:
1550:
1546:
1545:Connes, Alain
1542:
1539:
1533:
1529:
1524:
1523:
1517:
1516:Connes, Alain
1513:
1512:
1500:
1495:
1487:
1483:
1479:
1475:
1471:
1467:
1463:
1459:
1458:
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1438:
1433:
1428:
1424:
1420:
1416:
1412:
1411:
1403:
1396:
1389:
1388:0-8247-0424-X
1385:
1379:
1373:
1369:
1365:
1361:
1357:
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1347:
1339:
1335:
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1208:
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1200:
1196:
1192:
1188:
1184:
1180:
1175:
1170:
1166:
1162:
1161:
1153:
1147:, p. 21.
1146:
1141:
1134:
1129:
1125:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1095:Moyal product
1093:
1091:
1088:
1086:
1083:
1081:
1080:Commutativity
1078:
1077:
1071:
1057:
1054:
1051:
1048:
1045:
1042:
1036:
1028:
1020:
1014:
1011:
1003:
991:
972:
967:
957:
953:
949:
943:
940:
930:
929:
928:
926:
922:
918:
908:
906:
902:
901:Joachim Cuntz
898:
894:
890:
886:
875:
866:
862:
859:This section
857:
854:
850:
849:
835:
826:
822:
819:This section
817:
814:
810:
809:
798:
794:
793:number theory
791:arising from
790:
786:
783:
779:
776:
773:
769:
766:
762:
758:
755:
751:
747:
743:
740:
736:
732:
731:
725:
723:
719:
715:
711:
707:
703:
698:
696:
692:
688:
684:
674:
672:
668:
664:
659:
657:
656:Serre duality
652:
650:
649:Michael Artin
646:
641:
636:
634:
630:
626:
622:
612:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
569:
567:
563:
559:
555:
551:
547:
543:
539:
535:
531:
527:
523:
519:
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511:
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499:
495:
491:
487:
484:
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471:
465:
462:
458:
454:
450:
445:
443:
439:
435:
431:
428:
424:
420:
416:
412:
409:
399:
397:
394:would become
393:
392:group actions
389:
385:
384:George Mackey
381:
377:
367:
365:
361:
357:
353:
343:
341:
337:
333:
328:
324:
320:
318:
314:
310:
306:
288:
284:
263:
255:
251:
247:
243:
239:
235:
230:
228:
224:
220:
216:
212:
208:
205:
201:
197:
193:
190:
186:
183:
179:
175:
171:
167:
163:
162:
151:
149:
145:
141:
137:
133:
132:Hilbert space
129:
125:
120:
118:
114:
110:
106:
90:
87:
67:
64:
56:
52:
48:
44:
40:
36:
32:
19:
2189:math/0501166
2103:
2094:math/0409520
2079:math/0408416
2064:math/0506603
2026:
1998:
1955:
1899:
1895:
1860:
1854:
1831:
1798:
1792:
1762:
1758:
1723:
1675:
1657:10.1142/6422
1647:
1628:
1594:
1562:math/0601054
1552:
1521:
1494:
1461:
1455:
1449:
1414:
1408:
1395:
1378:
1370:(Feb 2000):
1367:
1346:
1319:
1313:
1303:
1276:
1270:
1260:
1225:
1221:
1215:
1164:
1158:
1152:
1140:
1128:
1085:Fuzzy sphere
990:Leibniz rule
987:
924:
920:
916:
914:
897:Alain Connes
884:
882:
869:
865:adding to it
860:
829:
825:adding to it
820:
777:Snyder space
699:
687:Alain Connes
680:
670:
662:
660:
653:
644:
637:
632:
618:
608:
604:
600:
596:
592:
588:
584:
580:
576:
570:
565:
561:
553:
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
501:
497:
493:
485:
480:
467:
446:
437:
433:
405:
387:
386:to create a
376:Alain Connes
373:
349:
329:
325:
321:
231:
226:
222:
218:
214:
210:
199:
195:
191:
177:
173:
159:
157:
121:
46:
34:
30:
29:
797:Gauss shift
742:phase space
718:JLO cocycle
466:are called
419:commutative
411:C*-algebras
332:C*-algebras
144:connections
55:commutative
39:mathematics
2226:Categories
2019:1245.00040
1801:: 41â144.
1508:References
1167:(2): 003.
911:Definition
889:connection
804:Connection
782:foliations
739:symplectic
305:P. Gabriel
303:-modules (
182:continuous
154:Motivation
2204:0910.1515
1909:1106.1512
1879:0894-0347
1823:122740195
1815:1618-1913
1499:Vale 2009
1486:0031-899X
1441:0022-4049
1338:0002-9939
1295:0001-8708
1235:0810.2088
1199:1029-8479
1121:Citations
1052:⊗
1025:∇
1000:∇
964:Ω
954:⊗
947:→
938:∇
481:A smooth
234:Hausdorff
166:functions
148:curvature
2148:15595586
1592:(2008),
1518:(1994),
1366:lecture
1252:17287100
1228:: 1â82.
1074:See also
919:-module
872:May 2023
832:May 2023
750:deformed
706:K-theory
623:between
447:For the
364:M-theory
113:topology
2118:Bibcode
2106:: 111.
1934:5946411
1914:Bibcode
1902:: 006.
1887:2152819
1716:1482228
1694:Bibcode
1621:2371808
1579:2408150
1567:Bibcode
1466:Bibcode
1207:7562354
1179:Bibcode
733:In the
621:duality
583:,
579:,
552:,
524:,
449:duality
204:compact
185:complex
150:, etc.
2146:
2136:
2033:
2017:
2007:
1970:
1932:
1885:
1877:
1838:
1821:
1813:
1734:
1714:
1704:
1663:
1635:
1619:
1609:
1577:
1534:
1484:
1439:
1386:
1336:
1293:
1250:
1205:
1197:
440:; see
161:spaces
47:spaces
2199:arXiv
2184:arXiv
2157:arXiv
2144:S2CID
2108:arXiv
2089:arXiv
2074:arXiv
2059:arXiv
1947:(PDF)
1930:S2CID
1904:arXiv
1883:JSTOR
1819:S2CID
1772:arXiv
1684:arXiv
1599:(PDF)
1557:arXiv
1405:(PDF)
1372:video
1248:S2CID
1230:arXiv
1203:S2CID
1169:arXiv
640:Serre
512:over
488:is a
313:topos
213:from
202:is a
198:, if
180:) of
130:on a
2134:ISBN
2031:ISBN
2005:ISBN
2000:2009
1968:ISBN
1875:ISSN
1836:ISBN
1811:ISSN
1732:ISBN
1702:ISBN
1661:ISBN
1633:ISBN
1607:ISBN
1532:ISBN
1482:ISSN
1437:ISSN
1384:ISBN
1364:MSRI
1334:ISSN
1291:ISSN
1195:ISSN
1165:1998
903:and
770:The
759:The
708:and
627:and
423:dual
358:and
225:has
196:e.g.
117:norm
2126:doi
2015:Zbl
1960:doi
1922:doi
1865:doi
1803:doi
1763:290
1653:doi
1474:doi
1427:hdl
1419:doi
1415:104
1356:dvi
1352:doi
1324:doi
1320:125
1281:doi
1277:109
1240:doi
1187:doi
891:in
867:.
827:.
748:is
744:of
647:by
599:on
544:in
425:to
115:or
35:NCG
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2013:,
1992:;
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1940:*
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1920:.
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1859:.
1853:.
1817:.
1809:.
1799:62
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1425:.
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1362:;
1360:ps
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870:(
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784:.
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609:A
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