1659:
Ton-That and Tran have investigated the history of the theorem. They have found out that the majority of the sources before
Bourbaki's 1960 book call it Birkhoff-Witt theorem. Following this old tradition, Fofanova in her encyclopaedic entry says that Poincaré obtained the first variant of the
1173:
407:
643:
519:
872:
in a product. This fact, modulo an inductive argument on the degree of (non-canonical) monomials, shows one can always achieve products where the factors are ordered in a non-decreasing fashion.
1579:
840:
1660:
theorem. She further says that the theorem was subsequently completely demonstrated by Witt and
Birkhoff. It appears that pre-Bourbaki sources were not familiar with Poincaré's paper.
1642:
1245:
1019:
1060:
758:
1052:
1268:
275:
1386:). Then, the (graded) PBW theorem can be reformulated as the statement that, under certain hypotheses, this final morphism is an isomorphism
2119:
1958:
687:). Note that the unit element 1 corresponds to the empty canonical monomial. The theorem then asserts that these monomials form a basis for
562:
1401:(see, for example, the last section of Cohn's 1961 paper), but is true in many cases. These include the aforementioned ones, where either
932:-module, one needs to make a reformulation that does not use bases. This involves replacing the space of monomials in some basis with the
423:
2033:
1885:
2090:
1906:
20:
2251:
1417:
contains the field of rational numbers. More generally, the PBW theorem as formulated above extends to cases such as where (1)
1456:
Finally, it is worth noting that, in some of these cases, one also obtains the stronger statement that the canonical morphism
2205:
2017:
901:) is injective. In particular, any Lie algebra over a field is isomorphic to a Lie subalgebra of an associative algebra.
152:
43:
1696:"Poincaré had discovered and completely demonstrated this theorem at least thirty-seven years before Witt and Birkhoff"
2012:
1531:
766:
1430:
1702:. Their own proofs of all the steps are rather long according to their admission. Borel states that Poincaré "
1604:
1200:
974:
2007:
2182:
97:
1601:
proved, in different terminology, what is now known as the
Poincaré–Birkhoff–Witt theorem in the case of
1168:{\displaystyle {\frac {1}{n!}}\sum _{\sigma \in S_{n}}v_{\sigma (1)}v_{\sigma (2)}\cdots v_{\sigma (n)}.}
875:
The
Poincaré–Birkhoff–Witt theorem can be interpreted as saying that the end result of this reduction is
2246:
144:
1880:. History of Mathematics. Vol. 21. American mathematical society and London mathematical society.
1504:
contains the field of rational numbers, using the construction outlined here (in fact, the result is a
1589:). This stronger statement, however, might not extend to all of the cases in the previous paragraph.
2174:
1645:
2187:
116:. In the formulation of Poincaré–Birkhoff–Witt theorem we consider bases of which the elements are
715:
93:
1318:(for example, by the universal property of tensor algebras), and this is a filtered map equipping
2225:
2164:
2096:
1937:
1864:
725:
109:
1024:
1847:
Birkhoff, Garrett (April 1937). "Representability of Lie algebras and Lie groups by matrices".
2136:
2115:
2086:
2029:
1954:
1902:
1881:
933:
47:
2217:
2192:
2056:
1994:
1929:
1856:
1680:
1676:
1668:
1250:
1194:
70:
51:
2196:
1598:
1450:
2178:
2149:
1694:
It is not clear whether
Poincaré's result was complete. Ton-That and Tran conclude that
1492:), without taking associated graded. This is true in the first cases mentioned, where
1311:
74:
1969:
1683:
presents a clear illustration of the shifting tradition. In his 1986 book he calls it
2240:
2229:
2061:
2044:
1941:
1434:
1338:) is graded). Then, passing to the associated graded, one gets a canonical morphism
849:
s to a linear combination of canonical monomials: The structure constants determine
1664:
1649:
402:{\displaystyle h(x_{1},x_{2},\ldots ,x_{n})=h(x_{1})\cdot h(x_{2})\cdots h(x_{n}).}
86:
2130:
2109:
2080:
1953:. Princeton Mathematical Series (PMS). Vol. 19. Princeton University Press.
1896:
128:
117:
31:
27:
1998:
1441:
is a direct sum of cyclic modules (or all its localizations at prime ideals of
2221:
2140:
955:
contains the field of rational numbers, one can consider the natural map from
55:
2100:
1505:
417:
2085:. Princeton Mathematical Series. Vol. 36. Princeton University Press.
2082:
Representation theory of semisimple groups. An overview based on examples
2028:. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer.
1663:
Birkhoff and Witt do not mention
Poincaré's work in their 1937 papers.
69:
may also refer to various analogues of the original theorem, comparing a
2026:
Lie Groups, Lie
Algebras and Representations: An Elementary Introduction
1933:
1868:
1656:, and he does not suggest that Poincaré was aware of Capelli's result.
879:
and does not depend on the order in which one swaps adjacent elements.
638:{\displaystyle y_{1}^{k_{1}}y_{2}^{k_{2}}\cdots y_{\ell }^{k_{\ell }}}
16:
Explicitly describes the universal enveloping algebra of a Lie algebra
2169:
1453:. See, for example, the 1969 paper by Higgins for these statements.
703:); the content of the theorem is that they are linearly independent.
1917:
1860:
1700:"Poincaré makes several statements without bothering to prove them"
868:, i.e. what to do in order to change the order of two elements of
514:{\displaystyle \{h(x_{1},\ldots ,x_{n})|x_{1}\leq ...\leq x_{n}\}}
1974:
Annali della Scuola
Normale Superiore di Pisa - Classe di Scienze
695:) as a vector space. It is easy to see that these monomials span
73:
to its associated graded algebra, in particular in the area of
1178:
Then, one has the theorem that this map is an isomorphism of
1985:
Cohn, P.M. (1963). "A remark on the
Birkhoff-Witt theorem".
1750:
1748:
679:, together with the multiplicative unit 1, form a basis for
420:
on the set of canonical monomials and the image of this set
1918:"Sur les Opérations dans la théorie des formes algébriques"
1679:
were the first to use all three names in their 1960 book.
2150:"Poincaré's proof of the so-called Birkhoff-Witt theorem"
1895:
Bourbaki, Nicolas (1960). "Chapitre 1: Algèbres de Lie".
1648:; while Poincaré later stated it more generally in 1900.
913:
could be replaced by any commutative ring, provided that
1878:
Essays in the
History of Lie groups and algebraic groups
1197:, equipped with the filtration given by specifying that
1704:
more or less proved the Poincaré-Birkhoff-Witt theorem
1607:
1534:
1253:
1203:
1185:
Still more generally and naturally, one can consider
1063:
1027:
977:
769:
728:
565:
426:
278:
2129:
Poincaré, Henri (1900). "Sur les groupes continus".
1528:) with their natural coalgebra structures such that
42:) is a result giving an explicit description of the
2132:
Transactions of the Cambridge Philosophical Society
19:For the Poincaré–Birkhoff fixed-point theorem, see
2135:. Vol. 18. University Press. pp. 220–5.
1636:
1573:
1262:
1239:
1167:
1046:
1013:
909:Already at its earliest stages, it was known that
845:This relation allows one to reduce any product of
834:
752:
637:
513:
401:
889:is a Lie algebra over a field, the canonical map
219:which is non-decreasing in the order ≤, that is,
1574:{\displaystyle \Delta (v)=v\otimes 1+1\otimes v}
552:is totally ordered by the induced ordering from
2045:"Baer Invariants and the Birkhoff-Witt theorem"
1798:
1687:, while in his later 1996 book he switches to
835:{\displaystyle =\sum _{x\in X}c_{u,v}^{x}\;x.}
1370:, and hence descends to a canonical morphism
8:
1970:"Remarques sur le théorème de Birkhoff–Witt"
1901:. Éléments de mathématique. Paris: Hermann.
1754:
508:
427:
1654:"completely forgotten for almost a century"
246:to all canonical monomials as follows: if (
1675:and attribute the complete proof to Witt.
825:
2186:
2168:
2060:
1949:Cartan, Henri; Eilenberg, Samuel (1956).
1735:
1733:
1731:
1698:. On the other hand, they point out that
1625:
1616:
1615:
1606:
1533:
1252:
1231:
1218:
1208:
1202:
1147:
1125:
1106:
1094:
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1064:
1062:
1032:
1026:
1005:
992:
982:
976:
819:
808:
792:
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733:
727:
627:
622:
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602:
597:
592:
580:
575:
570:
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502:
477:
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459:
440:
425:
387:
365:
343:
321:
302:
289:
277:
1809:
1776:
1765:
1652:says that these results of Capelli were
1715:
1637:{\displaystyle L={\mathfrak {gl}}_{n},}
1286:-modules canonically extends to a map
1240:{\displaystyle v_{1}v_{2}\cdots v_{n}}
1014:{\displaystyle v_{1}v_{2}\cdots v_{n}}
536:Stated somewhat differently, consider
1831:
1820:
1739:
921:-module, i.e., has a basis as above.
30:, more specifically in the theory of
7:
1787:
1722:
1512:-module isomorphism, equipping both
120:by some relation which we denote ≤.
46:of a Lie algebra. It is named after
1620:
1617:
2206:"Treue Darstellung Liescher Ringe"
2148:Ton-That, T.; Tran, T.-D. (1999).
1535:
14:
2111:Lie groups beyond an introduction
706:The multiplicative structure of
269:) is a canonical monomial, let
1689:Poincaré-Birkhoff-Witt Theorem
1597:In four papers from the 1880s
1544:
1538:
1508:isomorphism, and not merely a
1326:) with the filtration putting
1157:
1151:
1135:
1129:
1116:
1110:
782:
770:
469:
465:
433:
393:
380:
371:
358:
349:
336:
327:
282:
36:Poincaré–Birkhoff–Witt theorem
1:
2062:10.1016/0021-8693(69)90086-6
1445:have this property), or (4)
1354:), which kills the elements
722:, that is, the coefficients
153:universal enveloping algebra
44:universal enveloping algebra
2013:Encyclopedia of Mathematics
1799:Cartan & Eilenberg 1956
924:To extend to the case when
753:{\displaystyle c_{u,v}^{x}}
180:a totally ordered basis of
2268:
1898:Groupes et algèbres de Lie
1646:General linear Lie algebra
1047:{\displaystyle v_{i}\in L}
18:
2222:10.1515/crll.1937.177.152
1916:Capelli, Alfredo (1890).
1393:This is not true for all
1330:in degree one (actually,
104:such that any element of
21:Poincaré–Birkhoff theorem
2072:The Theory of Lie Groups
2006:Fofanova, T.S. (2001) ,
1999:10.1112/jlms/s1-38.1.197
1968:Cartier, Pierre (1958).
1755:Ton-That & Tran 1999
1409:-module (hence whenever
1247:lies in filtered degree
675:, and the exponents are
81:Statement of the theorem
2252:Theorems about algebras
2070:Hochschild, G. (1965).
2024:Hall, Brian C. (2015).
2008:"Birkhoff–Witt theorem"
1388:of commutative algebras
714:) is determined by the
556:. The set of monomials
192:is a finite sequence (
172:be a Lie algebra over
2108:Knapp, A. W. (2013) .
2079:Knapp, A. W. (2001) .
2043:Higgins, P.J. (1969).
1876:Borel, Armand (2001).
1638:
1593:History of the theorem
1575:
1264:
1263:{\displaystyle \leq n}
1241:
1169:
1048:
1015:
971:), sending a monomial
836:
754:
639:
515:
403:
1922:Mathematische Annalen
1849:Annals of Mathematics
1685:Birkhoff-Witt Theorem
1673:Poincaré-Witt Theorem
1639:
1576:
1302:) of algebras, where
1265:
1242:
1170:
1049:
1016:
905:More general contexts
837:
755:
640:
516:
404:
139:denote the canonical
108:is a unique (finite)
2210:J. Reine Angew. Math
2204:Witt, Ernst (1937).
1605:
1532:
1476:-module isomorphism
1251:
1201:
1061:
1025:
975:
928:is no longer a free
767:
726:
563:
424:
276:
2179:1999math......8139T
1987:J. London Math. Soc
1951:Homological Algebra
824:
749:
716:structure constants
634:
609:
587:
2157:Rev. Histoire Math
2049:Journal of Algebra
1934:10.1007/BF01206702
1634:
1571:
1260:
1237:
1165:
1101:
1054:, to the element
1044:
1011:
832:
804:
803:
750:
729:
635:
613:
588:
566:
521:forms a basis for
511:
399:
186:canonical monomial
110:linear combination
2121:978-1-4757-2453-0
1960:978-0-691-04991-5
1671:call the theorem
1079:
1077:
951:In the case that
934:symmetric algebra
788:
215:) of elements of
2259:
2233:
2216:(177): 152–160.
2200:
2190:
2172:
2154:
2144:
2125:
2104:
2075:
2066:
2064:
2039:
2020:
2002:
1981:
1964:
1945:
1912:
1891:
1872:
1834:
1829:
1823:
1818:
1812:
1807:
1801:
1796:
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1785:
1779:
1774:
1768:
1763:
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1752:
1743:
1737:
1726:
1720:
1643:
1641:
1640:
1635:
1630:
1629:
1624:
1623:
1580:
1578:
1577:
1572:
1413:is a field), or
1269:
1267:
1266:
1261:
1246:
1244:
1243:
1238:
1236:
1235:
1223:
1222:
1213:
1212:
1195:filtered algebra
1174:
1172:
1171:
1166:
1161:
1160:
1139:
1138:
1120:
1119:
1100:
1099:
1098:
1078:
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1065:
1053:
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1050:
1045:
1037:
1036:
1020:
1018:
1017:
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1010:
1009:
997:
996:
987:
986:
841:
839:
838:
833:
823:
818:
802:
759:
757:
756:
751:
748:
743:
671:are elements of
644:
642:
641:
636:
633:
632:
631:
621:
608:
607:
606:
596:
586:
585:
584:
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482:
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408:
406:
405:
400:
392:
391:
370:
369:
348:
347:
326:
325:
307:
306:
294:
293:
100:; this is a set
85:Recall that any
71:filtered algebra
63:PBW type theorem
52:Garrett Birkhoff
2267:
2266:
2262:
2261:
2260:
2258:
2257:
2256:
2237:
2236:
2203:
2188:10.1.1.489.7065
2152:
2147:
2128:
2122:
2107:
2093:
2078:
2069:
2042:
2036:
2023:
2005:
1984:
1967:
1961:
1948:
1915:
1909:
1894:
1888:
1875:
1861:10.2307/1968569
1846:
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1753:
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1721:
1717:
1712:
1614:
1603:
1602:
1599:Alfredo Capelli
1595:
1530:
1529:
1451:Dedekind domain
1249:
1248:
1227:
1214:
1204:
1199:
1198:
1143:
1121:
1102:
1090:
1069:
1059:
1058:
1028:
1023:
1022:
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988:
978:
973:
972:
907:
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858:
854:
765:
764:
724:
723:
670:
661:
654:
623:
598:
576:
561:
560:
533:-vector space.
498:
473:
455:
436:
422:
421:
383:
361:
339:
317:
298:
285:
274:
273:
268:
259:
252:
241:
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225:
214:
205:
198:
118:totally ordered
112:of elements of
83:
24:
17:
12:
11:
5:
2265:
2263:
2255:
2254:
2249:
2239:
2238:
2235:
2234:
2201:
2145:
2126:
2120:
2105:
2091:
2076:
2067:
2055:(4): 469–482.
2040:
2035:978-3319134666
2034:
2021:
2003:
1982:
1965:
1959:
1946:
1913:
1907:
1892:
1887:978-0821802885
1886:
1873:
1855:(2): 526–532.
1842:
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1813:
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1552:
1549:
1546:
1543:
1540:
1537:
1312:tensor algebra
1259:
1256:
1234:
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1226:
1221:
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1207:
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662:< ... <
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82:
79:
75:quantum groups
48:Henri Poincaré
15:
13:
10:
9:
6:
4:
3:
2:
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2106:
2102:
2098:
2094:
2092:0-691-09089-0
2088:
2084:
2083:
2077:
2074:. Holden-Day.
2073:
2068:
2063:
2058:
2054:
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2041:
2037:
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1908:9782705613648
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1817:
1814:
1811:
1810:Bourbaki 1960
1806:
1803:
1800:
1795:
1792:
1789:
1784:
1781:
1778:
1777:Birkhoff 1937
1773:
1770:
1767:
1766:Fofanova 2001
1762:
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1515:
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1507:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1472:) lifts to a
1471:
1467:
1463:
1459:
1454:
1452:
1448:
1444:
1440:
1436:
1435:abelian group
1432:
1428:
1425:-module, (2)
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1391:
1389:
1385:
1381:
1377:
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1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1257:
1254:
1232:
1228:
1224:
1219:
1215:
1209:
1205:
1196:
1192:
1188:
1183:
1181:
1162:
1154:
1148:
1144:
1140:
1132:
1126:
1122:
1113:
1107:
1103:
1095:
1091:
1087:
1084:
1080:
1073:
1070:
1066:
1057:
1056:
1055:
1041:
1038:
1033:
1029:
1006:
1002:
998:
993:
989:
983:
979:
970:
966:
962:
958:
954:
949:
947:
943:
939:
935:
931:
927:
922:
920:
916:
912:
904:
902:
900:
896:
892:
888:
884:
880:
878:
873:
871:
867:
848:
829:
826:
820:
815:
812:
809:
805:
799:
796:
793:
789:
785:
779:
776:
773:
763:
762:
761:
745:
740:
737:
734:
730:
721:
718:in the basis
717:
713:
709:
704:
702:
698:
694:
690:
686:
682:
678:
674:
669:
665:
658:
651:
628:
624:
618:
614:
610:
603:
599:
593:
589:
581:
577:
571:
567:
559:
558:
557:
555:
551:
547:
543:
539:
534:
532:
528:
524:
503:
499:
495:
492:
489:
486:
483:
478:
474:
460:
456:
452:
449:
446:
441:
437:
430:
419:
415:
396:
388:
384:
377:
374:
366:
362:
355:
352:
344:
340:
333:
330:
322:
318:
314:
311:
308:
303:
299:
295:
290:
286:
279:
272:
271:
270:
267:
263:
256:
249:
245:
240:
236:
229:
222:
218:
213:
209:
202:
195:
191:
187:
183:
179:
175:
171:
167:
163:
161:
157:
154:
150:
146:
142:
138:
134:
131:over a field
130:
126:
121:
119:
115:
111:
107:
103:
99:
95:
91:
88:
80:
78:
76:
72:
68:
64:
59:
57:
53:
49:
45:
41:
37:
33:
29:
22:
2247:Lie algebras
2213:
2209:
2170:math/9908139
2160:
2156:
2131:
2114:. Springer.
2110:
2101:j.ctt1bpm9sn
2081:
2071:
2052:
2048:
2025:
2011:
1990:
1986:
1977:
1973:
1950:
1925:
1921:
1897:
1877:
1852:
1848:
1827:
1816:
1805:
1794:
1783:
1772:
1761:
1718:
1703:
1699:
1695:
1693:
1688:
1684:
1672:
1662:
1658:
1653:
1650:Armand Borel
1596:
1586:
1582:
1525:
1521:
1517:
1513:
1509:
1501:
1500:-module, or
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1455:
1446:
1442:
1438:
1431:torsion-free
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1392:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1307:
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1190:
1186:
1184:
1179:
1177:
968:
964:
960:
956:
952:
950:
945:
941:
937:
929:
925:
923:
918:
914:
910:
908:
898:
894:
890:
886:
882:
881:
876:
874:
869:
850:
846:
844:
719:
711:
707:
705:
700:
696:
692:
688:
684:
680:
677:non-negative
676:
672:
667:
663:
656:
649:
647:
553:
549:
545:
541:
537:
535:
530:
526:
522:
413:
411:
265:
261:
254:
247:
243:
238:
234:
227:
220:
216:
211:
207:
200:
193:
189:
185:
181:
177:
173:
169:
165:
164:
159:
155:
148:
140:
136:
132:
124:
122:
113:
105:
101:
89:
87:vector space
84:
66:
62:
60:
39:
35:
32:Lie algebras
25:
2163:: 249–284.
1993:: 197–203.
1980:(1–2): 1–4.
1976:. Série 3.
1742:, p. 6
1725:Theorem 9.9
1706:" in 1900.
1270:. The map
129:Lie algebra
67:PBW theorem
40:PBW theorem
28:mathematics
2241:Categories
2197:0958.01012
2141:1026731418
1841:References
1832:Knapp 1996
1821:Knapp 1986
1740:Borel 2001
1496:is a free
1421:is a flat
1405:is a free
1182:-modules.
917:is a free
760:such that
145:linear map
61:The terms
56:Ernst Witt
2230:118046494
2183:CiteSeerX
2018:EMS Press
1942:121470841
1788:Witt 1937
1723:Hall 2015
1669:Eilenberg
1566:⊗
1554:⊗
1536:Δ
1506:coalgebra
1310:) is the
1255:≤
1225:⋯
1149:σ
1141:⋯
1127:σ
1108:σ
1088:∈
1085:σ
1081:∑
1039:∈
999:⋯
883:Corollary
859:– y
797:∈
790:∑
629:ℓ
619:ℓ
611:⋯
496:≤
484:≤
450:…
418:injective
375:⋯
353:⋅
312:…
242:. Extend
233:≤ ... ≤
151:into the
1928:: 1–37.
1677:Bourbaki
2175:Bibcode
1869:1968569
1193:) as a
529:) as a
260:, ...,
166:Theorem
135:, let
92:over a
2228:
2195:
2185:
2139:
2118:
2099:
2089:
2032:
1957:
1940:
1905:
1884:
1867:
1665:Cartan
1520:) and
1464:) → gr
1437:, (3)
1433:as an
1378:) → gr
1346:) → gr
1282:) of
1021:. for
944:), on
877:unique
648:where
168:. Let
96:has a
54:, and
34:, the
2226:S2CID
2165:arXiv
2153:(PDF)
2097:JSTOR
1938:S2CID
1865:JSTOR
1710:Notes
1681:Knapp
1449:is a
963:) to
885:. If
412:Then
206:...,
188:over
184:. A
147:from
127:is a
98:basis
94:field
2214:1937
2137:OCLC
2116:ISBN
2087:ISBN
2030:ISBN
1955:ISBN
1903:ISBN
1882:ISBN
1667:and
1644:the
1581:for
1484:) →
1397:and
1364:v, w
1362:for
1294:) →
655:<
176:and
65:and
38:(or
2218:doi
2193:Zbl
2057:doi
1995:doi
1930:doi
1857:doi
1429:is
1314:on
548:).
416:is
162:).
123:If
26:In
2243::
2224:.
2212:.
2208:.
2191:.
2181:.
2173:.
2159:.
2155:.
2095:.
2053:11
2051:.
2047:.
2016:,
2010:,
1991:38
1989:.
1978:12
1972:.
1936:.
1926:37
1924:.
1920:.
1863:.
1853:38
1851:.
1747:^
1730:^
1691:.
1585:∈
1390:.
1366:∈
1360:wv
1358:-
1356:vw
1274:→
948:.
936:,
893:→
847:y'
540:=
253:,
199:,
77:.
58:.
50:,
2232:.
2220::
2199:.
2177::
2167::
2161:5
2143:.
2124:.
2103:.
2065:.
2059::
2038:.
2001:.
1997::
1963:.
1944:.
1932::
1911:.
1890:.
1871:.
1859::
1632:,
1627:n
1621:l
1618:g
1612:=
1609:L
1587:L
1583:v
1569:v
1563:1
1560:+
1557:1
1551:v
1548:=
1545:)
1542:v
1539:(
1526:L
1524:(
1522:U
1518:L
1516:(
1514:S
1510:K
1502:K
1498:K
1494:L
1490:L
1488:(
1486:U
1482:L
1480:(
1478:S
1474:K
1470:L
1468:(
1466:U
1462:L
1460:(
1458:S
1447:K
1443:K
1439:L
1427:L
1423:K
1419:L
1415:K
1411:K
1407:K
1403:L
1399:L
1395:K
1384:L
1382:(
1380:U
1376:L
1374:(
1372:S
1368:L
1352:L
1350:(
1348:U
1344:L
1342:(
1340:T
1336:L
1334:(
1332:T
1328:L
1324:L
1322:(
1320:T
1316:L
1308:L
1306:(
1304:T
1300:L
1298:(
1296:U
1292:L
1290:(
1288:T
1284:K
1280:L
1278:(
1276:U
1272:L
1258:n
1233:n
1229:v
1220:2
1216:v
1210:1
1206:v
1191:L
1189:(
1187:U
1180:K
1163:.
1158:)
1155:n
1152:(
1145:v
1136:)
1133:2
1130:(
1123:v
1117:)
1114:1
1111:(
1104:v
1096:n
1092:S
1074:!
1071:n
1067:1
1042:L
1034:i
1030:v
1007:n
1003:v
994:2
990:v
984:1
980:v
969:L
967:(
965:U
961:L
959:(
957:S
953:K
946:L
942:L
940:(
938:S
930:K
926:L
919:K
915:L
911:K
899:L
897:(
895:U
891:L
887:L
870:Y
865:i
863:y
861:j
857:j
855:y
853:i
851:y
830:.
827:x
821:x
816:v
813:,
810:u
806:c
800:X
794:x
786:=
783:]
780:v
777:,
774:u
771:[
746:x
741:v
738:,
735:u
731:c
720:X
712:L
710:(
708:U
701:L
699:(
697:U
693:L
691:(
689:U
685:L
683:(
681:U
673:Y
668:n
664:y
660:2
657:y
653:1
650:y
625:k
615:y
604:2
600:k
594:2
590:y
582:1
578:k
572:1
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554:X
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546:X
544:(
542:h
538:Y
531:K
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525:(
523:U
509:}
504:n
500:x
493:.
490:.
487:.
479:1
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447:,
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434:(
431:h
428:{
414:h
397:.
394:)
389:n
385:x
381:(
378:h
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356:h
350:)
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337:(
334:h
331:=
328:)
323:n
319:x
315:,
309:,
304:2
300:x
296:,
291:1
287:x
283:(
280:h
266:n
262:x
258:2
255:x
251:1
248:x
244:h
239:n
235:x
231:2
228:x
226:≤
224:1
221:x
217:X
212:n
208:x
204:2
201:x
197:1
194:x
190:X
182:L
178:X
174:K
170:L
160:L
158:(
156:U
149:L
143:-
141:K
137:h
133:K
125:L
114:S
106:V
102:S
90:V
23:.
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