322:, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any eigenbasis for this subspace can be extended to an eigenbasis of the full space.
100:
of the base field does not divide the order of the group). So in the case of finite groups with this condition, every finite-dimensional representation is semi-simple. Especially in algebra and representation theory, "semi-simplicity" is also called
636:
1307:
of a group or a Lie algebra is semisimple, that is, whether every finite-dimensional representation decomposes as a direct sum of irreducible representations. The answer, in general, is no. For example, the representation of
1133:
1399:
851:
776:
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1224:
1170:. The presence of this so-called polarization causes the category of polarizable Hodge structures to be semi-simple. Another example from algebraic geometry is the category of
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is a collection of objects and maps between such objects, the idea being that the maps between the objects preserve some structure inherent in these objects. For example,
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1893:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , 52. Springer-Verlag, Berlin, 2008. xiv+470 pp.
55:
objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context.
1298:
106:
1703:. Thus, the just-mentioned result about representations of compact groups applies. It is also possible to prove semisimplicity of representations of
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318:. This is because such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant
296:
1849:
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is not a direct sum of irreducibles. (There is precisely one nontrivial invariant subspace, the span of the first basis element,
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is semi-simple if and only if it is the direct sum of simple modules (the trivial module is the empty direct sum). Finally,
246:
vector spaces are the simple ones. So it is a basic result of linear algebra that any finite-dimensional vector space is the
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472:
96:
says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the
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Examples of semi-simple rings include fields and, more generally, finite direct products of fields. For a finite group
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As indicated above, the theory of semi-simple rings is much more easy than the one of general rings. For example, any
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in a semi-simple category is a product of matrix rings over division rings, i.e., semi-simple.
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of simple vector spaces; in other words, all finite-dimensional vector spaces are semi-simple.
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631:{\displaystyle M_{n_{1}}(D_{1})\times M_{n_{2}}(D_{2})\times \cdots \times M_{n_{r}}(D_{r})}
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This article is about mathematical use. For the philosophical reduction thinking, see
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Nilpotence, radicaux et structures monoĂŻdales. With an appendix by Peter O'Sullivan
1844:. Graduate texts in mathematics. Vol. 131 (2 ed.). Springer. p. 27.
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is a complex semisimple Lie algebra, every finite-dimensional representation of
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Bondarko, Mikhail V. (2012), "Weight structures and 'weights' on the hearts of
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is a linear algebraic group whose radical of the identity component is trivial.
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Many of the above notions of semi-simplicity are recovered by the concept of a
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319:
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or, equivalently, product) of finitely many simple objects. It follows from
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1005:
116:
1257:, this category is semi-simple if and only if the equivalence relation is
2015:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
1128:{\displaystyle \operatorname {End} _{C}(X)=\operatorname {Hom} _{C}(X,X)}
2017:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer
1394:{\displaystyle \Pi (x)={\begin{pmatrix}1&x\\0&1\end{pmatrix}}}
211:
These notions of semi-simplicity can be unified using the language of
1877:
411:-module. As it turns out, this is equivalent to requiring that any
1261:. This fact is a conceptual cornerstone in the theory of motives.
204:; if the field is algebraically closed, this is the same as being
2032:
1865:
More generally, the same definition of semi-simplicity works for
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is a sum of irreducibles. Weyl's original proof of this used the
1264:
Semisimple abelian categories also arise from a combination of a
1145:
is semi-simple if and only if the category of finitely generated
971:
is called semi-simple if there is a collection of simple objects
2041:
1727:
directly by algebraic means, as in
Section 10.3 of Hall's book.
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is semi-simple in the sense above if and only if the subalgebra
387:(the trivial module 0 is semi-simple, but not simple). For an
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is a Lie algebra that is a direct sum of simple Lie algebras.
84:
if the only subrepresentations it contains are either {0} or
1809:(2nd ed.). Englewood Cliffs, N.J.: Prentice-Hall, Inc.
471:-modules, this fact is an important dichotomy, which causes
2033:
Are abelian non-degenerate tensor categories semisimple?
957:-linear maps between them form a category, for any ring
1876:. Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291.
1163:, i.e., pure Hodge structures equipped with a suitable
495:| does not divide the characteristic, in particular if
192:), then the only simple matrices are of size 1-by-1. A
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771:{\displaystyle F\subseteq \operatorname {End} _{F}(V)}
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of modules over a semi-simple ring must split, i.e.,
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is simple, if it has no submodules other than 0 and
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Motives, numerical equivalence, and semi-simplicity
1539:decomposes as a sum of irreducibles. Similarly, if
510:is semisimple if and only if it is (isomorphic to)
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165:a finite-dimensional vector space) is said to be
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295:-invariant subspace. This is equivalent to the
1455:, then every finite-dimensional representation
1219:{\displaystyle \operatorname {Mot} (k)_{\sim }}
314:, semi-simplicity of a matrix is equivalent to
27:is a widespread concept in disciplines such as
1889:Peters, Chris A. M.; Steenbrink, Joseph H. M.
1780:
1778:
1305:category of finite-dimensional representations
778:generated by the powers (i.e., iterations) of
109:says a finite-dimensional representation of a
8:
51:is one that can be decomposed into a sum of
902:, this means that there are no non-trivial
1872:. See for example Yves André, Bruno Kahn:
499:is a field of characteristic zero. By the
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491:to be more difficult than the case when |
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1283:Semi-simplicity in representation theory
846:{\displaystyle 0\to M'\to M\to M''\to 0}
1774:
1299:Weyl's theorem on complete reducibility
107:Weyl's theorem on complete reducibility
69:, then a nontrivial finite-dimensional
1841:A first course in noncommutative rings
226:Introductory example of vector spaces
7:
2001:
1989:
1977:
1965:
1916:, Invent. math. 107, 447~452 (1992)
1712:
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891:{\displaystyle M\cong M'\oplus M''}
1878:https://arxiv.org/abs/math/0203273
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14:
1801:(1971). "Semi-Simple operators".
455:. Since the theory of modules of
218:, and generalized to semi-simple
997:{\displaystyle X_{\alpha }\in C}
910:of integers is not semi-simple:
1720:{\displaystyle {\mathfrak {g}}}
1696:{\displaystyle {\mathfrak {g}}}
1608:{\displaystyle {\mathfrak {g}}}
1580:{\displaystyle {\mathfrak {g}}}
1556:{\displaystyle {\mathfrak {g}}}
1856:"(2.5) Theorem and Definition"
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443:is semi-simple if and only if
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1:
1228:adequate equivalence relation
473:modular representation theory
326:Semi-simple modules and rings
202:direct sum of simple matrices
1323:{\displaystyle \mathbb {R} }
898:. From the point of view of
1945:10.4310/HHA.2012.v14.n1.a12
1032:{\displaystyle X_{\alpha }}
407:if it is semi-simple as an
90:irreducible representations
2091:
1752:semisimple algebraic group
1292:
1286:
383:-module direct summand of
329:
306:For vector spaces over an
15:
1763:Semisimple representation
1519:is unitary, showing that
1431:.) On the other hand, if
1289:Semisimple representation
1271:and a (suitably related)
914:is not the direct sum of
713:matrices with entries in
1734:(which are semisimple).
1303:One can ask whether the
1249:. As was conjectured by
1149:-modules is semisimple.
698:{\displaystyle M_{n}(D)}
501:Artin–Wedderburn theorem
269:on a finite-dimensional
154:{\displaystyle T:V\to V}
2013:Hall, Brian C. (2015),
1932:Homology Homotopy Appl.
1838:Lam, Tsit-Yuen (2001).
475:, i.e., the case when |
88:(these are also called
1891:Mixed Hodge structures
1745:semisimple Lie algebra
1721:
1697:
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1499:with respect to which
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933:Semi-simple categories
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18:Reduction (philosophy)
2065:Representation theory
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1424:{\displaystyle e_{1}}
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1277:triangulated category
1259:numerical equivalence
1244:
1242:{\displaystyle \sim }
1221:
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1034:
1004:, i.e., ones with no
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658:{\displaystyle D_{i}}
633:
461:representation theory
330:Further information:
230:If one considers all
156:
103:complete reducibility
37:representation theory
1901:; see Corollary 2.12
1707:
1683:
1663:
1643:
1623:
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1532:{\displaystyle \Pi }
1523:
1512:{\displaystyle \Pi }
1503:
1479:
1468:{\displaystyle \Pi }
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1337:
1312:
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1182:projective varieties
1069:
1016:
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795:short exact sequence
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447:is semi-simple and |
308:algebraically closed
262:or, equivalently, a
254:Semi-simple matrices
186:algebraically closed
133:
2038:Semisimple category
1870:additive categories
1156:is the category of
900:homological algebra
782:inside the ring of
459:is the same as the
451:| is invertible in
303:being square-free.
2075:Algebraic geometry
1797:Hoffman, Kenneth;
1758:Semisimple algebra
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413:finitely generated
297:minimal polynomial
286:invariant subspace
194:semi-simple matrix
184:. If the field is
151:
126:(in other words a
49:semi-simple object
45:algebraic geometry
1899:978-3-540-77015-2
1672:{\displaystyle K}
1652:{\displaystyle K}
1632:{\displaystyle K}
1488:{\displaystyle G}
1444:{\displaystyle G}
1295:Maschke's theorem
1165:positive definite
1158:polarizable pure
1141:Moreover, a ring
1061:endomorphism ring
432:asserts that the
430:Maschke's theorem
332:Semisimple module
316:diagonalizability
94:Maschke's theorem
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1152:An example from
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405:semi-simple ring
174:linear subspaces
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58:For example, if
33:abstract algebra
23:In mathematics,
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1732:Fusion category
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439:over some ring
347:, a nontrivial
338:
336:Semisimple ring
328:
264:linear operator
256:
228:
196:is one that is
190:complex numbers
131:
130:
128:linear operator
119:is semisimple.
105:. For example,
41:category theory
25:semi-simplicity
21:
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2023:External links
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2020:
2019:
2007:
2006:
1994:
1982:
1970:
1958:
1938:(1): 239–261,
1930:-structures",
1918:
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1805:Linear algebra
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1287:Main article:
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238:, such as the
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206:diagonalizable
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1851:0-387-95183-0
1847:
1843:
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1816:9780135367971
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1591:: Every such
1590:
1498:
1497:inner product
1482:
1454:
1438:
1416:
1412:
1386:
1380:
1375:
1368:
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1260:
1256:
1253:and shown by
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1236:
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1203:
1197:
1194:
1187:
1184:over a field
1183:
1180:
1176:
1175:
1169:
1168:bilinear form
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1098:
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1077:
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1062:
1058:
1057:Schur's lemma
1054:
1050:
1046:
1042:
1024:
1020:
1011:
1007:
991:
988:
983:
979:
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953:-modules and
952:
948:
945:. Briefly, a
944:
940:
932:
930:
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917:
913:
909:
905:
901:
884:
881:
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840:
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830:
823:
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813:
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784:endomorphisms
781:
762:
756:
751:
747:
743:
737:
731:
723:
718:
716:
712:
708:
689:
681:
677:
668:
667:division ring
650:
646:
638:, where each
620:
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586:
578:
574:
563:
559:
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550:
542:
538:
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509:
506:
505:Artinian ring
502:
498:
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346:
343:
337:
333:
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323:
321:
317:
313:
309:
304:
302:
298:
294:
291:
290:complementary
287:
283:
279:
275:
272:
268:
265:
261:
260:square matrix
253:
251:
249:
245:
241:
237:
233:
232:vector spaces
225:
223:
221:
217:
214:
209:
207:
203:
199:
195:
191:
188:(such as the
187:
183:
179:
175:
172:
168:
164:
148:
142:
139:
136:
129:
125:
124:square matrix
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118:
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108:
104:
99:
95:
91:
87:
83:
79:
75:
72:
68:
65:
61:
56:
54:
50:
46:
42:
38:
34:
30:
26:
19:
2042:
2029:MathOverflow
2014:
1997:
1992:Theorem 10.9
1985:
1980:Theorem 4.28
1973:
1968:Example 4.25
1961:
1935:
1931:
1927:
1921:
1911:
1906:
1890:
1885:
1873:
1861:
1840:
1833:
1804:
1792:
1784:Lam (2001),
1729:
1403:
1302:
1266:
1263:
1251:Grothendieck
1185:
1171:
1157:
1154:Hodge theory
1151:
1146:
1142:
1140:
1137:
1044:
1040:
968:
963:
958:
954:
950:
942:
938:
936:
926:
922:
918:
915:
911:
907:
855:
792:
787:
779:
721:
720:An operator
719:
714:
710:
706:
507:
496:
492:
488:
480:
476:
468:
464:
456:
452:
448:
444:
440:
436:
426:
424:
419:
415:
408:
403:is called a
400:
396:
392:
388:
384:
380:
376:
372:
364:
360:
356:
352:
348:
344:
340:For a fixed
339:
311:
305:
300:
292:
281:
277:
273:
271:vector space
266:
257:
240:real numbers
229:
210:
193:
181:
180:are {0} and
177:
169:if its only
166:
162:
121:
102:
85:
81:
73:
59:
57:
52:
48:
24:
22:
2070:Ring theory
2004:Theorem 5.6
1010:zero object
939:semi-simple
906:. The ring
503:, a unital
483:divide the
369:semi-simple
278:semi-simple
244:dimensional
213:semi-simple
2054:Categories
1954:1251.18006
1799:Kunze, Ray
1769:References
1730:See also:
1495:admits an
1293:See also:
1269:-structure
1226:modulo an
1049:direct sum
904:extensions
434:group ring
320:hyperplane
276:is called
248:direct sum
220:categories
111:semisimple
2002:Hall 2015
1990:Hall 2015
1978:Hall 2015
1966:Hall 2015
1527:Π
1507:Π
1463:Π
1341:Π
1330:given by
1237:∼
1212:∼
1198:
1108:
1083:
1059:that the
1053:coproduct
1025:α
1006:subobject
989:∈
984:α
941:category
878:⊕
867:≅
838:→
827:→
821:→
810:→
757:
744:⊆
593:×
590:⋯
587:×
551:×
371:if every
280:if every
171:invariant
146:→
117:Lie group
1738:See also
1639:. Since
947:category
885:″
874:′
834:″
817:′
418:-module
391:-module
363:-module
351:-module
234:(over a
2040:at the
1825:0276251
1679:and of
1453:compact
1255:Jannsen
1174:motives
1051:(i.e.,
1047:is the
1043:object
216:modules
198:similar
114:compact
92:). Now
76:over a
1952:
1897:
1848:
1823:
1813:
1179:smooth
1012:0 and
379:is an
310:field
288:has a
176:under
167:simple
82:simple
64:finite
53:simple
43:, and
1786:p. 39
1275:on a
1172:pure
665:is a
359:. An
236:field
200:to a
161:with
78:field
67:group
62:is a
1895:ISBN
1846:ISBN
1811:ISBN
1297:and
921:and
709:-by-
669:and
481:does
342:ring
334:and
47:. A
2045:Lab
1950:Zbl
1940:doi
1475:of
1451:is
1195:Mot
1177:of
1099:Hom
1074:End
1041:any
964:An
786:of
748:End
487:of
467:on
463:of
367:is
299:of
2056::
1948:,
1936:14
1934:,
1821:MR
1819:.
1777:^
1750:A
1743:A
1279:.
961:.
929:.
717:.
479:|
395:,
258:A
222:.
208:.
122:A
39:,
35:,
31:,
2043:n
2031::
1942::
1928:t
1880:.
1854:.
1827:.
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1689:g
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1381:1
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1364:1
1358:(
1353:=
1350:)
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1344:(
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1204:k
1201:(
1186:k
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1120:X
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1114:X
1111:(
1103:C
1095:=
1092:)
1089:X
1086:(
1078:C
1045:X
1021:X
992:C
980:X
969:C
959:R
955:R
951:R
943:C
927:n
925:/
923:Z
919:Z
916:n
912:Z
908:Z
882:M
871:M
864:M
841:0
831:M
824:M
814:M
807:0
788:V
780:T
766:)
763:V
760:(
752:F
741:]
738:T
735:[
732:F
722:T
715:D
711:n
707:n
693:)
690:D
687:(
682:n
678:M
651:i
647:D
626:)
621:r
617:D
613:(
606:r
602:n
597:M
584:)
579:2
575:D
571:(
564:2
560:n
555:M
548:)
543:1
539:D
535:(
528:1
524:n
519:M
508:R
497:R
493:G
489:R
477:G
469:R
465:G
457:R
453:R
449:G
445:R
441:R
437:R
427:G
420:M
416:R
409:R
401:R
397:M
393:M
389:R
385:M
381:R
377:M
373:R
365:M
361:R
357:M
353:M
349:R
345:R
312:F
301:T
293:T
284:-
282:T
274:V
267:T
182:V
178:T
163:V
149:V
143:V
140::
137:T
86:V
74:V
60:G
20:.
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