652:
515:
115:
253:
187:
797:
552:
368:
335:
587:
802:
773:
683:
Kolchin, E. R. (1948), "Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations",
598:
445:
403:
676:
671:
414:
259:
289:
522:
76:
518:
417:
60:
32:
666:
215:
792:
757:
685:
28:
121:
64:
145:
67:
204:
therefore acts through a one-dimensional representation. This is equivalent to the statement that
710:
42:
769:
702:
590:
528:
344:
311:
761:
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694:
557:
387:
37:
722:
730:
718:
53:
741:
440:
410:
391:
57:
786:
278:
262:
finite-dimensional representation of a connected and solvable linear algebraic group
762:"10. Nilpotent and Solvable Groups §10.2 The Lie-Kolchin Triangularization Theorem"
125:
266:
has dimension one. In fact, this is another way to state the Lie–Kolchin theorem.
517:
of absolute value one is a one-dimensional commutative (and therefore solvable)
436:
20:
737:
270:
706:
521:
over the real numbers which has a two-dimensional representation into the
768:, Graduate texts in mathematics, vol. 66, Springer, pp. 74–75,
304:
because by induction it implies that with respect to a suitable basis of
714:
698:
647:{\displaystyle {\begin{pmatrix}x&y\\-y&x\end{pmatrix}}.}
435:
is not algebraically closed, the theorem can fail. The standard
16:
Theorem in the representation theory of linear algebraic groups
510:{\displaystyle \{x+iy\in \mathbb {C} \mid x^{2}+y^{2}=1\}}
525:
SO(2) without an invariant (real) line. Here the image
607:
601:
560:
531:
448:
347:
314:
218:
148:
79:
212:
that is a common (simultaneous) eigenvector for all
742:"Theorie der Transformationsgruppen. Abhandlung II"
646:
581:
546:
509:
362:
329:
247:
181:
131:, then there is a one-dimensional linear subspace
109:
300:Sometimes the theorem is also referred to as the
8:
504:
449:
269:The result for Lie algebras was proved by
798:Representation theory of algebraic groups
602:
600:
559:
530:
492:
479:
468:
467:
447:
346:
313:
277:) and for algebraic groups was proved by
235:
234:
217:
147:
78:
746:Archiv for Mathematik og Naturvidenskab
409:The theorem applies in particular to a
282:
110:{\displaystyle \rho \colon G\to GL(V)}
302:Lie–Kolchin triangularization theorem
292:generalizes the Lie–Kolchin theorem.
7:
766:Introduction to Affine Group Schemes
274:
248:{\displaystyle \rho (g),\,\,g\in G}
386:) to a subgroup of the group T of
341:; in other words, the image group
14:
803:Theorems in representation theory
124:on a nonzero finite-dimensional
404:simultaneously triangularizable
258:It follows directly that every
541:
535:
357:
351:
324:
318:
228:
222:
182:{\displaystyle \rho (G)(L)=L.}
167:
161:
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104:
98:
89:
1:
665:Gorbatsevich, V.V. (2001) ,
672:Encyclopedia of Mathematics
819:
208:contains a nonzero vector
290:Borel fixed point theorem
547:{\displaystyle \rho (z)}
523:special orthogonal group
363:{\displaystyle \rho (G)}
330:{\displaystyle \rho (G)}
196:) has an invariant line
439:, viewed as the set of
390:matrices, the standard
33:linear algebraic groups
758:Waterhouse, William C.
648:
583:
582:{\displaystyle z=x+iy}
548:
519:linear algebraic group
511:
418:linear algebraic group
364:
331:
249:
183:
111:
61:linear algebraic group
686:Annals of Mathematics
667:"Lie-Kolchin theorem"
649:
584:
549:
512:
365:
332:
250:
184:
112:
29:representation theory
599:
558:
529:
446:
345:
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216:
146:
77:
65:algebraically closed
27:is a theorem in the
370:is conjugate in GL(
43:linear Lie algebras
25:Lie–Kolchin theorem
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635:
579:
544:
507:
360:
327:
245:
179:
107:
48:It states that if
41:is the analog for
775:978-1-4612-6217-6
689:, Second Series,
591:orthogonal matrix
296:Triangularization
279:Ellis Kolchin
810:
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402:): the image is
388:upper triangular
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339:triangular shape
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188:
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116:
114:
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108:
63:defined over an
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776:
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736:
699:10.2307/1969111
682:
664:
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619:
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603:
597:
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556:
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527:
526:
488:
475:
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443:
441:complex numbers
429:
427:Counter-example
343:
342:
310:
309:
298:
285:, p.19).
214:
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17:
12:
11:
5:
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491:
487:
482:
478:
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451:
428:
425:
411:Borel subgroup
392:Borel subgroup
359:
356:
353:
350:
326:
323:
320:
317:
297:
294:
271:Sophus Lie
244:
241:
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233:
230:
227:
224:
221:
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189:
178:
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169:
166:
163:
160:
157:
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122:representation
118:
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15:
13:
10:
9:
6:
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747:
743:
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724:
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604:
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573:
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538:
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524:
520:
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498:
493:
489:
485:
480:
476:
472:
464:
461:
458:
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452:
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438:
434:
431:If the field
426:
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134:
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38:Lie's theorem
34:
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26:
22:
793:Lie algebras
765:
749:
745:
690:
684:
670:
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301:
299:
287:
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193:
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126:vector space
119:
49:
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738:Lie, Sophus
693:(1): 1–42,
437:unit circle
260:irreducible
200:, on which
192:That is, ρ(
21:mathematics
787:Categories
731:0037.18701
658:References
415:semisimple
308:the image
139:such that
760:(2012) ,
752:: 152–193
707:0003-486X
677:EMS Press
623:−
533:ρ
473:∣
465:∈
378:) (where
349:ρ
316:ρ
240:∈
220:ρ
150:ρ
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81:ρ
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740:(1876),
58:solvable
723:0024884
715:1969111
589:is the
281: (
273: (
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721:
713:
705:
394:of GL(
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337:has a
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711:JSTOR
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52:is a
770:ISBN
703:ISSN
288:The
283:1948
275:1876
70:and
56:and
727:Zbl
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554:of
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