135:... it is perhaps surprising that the original impetus for our work did not come from the theory of hereditary artin algebras or those stably equivalent to hereditary artin algebras. Rather, the research came from an effort to explain a much older result of Gabriel and Roiter ... concerning artin algebras of finite representation type in terms of the technics and ideas developed by Auslander and Reiten in connection with almost split sequences and irreducible morphisms ...
155:, an important class of matrix problems with many applications in mathematics, such as the representation theory of finite dimensional algebras. (In 2005 they with M. N. Smirnova proved a theorem about antimonotone quadratic forms and partially ordered sets.) Also in the 1970s Roiter in three papers, two of which were joint work with Mark Kleiner, introduced representations of bocses, a very large class of matrix problems.
127:
and Sverre Olaf Smalø in a 1980 paper. Auslander and Smalø's paper and its follow-ups by several researchers introduced, among other things, covariantly and contravariantly finite subcategories of the category of finitely generated modules over an Artin algebra, which led to the theory of almost
178:
directed multigraphs) in
Hilbert spaces. In their 2006 paper they constructed for such representations Coxeter functors analogous to Bernstein-Gelfand-Ponomarev functors and applied the new functors to the study of locally scalar representations. In particular, they proved that a graph has only
186:
In 1961 Roiter started in Kyiv a seminar on the theory of representations. The seminar became the foundation of the highly esteemed Kyiv school of the representation theory. He was the supervisor for 13 Candidate of
Sciences degrees (PhDs). In 2007 A. V. Roiter was posthumously awarded the
147:
and
Vladimir V. Kirichenko on hereditary and Bass orders and the Drozd-Roiter criterion for a commutative order to have finitely many non-isomorphic indecomposable representations. An important tool in this research was his theory of divisibility of modules.
122:
Roiter proved the first Brauer-Thrall conjecture for finite-dimensional algebras; his paper never mentioned Artin algebras, but his techniques work for Artin algebras as well. There is an important line of research inspired by the paper and started by
429:"A. V. Roiter, "Unbounded dimensionality of indecomposable representations of an algebra with an infinite number of indecomposable representations", Izv. Akad. Nauk SSSR Ser. Mat., 32:6 (1968), 1275–1282; Math. USSR-Izv., 2:6 (1968), 1223–1230"
166:), published by Springer in 1992 in English translation, is important for its influence on the theory of representations of finite-dimensional algebras and the theory of matrix problems. There is a 1997 reprint of the English translation.
169:
In the years shortly before his death, Roiter did research on representations in
Hilbert spaces. In two papers, he with his wife and Stanislav A. Kruglyak introduced the notion of locally scalar representations of
119:
In a 1966 paper he proved an important theorem in the theory of the integral representation of rings. In a famous 1968 paper he proved the first Brauer-Thrall conjecture.
1189:
188:
1179:
1174:
98:
42:
80:
In his first published paper, Roiter in 1960 proved an important result that eventually led several other mathematicians to establish that a finite group
74:
1194:
41:
A. V. Roiter's father was the
Ukrainian physical chemist V. A. Roiter, a leading expert on catalysis. In 1955 Andrei V. Roiter matriculated at
885:
547:
748:
989:
428:
282:
951:
388:
65:, who also supervised Ludmila Nazarova's PhD. A. V. Roiter was hired in 1961 as a researcher at the Institute of Mathematics of the
57:). They married and began a lifelong collaboration on representation theory. He received in 1960 his Diploma (M.S.) and in 1963 his
50:
66:
62:
325:
310:
666:
Drozd, Yu. A.; Roĭter, A V (1967). "Commutative rings with a finite number of indecomposable integral representations".
523:
1209:
563:
Roiter, A. V. (1966). "An analogue of the theorem of Bass for modules of representations of non-commutative orders".
54:
1023:
Kruglyak, S. A.; Roiter, A. V. (2005). "Locally Scalar Graph
Representations in the Category of Hilbert Spaces".
1011:
Kruglyak, S.A.; Roiter, A. V. (2005). "Locally scalar graph representations in the category of
Hilbert spaces".
69:, where he worked until his death in 2006 and since 1991 was Head of the Department of Algebra. He received his
179:
finitely many indecomposable locally scalar representations (up to unitary isomorphism) if and only if it is a
728:
Roiter, A. V. (1965). "Divisibility in the category of representations over a complete local
Dedekind ring".
352:
Roiter, A. V. (1960). "On the representations of the cyclic group of fourth order by integral matrices".
1066:
Bernstein, I. N.; Gelfand, I. M.; Ponomarev, V. A. (1973). "Coxeter functors and a theorem of
Gabriel".
436:
152:
100:
has finitely many non-isomorphic indecomposable integral representations if and only if, for each prime
58:
844:
Kleiner, M. M.; Roiter, A. V. (1977). "Representations of differential graded categories. (Russian)".
1204:
1199:
1184:
675:
619:
591:
171:
1150:
Külshammer, Julian (2016). "In the bocs seat: Quasi-hereditary algebras and representation type".
1151:
1128:
1102:
1048:
968:
922:
646:
464:
229:
70:
1119:
Conde, Teresa (2020). "Gabriel–Roiter
Measure, Representation Dimension and Rejective Chains".
381:
Representation Theory, Group Rings, and Coding Theory: Papers in Honor of SD Berman (1922-1987)
1040:
947:
881:
880:. Encyclopedia of Mathematical Sciences. Vol. 73. Springer Science & Business Media.
793:
691:
543:
384:
371:
Isaacs, M.; Lichtman, A.; Passman, D.; Sehgal, S.; Sloane, N. J. A.; Zassenhaus, Hans (1989).
124:
46:
967:
Roiter, A. V.; Kruglyak, S. A.; Nazarova, L. A. (2006). "Matrix Problems in Hilbert Spaces".
372:
1138:
1094:
1032:
914:
783:
683:
627:
599:
503:
295:
281:
Drozd, Yu.; Kirichenko, V.; Krugliak, S.; Kleiner, M.; Bondarenko, V.; Ovsienko, S. (2012).
221:
829:
Roiter, A. V.; Kleiner, M. M. (1975). "Representations of differential graded categories".
517:
183:. Their result is analogous to that of Gabriel for the “usual” representations of quivers.
811:
513:
449:
163:
679:
623:
595:
831:
Representations of algebras (Proc. Internat. Conf., Carleton Univ., Ottawa, Ont., 1974)
610:
Drozd, Ju A.; Kiričenko, V. V.; Roĭter, A. V. (1967). "On hereditary and Bass orders".
159:
105:
83:
1168:
1106:
1052:
582:
Drozd, Y. A.; Kirichenko, V. V.; Roiter, V. A. (1967). "Hereditary and Bass orders".
508:
233:
687:
631:
603:
522:(Note: the word "technic" is a jargon term sometimes used by algebraists working in
375:
SD Herman's Contribution to the Theory of Integral Representations of Finite Groups
180:
788:
771:
941:
875:
859:
Roiter, A. V. (1979). "Matrix problems and representations of BOCSs. (Russian)".
537:
144:
709:
Roiter, A. V. (1963). "Categories with division and integral representations".
488:
404:
Roiter, A. V. (1966). "Integer-valued representations belonging to one genus".
1036:
225:
1044:
797:
695:
645:
Yang, Tse-Chung; Yu, Chia-Fu (2013). "Monomial, Gorenstein and Bass Orders".
1142:
321:
306:
833:. Lecture Notes in Math. Vol. 488. Berlin: Springer. pp. 316–339.
339:
Proceedings of the International Congress of Mathematicians, 1978, Helsinki
990:"Orthoscalar representations of quivers in the category of Hilbert spaces"
863:. Vol. 154. Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev. pp. 3–38.
73:
degree (habilitation) in 1969. In 1978 he was an invited speaker at the
1098:
926:
973:
536:
Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (21 August 1997).
31:
918:
209:
1156:
1133:
651:
469:
38:, Latvia) was a Ukrainian mathematician, specializing in algebra.
35:
463:
Krause, Henning (2011). "Notes on the Gabriel-Roiter measure".
903:
Algebra VIII. Representations of Finite-Dimensional Algebras
151:
In 1972 Nazarova and Roiter introduced representations of
848:. Akad. Nauk Ukrain. SSR Inst. Mat., Kiev. pp. 5–70.
772:"Antimonotone quadratic forms and partially ordered sets"
988:
Kruglyak, S. A.; Nazarova, L. A.; Roiter, A. V. (2006).
770:
Nazarova, L. A.; Roĭter, A. V.; Smirnova, M. N. (2006).
423:
421:
419:
940:
Gabriel, Peter; Roiter, Andrei V. (12 September 1997).
383:. Vol. 93. American Mathematical Soc. p. 27.
283:"Andrei Vladimirovich Roiter. To the 75th anniversary"
143:-adic representations, especially his 1967 paper with
1085:
Gabriel, P. (1972). "Unzerlegbare Darstellungen I.".
86:
874:
Gabriel, Peter; Roiter, Andrei V. (8 October 1992).
92:
30:: Андрій Володимирович Ройтер, November 30, 1937,
482:
480:
189:State Prize of Ukraine in Science and Technology
749:"Representations of the partially ordered sets"
238:(with list of Roiter's publications; 67 titles)
133:
61:degree (PhD). His PhD thesis was supervised by
943:Representations of Finite-Dimensional Algebras
877:Representations of Finite-Dimensional Algebras
210:"To the memory of Andrei Vladimirovich Roiter"
861:Representations and quadratic forms (Russian)
8:
43:Taras Shevchenko National University of Kyiv
489:"Preprojective modules over Artin algebras"
191:for his research on representation theory.
296:"In Memory of Andrei Vladimirovich Roiter"
45:, where he met a fellow mathematics major
1155:
1132:
972:
901:Denton, Brian H. (1993). "Reviewed work:
787:
650:
507:
468:
85:
49:. In 1958 he and Nazarova transferred to
1190:Saint Petersburg State University alumni
1025:Functional Analysis and Its Applications
812:"Mark Kleiner, Professor of Mathematics"
75:International Congress of Mathematicians
747:Nazarova, L. A.; Roiter, A. V. (1972).
539:Representation Theory of Artin Algebras
200:
445:
434:
276:
274:
272:
270:
268:
266:
264:
262:
260:
258:
1180:21st-century Ukrainian mathematicians
1175:20th-century Ukrainian mathematicians
7:
1121:The Quarterly Journal of Mathematics
487:Auslander, M.; Smalø, S. O. (1980).
776:St. Petersburg Mathematical Journal
131:According to Auslander and Smalø:
128:split sequences in subcategories.
14:
668:Mathematics of the USSR-Izvestiya
337:Roiter, A. V. "Matrix problems".
139:Roiter did important research on
51:Saint Petersburg State University
753:Zapiski Nauchnykh Seminarov POMI
341:. Vol. 1. pp. 319–322.
214:Journal of Mathematical Sciences
688:10.1070/IM1967v001n04ABEH000588
632:10.1070/IM1967v001n06ABEH000625
604:10.1070/IM1967v001n06ABEH000625
542:. Cambridge University Press.
322:Dmitry Konstantinovich Faddeev
67:Academy of Sciences of Ukraine
63:Dmitry Konstantinovich Faddeev
26:: Андрей Владимирович Ройтер;
1:
1195:NASU Institute of Mathematics
789:10.1090/S1061-0022-06-00938-1
584:Izv. Akad. Nauk SSSR Ser. Mat
406:Izv. Akad. Nauk SSSR Ser. Mat
326:Mathematics Genealogy Project
311:Mathematics Genealogy Project
249:V. A. Roiter, Selected Works
816:Faculty, Syracuse University
509:10.1016/0021-8693(80)90113-1
251:. Kiev: Naukova Dumka. 1976.
158:The monograph by Roiter and
112:is cyclic of order at most
20:Andrei Vladimirovich Roiter
1226:
55:Leningrad State University
1037:10.1007/s10688-005-0022-8
846:Matrix problems (Russian)
377:by Alexander I. Lichtman"
226:10.1007/s10958-007-0316-x
994:Zap. Nauchn. Semin. POMI
907:The Mathematical Gazette
208:Yakovlev, A. V. (2007).
162:(with a contribution by
524:Auslander–Reiten theory
354:Vestnik Leningrad. Univ
16:Ukrainian mathematician
444:Cite journal requires
153:partially ordered sets
137:
94:
1143:10.1093/qmathj/haz062
1013:Funkts. Anal. Prilozh
711:Dokl. Akad. Nauk SSSR
565:Dokl. Akad. Nauk SSSR
287:Algebra Discrete Math
95:
59:Candidate of Sciences
612:Math. USSR Izvestiya
84:
680:1967IzMat...1..757D
624:1967IzMat...1.1357D
596:1967IzMat...1.1357D
1210:People from Dnipro
1099:10.1007/BF01298413
1021:Eng. translation:
496:Journal of Algebra
90:
71:Doctor of Sciences
1068:Uspekhi Mat. Nauk
887:978-3-540-53732-8
608:AMS copyrighted:
549:978-0-521-59923-8
125:Maurice Auslander
93:{\displaystyle G}
47:Lyudmyla Nazarova
34:– July 26, 2006,
1217:
1161:
1159:
1146:
1136:
1111:
1110:
1087:Manuscripta Math
1082:
1076:
1075:
1063:
1057:
1056:
1020:
1008:
1002:
1001:
985:
979:
978:
976:
964:
958:
957:
937:
931:
930:
913:(480): 386–387.
898:
892:
891:
871:
865:
864:
856:
850:
849:
841:
835:
834:
826:
820:
819:
808:
802:
801:
791:
782:(6): 1015–1030.
767:
761:
760:
744:
738:
737:
725:
719:
718:
706:
700:
699:
663:
657:
656:
654:
642:
636:
635:
618:(6): 1357–1376.
607:
590:(6): 1415–1436.
579:
573:
572:
560:
554:
553:
533:
527:
521:
511:
493:
484:
475:
474:
472:
460:
454:
453:
447:
442:
440:
432:
425:
414:
413:
401:
395:
394:
368:
362:
361:
349:
343:
342:
334:
328:
319:
313:
307:Andrei V. Roiter
304:
298:
294:
278:
253:
252:
245:
239:
237:
220:(1): 4831–4835.
205:
99:
97:
96:
91:
1225:
1224:
1220:
1219:
1218:
1216:
1215:
1214:
1165:
1164:
1149:
1118:
1115:
1114:
1084:
1083:
1079:
1065:
1064:
1060:
1022:
1010:
1009:
1005:
987:
986:
982:
966:
965:
961:
954:
939:
938:
934:
919:10.2307/3619799
900:
899:
895:
888:
873:
872:
868:
858:
857:
853:
843:
842:
838:
828:
827:
823:
810:
809:
805:
769:
768:
764:
746:
745:
741:
727:
726:
722:
708:
707:
703:
665:
664:
660:
644:
643:
639:
609:
581:
580:
576:
562:
561:
557:
550:
535:
534:
530:
491:
486:
485:
478:
462:
461:
457:
443:
433:
427:
426:
417:
403:
402:
398:
391:
370:
369:
365:
351:
350:
346:
336:
335:
331:
320:
316:
305:
301:
280:
279:
256:
247:
246:
242:
207:
206:
202:
197:
164:Bernhard Keller
82:
81:
17:
12:
11:
5:
1223:
1221:
1213:
1212:
1207:
1202:
1197:
1192:
1187:
1182:
1177:
1167:
1166:
1163:
1162:
1147:
1127:(2): 619–635.
1113:
1112:
1077:
1058:
1003:
980:
959:
952:
932:
893:
886:
866:
851:
836:
821:
803:
762:
739:
730:Ukrain. Mat. J
720:
701:
674:(4): 757–772.
658:
637:
574:
555:
548:
528:
476:
455:
446:|journal=
415:
396:
389:
363:
344:
329:
314:
299:
254:
240:
199:
198:
196:
193:
89:
15:
13:
10:
9:
6:
4:
3:
2:
1222:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1176:
1173:
1172:
1170:
1158:
1153:
1148:
1144:
1140:
1135:
1130:
1126:
1122:
1117:
1116:
1108:
1104:
1100:
1096:
1092:
1088:
1081:
1078:
1073:
1069:
1062:
1059:
1054:
1050:
1046:
1042:
1038:
1034:
1031:(2): 91–105.
1030:
1026:
1018:
1014:
1007:
1004:
999:
995:
991:
984:
981:
975:
970:
963:
960:
955:
953:9783540629900
949:
945:
944:
936:
933:
928:
924:
920:
916:
912:
908:
904:
897:
894:
889:
883:
879:
878:
870:
867:
862:
855:
852:
847:
840:
837:
832:
825:
822:
817:
813:
807:
804:
799:
795:
790:
785:
781:
777:
773:
766:
763:
758:
754:
750:
743:
740:
736:(4): 124–129.
735:
731:
724:
721:
716:
712:
705:
702:
697:
693:
689:
685:
681:
677:
673:
669:
662:
659:
653:
648:
641:
638:
633:
629:
625:
621:
617:
613:
605:
601:
597:
593:
589:
585:
578:
575:
570:
566:
559:
556:
551:
545:
541:
540:
532:
529:
525:
519:
515:
510:
505:
502:(1): 61–122.
501:
497:
490:
483:
481:
477:
471:
466:
459:
456:
451:
438:
430:
424:
422:
420:
416:
411:
407:
400:
397:
392:
390:9780821850985
386:
382:
378:
376:
367:
364:
359:
355:
348:
345:
340:
333:
330:
327:
323:
318:
315:
312:
308:
303:
300:
297:
292:
288:
284:
277:
275:
273:
271:
269:
267:
265:
263:
261:
259:
255:
250:
244:
241:
235:
231:
227:
223:
219:
215:
211:
204:
201:
194:
192:
190:
184:
182:
177:
173:
167:
165:
161:
156:
154:
149:
146:
142:
136:
132:
129:
126:
120:
117:
115:
111:
109:
103:
87:
78:
77:in Helsinki.
76:
72:
68:
64:
60:
56:
52:
48:
44:
39:
37:
33:
29:
25:
21:
1124:
1120:
1090:
1086:
1080:
1071:
1067:
1061:
1028:
1024:
1016:
1012:
1006:
997:
993:
983:
974:math/0605728
962:
946:. Springer.
942:
935:
910:
906:
902:
896:
876:
869:
860:
854:
845:
839:
830:
824:
815:
806:
779:
775:
765:
756:
752:
742:
733:
729:
723:
714:
710:
704:
671:
667:
661:
640:
615:
611:
587:
583:
577:
571:: 1261–1264.
568:
564:
558:
538:
531:
499:
495:
458:
437:cite journal
412:: 1315–1324.
409:
405:
399:
380:
374:
366:
357:
353:
347:
338:
332:
317:
302:
290:
286:
248:
243:
217:
213:
203:
185:
181:Dynkin graph
175:
168:
157:
150:
140:
138:
134:
130:
121:
118:
113:
107:
101:
79:
53:(then named
40:
27:
23:
19:
18:
1205:2006 deaths
1200:1937 births
1185:Algebraists
1019:(2): 13–30.
145:Yuriy Drozd
1169:Categories
1157:1601.03899
1134:1903.05555
1093:: 71–103.
1000:: 180–201.
195:References
160:P. Gabriel
1107:119425731
1053:121930940
1045:0016-2663
798:1061-0022
696:0025-5726
652:1308.6017
470:1107.2631
293:(2): C–H.
234:123095732
110:-subgroup
28:Ukrainian
1074:: 19–33.
717:: 46–48.
360:: 65–74.
927:3619799
759:: 5–31.
676:Bibcode
620:Bibcode
592:Bibcode
518:0591246
324:at the
309:at the
172:quivers
24:Russian
1105:
1051:
1043:
950:
925:
884:
796:
694:
546:
516:
387:
232:
106:Sylow
104:, its
32:Dnipro
1152:arXiv
1129:arXiv
1103:S2CID
1049:S2CID
969:arXiv
923:JSTOR
647:arXiv
492:(PDF)
465:arXiv
230:S2CID
1041:ISSN
948:ISBN
882:ISBN
794:ISSN
692:ISSN
544:ISBN
450:help
385:ISBN
176:i.e.
36:Riga
1139:doi
1095:doi
1033:doi
998:338
915:doi
905:".
784:doi
715:153
684:doi
628:doi
600:doi
569:168
504:doi
222:doi
218:145
1171::
1137:.
1125:71
1123:.
1101:.
1089:.
1072:28
1070:.
1047:.
1039:.
1029:39
1027:.
1017:39
1015:.
996:.
992:.
921:.
911:77
909:.
814:.
792:.
780:17
778:.
774:.
757:28
755:.
751:.
734:17
732:.
713:.
690:.
682:.
670:.
626:.
614:.
598:.
588:31
586:.
567:.
526:.)
514:MR
512:.
500:66
498:.
494:.
479:^
441::
439:}}
435:{{
418:^
410:30
408:.
379:.
358:15
356:.
291:14
289:.
285:.
257:^
228:.
216:.
212:.
116:.
1160:.
1154::
1145:.
1141::
1131::
1109:.
1097::
1091:6
1055:.
1035::
977:.
971::
956:.
929:.
917::
890:.
818:.
800:.
786::
698:.
686::
678::
672:1
655:.
649::
634:.
630::
622::
616:1
606:.
602::
594::
552:.
520:.
506::
473:.
467::
452:)
448:(
431:.
393:.
373:"
236:.
224::
174:(
141:p
114:p
108:p
102:p
88:G
22:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.