73:. Supermodules often play a more prominent role in super linear algebra than do super vector spaces. These reason is that it is often necessary or useful to extend the field of scalars to include odd variables. In doing so one moves from fields to commutative superalgebras and from vector spaces to modules.
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1285:{\displaystyle {\begin{aligned}(a\cdot \phi )(x)&=a\cdot \phi (x)\\(\phi \cdot a)(x)&=\phi (a\cdot x).\end{aligned}}}
647:
1136:
That is, the even homomorphisms are both right and left linear whereas the odd homomorphism are right linear but left
1490:
1467:
1448:
911:
580:
407:
705:
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193:
794:
In many cases, it is necessary or convenient to consider a larger class of morphisms between supermodules. Let
1514:
1412:
137:
1126:{\displaystyle \phi (x\cdot a)=\phi (x)\cdot a\qquad \phi (a\cdot x)=(-1)^{|a||\phi |}a\cdot \phi (x).}
541:
1415:
59:
1509:
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1419:
44:
8:
520:
67:
52:
63:
1486:
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394:
1400:
1439:; John W. Morgan (1999). "Notes on Supersymmetry (following Joseph Bernstein)".
1436:
83:
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32:
389:
whose scalar multiplications respect the gradings in the obvious manner. If
516:
113:
101:
40:
378:
79:
17:
1137:
1311:
whose even part is the set of all ordinary supermodule homomorphisms
802:
be regarded as superbimodules in a natural fashion. For supermodules
36:
1411:
with supermodule homomorphisms as the morphisms. This category is a
850:) where the even homomorphisms are those that preserve the grading
382:
1443:. Vol. 1. American Mathematical Society. pp. 41โ97.
905:
and the odd homomorphisms are those that reverse the grading
1389:{\displaystyle \mathbf {Hom} _{0}(E,F)=\mathrm {Hom} (E,F).}
47:
which is a mathematical framework for studying the concept
798:
be a supercommutative algebra. Then all supermodules over
507:
is purely even this reduces to the ordinary definition.
1441:
Quantum Fields and
Strings: A Course for Mathematicians
1435:
1320:
1168:
998:
914:
859:
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544:
410:
196:
140:
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958:
894:
826:A-linear maps (i.e. all module homomorphisms from
744:
693:
635:
563:
476:
238:
172:
78:In this article, all superalgebras are assumed be
1479:Supersymmetry for Mathematicians: An Introduction
694:{\displaystyle \phi (x\cdot a)=\phi (x)\cdot a\,}
1501:
1155:) can be given the structure of a bimodule over
959:{\displaystyle \phi (E_{i})\subseteq F_{1-i}.}
636:{\displaystyle \phi (x+y)=\phi (x)+\phi (y)\,}
477:{\displaystyle a\cdot x=(-1)^{|a||x|}x\cdot a}
401:may be regarded as a superbimodule by setting
745:{\displaystyle \phi (E_{i})\subseteq F_{i}\,}
1140:(with respect to the grading automorphism).
397:, then every left or right supermodule over
1476:
895:{\displaystyle \phi (E_{i})\subseteq F_{i}}
775:. The set of all module homomorphisms from
308:|, is 0 or 1 according to whether it is in
239:{\displaystyle E_{i}A_{j}\subseteq E_{i+j}}
838:-modules). There is a natural grading on
741:
690:
632:
560:
1462:((2nd ed.) ed.). Berlin: Springer.
183:such that multiplication by elements of
1481:. Courant Lecture Notes in Mathematics
1460:Gauge Field Theory and Complex Geometry
1502:
322:. Elements of parity 0 are said to be
1457:
1418:under the super tensor product whose
1403:, the class of all supermodules over
91:
62:can be viewed as generalizations of
173:{\displaystyle E=E_{0}\oplus E_{1}}
13:
1364:
1361:
1358:
574:is a supermodule homomorphism if
14:
1526:
1485:. American Mathematical Society.
503:, and extending by linearity. If
1329:
1326:
1323:
523:that preserves the grading. Let
510:
1038:
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564:{\displaystyle \phi :E\to F\,}
554:
459:
451:
446:
438:
433:
423:
1:
1429:
1307:) becomes a supermodule over
834:considered as ungraded right
338:is a homogeneous element of
334:is a homogeneous scalar and
326:and those of parity 1 to be
7:
1477:Varadarajan, V. S. (2004).
531:be right supermodules over
10:
1531:
822:) denote the space of all
519:between supermodules is a
487:for homogeneous elements
300:of a homogeneous element
1416:monoidal closed category
86:unless stated otherwise.
60:commutative superalgebra
43:. Supermodules arise in
1295:With the above grading
1390:
1286:
1127:
960:
896:
746:
695:
637:
565:
478:
350:| is homogeneous and |
240:
174:
1458:Manin, Y. I. (1997).
1391:
1287:
1128:
989:are homogeneous then
961:
897:
747:
696:
638:
566:
479:
241:
175:
127:decomposition (as an
66:over a (purely even)
1515:Super linear algebra
1420:internal Hom functor
1318:
1166:
996:
912:
857:
706:
648:
581:
542:
408:
194:
138:
58:Supermodules over a
45:super linear algebra
1399:In the language of
521:module homomorphism
64:super vector spaces
53:theoretical physics
1386:
1282:
1280:
1123:
956:
892:
783:is denoted by Hom(
742:
691:
633:
561:
474:
236:
170:
371:left supermodules
106:right supermodule
92:Formal definition
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395:supercommutative
283:The elements of
264:. The subgroups
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1437:Deligne, Pierre
1432:
1401:category theory
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409:
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377:are defined as
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292:are said to be
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279:
273:are then right
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375:superbimodules
319:
312:
304:, denoted by |
287:
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28:
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3:
2:
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1511:
1510:Module theory
1508:
1507:
1505:
1494:
1492:0-8218-3574-2
1488:
1484:
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1471:
1469:3-540-61378-1
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1450:0-8218-2012-5
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976:
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948:
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837:
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792:
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786:
782:
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766:
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709:
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687:
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672:
669:
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537:
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530:
526:
522:
518:
511:Homomorphisms
508:
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231:
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190:
189:
188:
186:
165:
161:
157:
152:
148:
144:
141:
134:
133:
132:
130:
129:abelian group
126:
122:
118:
115:
111:
107:
103:
99:
87:
85:
81:
76:
75:
74:
72:
69:
65:
61:
56:
54:
50:
49:supersymmetry
46:
42:
38:
34:
33:graded module
27:
23:
19:
1482:
1478:
1459:
1440:
1423:
1422:is given by
1404:
1398:
1308:
1304:
1300:
1296:
1294:
1156:
1152:
1148:
1144:
1142:
1135:
986:
982:
978:
974:
970:
968:
904:
847:
843:
839:
835:
831:
827:
823:
819:
815:
811:
807:
803:
799:
795:
793:
788:
784:
780:
776:
772:
768:
764:
760:
756:
754:
573:
532:
528:
524:
517:homomorphism
514:
504:
500:
496:
492:
488:
486:
398:
390:
386:
379:left modules
374:
370:
368:
363:
359:
355:
351:
347:
343:
339:
335:
331:
327:
323:
316:
309:
305:
301:
297:
293:
288:
284:
282:
274:
269:
265:
258:
254:
250:
248:
184:
182:
120:
116:
114:right module
109:
105:
102:superalgebra
97:
95:
77:
70:
57:
41:superalgebra
25:
21:
15:
1159:by setting
294:homogeneous
100:be a fixed
80:associative
22:supermodule
18:mathematics
1504:Categories
1430:References
1138:antilinear
369:Likewise,
280:-modules.
187:satisfies
125:direct sum
1413:symmetric
1267:⋅
1258:ϕ
1233:⋅
1230:ϕ
1211:ϕ
1208:⋅
1183:ϕ
1180:⋅
1109:ϕ
1106:⋅
1093:ϕ
1064:−
1049:⋅
1040:ϕ
1033:⋅
1021:ϕ
1009:⋅
1000:ϕ
946:−
935:⊆
916:ϕ
880:⊆
861:ϕ
729:⊆
710:ϕ
685:⋅
673:ϕ
661:⋅
652:ϕ
621:ϕ
606:ϕ
585:ϕ
555:→
546:ϕ
469:⋅
427:−
415:⋅
383:bimodules
218:⊆
158:⊕
37:superring
1409:category
1407:forms a
1143:The set
763:and all
755:for all
535:. A map
249:for all
969:If ฯ โ
123:with a
35:over a
1489:
1466:
1447:
981:) and
810:, let
342:then |
298:parity
296:. The
84:unital
824:right
385:over
362:| + |
358:| = |
330:. If
119:over
112:is a
108:over
68:field
24:is a
1487:ISBN
1464:ISBN
1445:ISBN
806:and
527:and
495:and
373:and
324:even
253:and
104:. A
96:Let
82:and
20:, a
1424:Hom
1297:Hom
1145:Hom
971:Hom
840:Hom
830:to
812:Hom
791:).
779:to
393:is
381:or
366:|.
328:odd
315:or
257:in
51:in
39:or
16:In
1506::
1483:11
1426:.
1303:,
1151:,
985:โ
977:,
846:,
818:,
787:,
515:A
499:โ
491:โ
131:)
55:.
1495:.
1472:.
1453:.
1405:A
1384:.
1381:)
1378:F
1375:,
1372:E
1369:(
1365:m
1362:o
1359:H
1355:=
1352:)
1349:F
1346:,
1343:E
1340:(
1335:0
1330:m
1327:o
1324:H
1309:A
1305:F
1301:E
1299:(
1276:.
1273:)
1270:x
1264:a
1261:(
1255:=
1248:)
1245:x
1242:(
1239:)
1236:a
1227:(
1220:)
1217:x
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1205:a
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1195:)
1192:x
1189:(
1186:)
1177:a
1174:(
1157:A
1153:F
1149:E
1147:(
1121:.
1118:)
1115:x
1112:(
1103:a
1097:|
1089:|
1084:|
1080:a
1076:|
1071:)
1067:1
1061:(
1058:=
1055:)
1052:x
1046:a
1043:(
1036:a
1030:)
1027:x
1024:(
1018:=
1015:)
1012:a
1006:x
1003:(
987:A
983:a
979:F
975:E
973:(
954:.
949:i
943:1
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927:i
923:E
919:(
888:i
884:F
877:)
872:i
868:E
864:(
848:F
844:E
842:(
836:A
832:F
828:E
820:F
816:E
814:(
808:F
804:E
800:A
796:A
789:F
785:E
781:F
777:E
773:E
771:โ
769:y
767:,
765:x
761:A
759:โ
757:a
737:i
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721:i
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713:(
688:a
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679:x
676:(
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664:a
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655:(
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627:y
624:(
618:+
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612:x
609:(
603:=
600:)
597:y
594:+
591:x
588:(
558:F
552:E
549::
533:A
529:F
525:E
505:A
501:E
497:x
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489:a
472:a
466:x
460:|
456:x
452:|
447:|
443:a
439:|
434:)
430:1
424:(
421:=
418:x
412:a
399:A
391:A
387:A
364:a
360:x
356:a
354:ยท
352:x
348:a
346:ยท
344:x
340:E
336:x
332:a
320:1
317:E
313:0
310:E
306:x
302:x
289:i
285:E
278:0
275:A
270:i
266:E
262:2
259:Z
255:j
251:i
232:j
229:+
226:i
222:E
213:j
209:A
203:i
199:E
185:A
166:1
162:E
153:0
149:E
145:=
142:E
121:A
117:E
110:A
98:A
71:K
31:-
29:2
26:Z
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