616:
into even and odd components, then two distinct (but essentially equivalent) sign conventions can be found in the literature. These can be called the "cohomological sign convention" and the "super sign convention". They differ in how the antipode (exchange of two elements) behaves. In the first case,
2276:
2632:
3528:
2305:
One can easily generalize the definition of superalgebras to include superalgebras over a commutative superring. The definition given above is then a specialization to the case where the base ring is purely even.
2040:
2467:
1496:
3573:
1130:
1422:
1788:
2289:
is purely even, this is equivalent to the ordinary ungraded tensor product (except that the result is graded). However, in general, the super tensor product is distinct from the tensor product of
3578:
2819:
2569:
286:
681:
1669:
1608:
896:
2095:
568:
3197:
1901:
208:
3566:
3202:
1189:
1158:
971:
935:
475:. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.
454:
973:
the parity. This is more often seen in physics texts, and requires a parity functor to be judiciously employed to track isomorphisms. Detailed arguments are provided by
3796:
2812:
796:
716:
585:. There are superalgebras that are commutative in the ordinary sense, but not in the superalgebra sense. For this reason, commutative superalgebras are often called
3892:
2926:
841:
This convention is commonly seen in conventional mathematical settings, such as differential geometry and differential topology. The other convention is to take
3644:
3561:
839:
3943:
2564:
816:
764:
744:
3928:
3918:
3379:
2805:
2728:
3424:
3897:
3583:
2931:
3340:
3938:
2941:
2788:
2738:
1955:
3948:
3507:
1053:
together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.
2961:
2956:
3637:
2353:
1433:
2762:
2708:
2778:
4064:
3953:
3677:
3374:
3058:
1069:
3991:
3986:
3121:
2881:
1354:
3923:
3048:
1705:
4226:
3630:
3053:
2946:
3607:
2951:
1250:
3887:
3811:
3971:
3856:
3672:
2993:
3871:
3611:
3075:
2070:
1695:(in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of
3841:
3725:
3471:
3404:
3399:
3362:
2893:
2866:
2836:
235:
3866:
3328:
3323:
3232:
3192:
2908:
2543:
1533:
1039:
480:
125:
623:
80:, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called
2271:{\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(-1)^{|b_{1}||a_{2}|}(a_{1}a_{2}\otimes b_{1}b_{2}).}
3517:
1619:
1549:
1050:
847:
65:
with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
3976:
3826:
3730:
3522:
3512:
3486:
3308:
3303:
3298:
3293:
3251:
3214:
3160:
2886:
2876:
2854:
2523:
498:
3913:
3851:
1814:
167:
4006:
4001:
3755:
3735:
3439:
3419:
3357:
3286:
3065:
3043:
3018:
2918:
1302:
1046:
158:
54:
3791:
3740:
3597:
3491:
3476:
3391:
3318:
3281:
3276:
3266:
3026:
2980:
2050:
148:
81:
1166:
1135:
940:
904:
4011:
3996:
3981:
3750:
3466:
3335:
3246:
3133:
3104:
2828:
121:
62:
35:
394:
4120:
3692:
3687:
3603:
3256:
3242:
3224:
3207:
3187:
3170:
3080:
2903:
1249:. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a
1196:
292:
97:
1015:
may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as
1285:, consisting of all even elements, is closed under multiplication and contains the identity of
1227:). This algebra may be identified with the algebra of endomorphisms of a free supermodule over
4221:
3429:
3165:
3143:
3070:
3031:
3008:
2784:
2774:
2758:
2734:
2704:
2531:
1692:
769:
689:
3821:
3720:
3707:
3347:
3138:
3126:
3109:
3087:
3036:
2988:
2849:
1242:
1056:
1031:
472:
113:
58:
4150:
4115:
3836:
3831:
3453:
3409:
3236:
3180:
3175:
3153:
3114:
2936:
2861:
2627:{\displaystyle {\begin{aligned}\mu &:A\otimes A\to A\\\eta &:R\to A\end{aligned}}}
2519:
1800:
1020:
89:
4180:
821:
4170:
4155:
3801:
3786:
3097:
2696:
1016:
1012:
974:
801:
749:
729:
613:
50:
2057:
as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of
4215:
4195:
4175:
4125:
3760:
3697:
3667:
3653:
3414:
3369:
3352:
2998:
2871:
2797:
2783:. Courant Lecture Notes in Mathematics. Vol. 11. American Mathematical Society.
2724:
1346:
1203:
485:
93:
85:
73:
4200:
4185:
4145:
4135:
4130:
4021:
3963:
3846:
3682:
3313:
3092:
3003:
2898:
1536:
1063:
214:
2680:
4190:
4165:
4160:
4110:
4089:
4054:
4049:
3776:
3545:
3461:
2322:
1246:
464:
317:
77:
31:
2699:; Morgan, J. W. (1999). "Notes on Supersymmetry (following Joseph Bernstein)".
17:
4094:
4084:
4069:
3861:
3781:
3745:
3481:
2750:
2720:
1290:
4140:
4044:
3933:
2844:
1276:
4079:
4074:
4059:
4034:
1327:
484:(or supercommutative algebra) is one which satisfies a graded version of
4039:
321:
2035:{\displaystyle \mathrm {Z} (A)=\{a\in A:=0{\text{ for all }}x\in A\}.}
3816:
3806:
3529:
The
Unreasonable Effectiveness of Mathematics in the Natural Sciences
3148:
2966:
2538:
serving as the unit object. An associative, unital superalgebra over
1038:
is a superalgebra. The exterior algebra is the standard example of a
4029:
1253:
of a Lie superalgebra which is a unital, associative superalgebra.
76:
in theoretical physics. Superalgebras and their representations,
3622:
2462:{\displaystyle r\cdot (xy)=(r\cdot x)y=(-1)^{|r||x|}x(r\cdot y)}
84:. Superalgebras also play an important role in related field of
3626:
3584:
European
Community on Computational Methods in Applied Sciences
2801:
1491:{\displaystyle \mu (x\otimes y)\cdot z=x\cdot \mu (y\otimes z)}
2703:. Vol. 1. American Mathematical Society. pp. 41â97.
954:
918:
3579:
International
Council for Industrial and Applied Mathematics
2733:. Memoirs of the AMS Series. Vol. 711. AMS Bookstore.
1125:{\displaystyle \mathbf {End} (V)\equiv \mathbf {Hom} (V,V)}
1520:. This follows from the associativity of the product in
1417:{\displaystyle \mu :A_{1}\otimes _{A_{0}}A_{1}\to A_{0}}
2701:
Quantum Fields and
Strings: A Course for Mathematicians
2347:
that respects the grading. Bilinearity here means that
1783:{\displaystyle A_{i}=\{x\in A:{\hat {x}}=(-1)^{i}x\}.}
1330:
whose scalar multiplication is just multiplication in
3198:
Numerical methods for ordinary differential equations
2567:
2356:
2098:
1958:
1817:
1708:
1622:
1552:
1436:
1357:
1169:
1138:
1072:
1059:
are superalgebras. They are generally noncommutative.
943:
907:
850:
824:
804:
772:
752:
732:
692:
626:
501:
397:
238:
170:
3574:
Société de Mathématiques
Appliquées et Industrielles
3567:
Japan
Society for Industrial and Applied Mathematics
3203:
Numerical methods for partial differential equations
4103:
4020:
3962:
3906:
3880:
3769:
3706:
3660:
3554:
3538:
3500:
3452:
3390:
3265:
3223:
3017:
2979:
2917:
2835:
2652:
989:may be regarded as a purely even superalgebra over
2626:
2461:
2270:
2034:
1895:
1782:
1663:
1602:
1490:
1416:
1183:
1152:
1124:
965:
929:
890:
833:
810:
790:
758:
738:
710:
675:
562:
448:
280:
202:
2780:Supersymmetry for Mathematicians: An Introduction
2495:Equivalently, one may define a superalgebra over
2730:Graded simple Jordan superalgebras of growth one
1239:and is the internal Hom of above for this space.
357:|, is 0 or 1 according to whether it is in
27:Algebraic structure used in theoretical physics
3562:Society for Industrial and Applied Mathematics
3638:
2813:
2550:-supermodules. That is, a superalgebra is an
8:
3380:Supersymmetric theory of stochastic dynamics
2026:
1976:
1906:on homogeneous elements, extended to all of
1774:
1722:
387:are both homogeneous then so is the product
2664:
295:2, i.e. they are thought of as elements of
281:{\displaystyle A_{i}A_{j}\subseteq A_{i+j}}
3645:
3631:
3623:
3387:
2914:
2820:
2806:
2798:
2301:Generalizations and categorical definition
2089:with a multiplication rule determined by:
1271:be a superalgebra over a commutative ring
2675:
2673:
2568:
2566:
2434:
2426:
2421:
2413:
2412:
2355:
2297:regarded as ordinary, ungraded algebras.
2256:
2246:
2233:
2223:
2209:
2203:
2194:
2189:
2183:
2174:
2173:
2148:
2135:
2119:
2106:
2097:
2012:
1959:
1957:
1880:
1872:
1867:
1859:
1858:
1816:
1765:
1738:
1737:
1713:
1707:
1655:
1642:
1624:
1623:
1621:
1590:
1582:
1581:
1554:
1553:
1551:
1543:. It is given on homogeneous elements by
1435:
1408:
1395:
1383:
1378:
1368:
1356:
1170:
1168:
1139:
1137:
1096:
1073:
1071:
957:
953:
942:
921:
917:
906:
873:
849:
823:
803:
771:
751:
731:
691:
649:
625:
559:
546:
538:
533:
525:
524:
500:
441:
433:
425:
417:
409:
398:
396:
266:
253:
243:
237:
194:
181:
169:
88:where they enter into the definitions of
2757:((2nd ed.) ed.). Berlin: Springer.
2503:together with an superring homomorphism
1945:which supercommute with all elements of
676:{\displaystyle xy\mapsto (-1)^{mn+pq}yx}
2755:Gauge Field Theory and Complex Geometry
2645:
2511:whose image lies in the supercenter of
1199:forms a superalgebra under composition.
2637:for which the usual diagrams commute.
1664:{\displaystyle {\hat {x}}=x_{0}-x_{1}}
1603:{\displaystyle {\hat {x}}=(-1)^{|x|}x}
371:. Elements of parity 0 are said to be
1258:Further definitions and constructions
891:{\displaystyle xy\mapsto (-1)^{pq}yx}
320:, is a superalgebra over the ring of
7:
2534:under the super tensor product with
985:Any algebra over a commutative ring
563:{\displaystyle yx=(-1)^{|x||y|}xy\,}
2049:is, in general, different than the
1896:{\displaystyle =xy-(-1)^{|x||y|}yx}
203:{\displaystyle A=A_{0}\oplus A_{1}}
2829:Industrial and applied mathematics
2518:One may also define superalgebras
2081:may be regarded as a superalgebra
1960:
1177:
1174:
1171:
1146:
1143:
1140:
604:grading arises as a "rollup" of a
25:
3059:Stochastic differential equations
2653:Kac, Martinez & Zelmanov 2001
3678:Supersymmetric quantum mechanics
3375:Supersymmetric quantum mechanics
1808:is the binary operator given by
1210:forms a superalgebra denoted by
1103:
1100:
1097:
1080:
1077:
1074:
3257:Stochastic variational calculus
3049:Ordinary differential equations
1539:on any superalgebra called the
463:is one whose multiplication is
3054:Partial differential equations
2927:Arbitrary-precision arithmetic
2614:
2591:
2456:
2444:
2435:
2427:
2422:
2414:
2409:
2399:
2390:
2378:
2372:
2363:
2313:be a commutative superring. A
2262:
2216:
2210:
2195:
2190:
2175:
2170:
2160:
2154:
2128:
2125:
2099:
2003:
1991:
1970:
1964:
1941:is the set of all elements of
1881:
1873:
1868:
1860:
1855:
1845:
1830:
1818:
1762:
1752:
1743:
1629:
1591:
1583:
1578:
1568:
1559:
1485:
1473:
1452:
1440:
1401:
1184:{\displaystyle \mathrm {Hom} }
1153:{\displaystyle \mathrm {Hom} }
1119:
1107:
1090:
1084:
966:{\displaystyle q=n{\bmod {2}}}
930:{\displaystyle p=m{\bmod {2}}}
870:
860:
857:
646:
636:
633:
547:
539:
534:
526:
521:
511:
442:
434:
426:
418:
410:
399:
291:where the subscripts are read
1:
2942:Interactive geometry software
2472:for all homogeneous elements
1683:are the homogeneous parts of
1613:and on arbitrary elements by
589:in order to avoid confusion.
573:for all homogeneous elements
471:is one with a multiplicative
1312:The set of all odd elements
1251:universal enveloping algebra
449:{\displaystyle |xy|=|x|+|y|}
375:and those of parity 1 to be
330:The elements of each of the
3673:Supersymmetric gauge theory
2994:Computational number theory
2957:Numerical-analysis software
901:with the parities given as
4243:
3972:Pure 4D N = 1 supergravity
2558:with two (even) morphisms
3872:Electricâmagnetic duality
3592:
3400:Algebra of physical space
2867:Automated theorem proving
2542:can then be defined as a
2335:-bilinear multiplication
347:of a homogeneous element
72:comes from the theory of
3893:HaagâĆopuszaĆskiâSohnius
3867:Little hierarchy problem
3193:Numerical linear algebra
1040:supercommutative algebra
791:{\displaystyle n=\deg y}
711:{\displaystyle m=\deg x}
617:one has an exchange map
481:commutative superalgebra
461:associative superalgebra
116:. In most applications,
3949:6D (2,0) superconformal
2932:Finite element analysis
2882:Constraint satisfaction
1910:by linearity. Elements
1301:. It forms an ordinary
1297:, naturally called the
1245:are a graded analog of
1051:alternating polynomials
3929:N = 4 super YangâMills
3919:N = 1 super YangâMills
3827:Supersymmetry breaking
3731:Superconformal algebra
3726:Super-Poincaré algebra
3487:Mathematical economics
3161:Multivariable calculus
3044:Differential equations
2887:Constraint programming
2877:Computational geometry
2628:
2530:-supermodules forms a
2463:
2272:
2036:
1897:
1784:
1665:
1604:
1492:
1418:
1289:and therefore forms a
1202:The set of all square
1185:
1154:
1126:
967:
931:
892:
835:
812:
792:
766:the parity. Likewise,
760:
740:
712:
677:
564:
450:
282:
204:
4007:Type IIB supergravity
4002:Type IIA supergravity
3977:4D N = 1 supergravity
3842:SeibergâWitten theory
3756:Super Minkowski space
3736:Supersymmetry algebra
3440:Supersymmetry algebra
3425:Representation theory
3420:Renormalization group
3066:Differential geometry
2947:Optimization software
2919:Mathematical software
2629:
2464:
2273:
2073:of two superalgebras
2037:
1898:
1785:
1666:
1605:
1532:There is a canonical
1493:
1419:
1186:
1155:
1132:, where the boldface
1127:
1047:symmetric polynomials
993:; that is, by taking
968:
932:
893:
836:
813:
793:
761:
741:
713:
678:
565:
451:
283:
205:
4227:Super linear algebra
3792:Short supermultiplet
3492:Mathematical finance
3477:Social choice theory
3392:Algebraic structures
3341:in quantum mechanics
3277:Analytical mechanics
3243:Stochastic processes
3215:Variational calculus
3027:Approximation theory
2952:Statistical software
2681:Deligne's discussion
2565:
2354:
2096:
2065:Super tensor product
1956:
1815:
1706:
1620:
1550:
1434:
1355:
1167:
1136:
1070:
941:
905:
848:
822:
802:
770:
750:
730:
690:
624:
499:
395:
236:
168:
82:super linear algebra
53:. That is, it is an
4012:Gauged supergravity
3997:Type I supergravity
3954:ABJM superconformal
3751:Harmonic superspace
3467:Operations research
3336:Perturbation theory
3134:Multilinear algebra
3105:Functional analysis
2962:Numerical libraries
2894:Computational logic
2683:of these two cases.
2546:in the category of
2045:The supercenter of
2014: for all
1030:In particular, any
469:unital superalgebra
351:, denoted by |
36:theoretical physics
3987:Higher dimensional
3982:N = 8 supergravity
3898:Nonrenormalization
3693:Super vector space
3688:Superstring theory
3604:Mathematics portal
3501:Other applications
3225:Probability theory
3208:Validated numerics
3188:Numerical analysis
3081:Geometric analysis
3071:Differential forms
2904:Information theory
2775:Varadarajan, V. S.
2624:
2622:
2459:
2268:
2032:
1893:
1794:Supercommutativity
1780:
1661:
1600:
1488:
1414:
1197:super vector space
1195:linear maps) of a
1181:
1160:is referred to as
1150:
1122:
963:
927:
888:
834:{\displaystyle q.}
831:
808:
788:
756:
736:
708:
673:
560:
492:is commutative if
446:
278:
200:
4209:
4208:
3852:WessâZumino gauge
3620:
3619:
3454:Decision sciences
3448:
3447:
3430:Spacetime algebra
3122:Harmonic analysis
3088:Dynamical systems
3032:Clifford analysis
3009:Discrete geometry
2975:
2974:
2790:978-0-8218-3574-6
2740:978-0-8218-2645-4
2532:monoidal category
2015:
1746:
1632:
1562:
1334:. The product in
1243:Lie superalgebras
1057:Clifford algebras
811:{\displaystyle y}
798:is the degree of
759:{\displaystyle p}
739:{\displaystyle x}
104:Formal definition
16:(Redirected from
4234:
3992:11D supergravity
3721:Lie superalgebra
3708:Supermathematics
3647:
3640:
3633:
3624:
3405:Feynman integral
3388:
3348:Potential theory
3237:random variables
3127:Fourier analysis
3110:Operator algebra
3037:Clifford algebra
2989:Computer algebra
2915:
2822:
2815:
2808:
2799:
2794:
2768:
2744:
2723:; Martinez, C.;
2714:
2684:
2677:
2668:
2665:Varadarajan 2004
2662:
2656:
2650:
2633:
2631:
2630:
2625:
2623:
2468:
2466:
2465:
2460:
2440:
2439:
2438:
2430:
2425:
2417:
2277:
2275:
2274:
2269:
2261:
2260:
2251:
2250:
2238:
2237:
2228:
2227:
2215:
2214:
2213:
2208:
2207:
2198:
2193:
2188:
2187:
2178:
2153:
2152:
2140:
2139:
2124:
2123:
2111:
2110:
2041:
2039:
2038:
2033:
2016:
2013:
1963:
1929:
1902:
1900:
1899:
1894:
1886:
1885:
1884:
1876:
1871:
1863:
1789:
1787:
1786:
1781:
1770:
1769:
1748:
1747:
1739:
1718:
1717:
1670:
1668:
1667:
1662:
1660:
1659:
1647:
1646:
1634:
1633:
1625:
1609:
1607:
1606:
1601:
1596:
1595:
1594:
1586:
1564:
1563:
1555:
1541:grade involution
1528:Grade involution
1497:
1495:
1494:
1489:
1423:
1421:
1420:
1415:
1413:
1412:
1400:
1399:
1390:
1389:
1388:
1387:
1373:
1372:
1206:with entries in
1190:
1188:
1187:
1182:
1180:
1159:
1157:
1156:
1151:
1149:
1131:
1129:
1128:
1123:
1106:
1083:
1032:exterior algebra
1021:polynomial rings
972:
970:
969:
964:
962:
961:
936:
934:
933:
928:
926:
925:
897:
895:
894:
889:
881:
880:
840:
838:
837:
832:
818:and with parity
817:
815:
814:
809:
797:
795:
794:
789:
765:
763:
762:
757:
745:
743:
742:
737:
717:
715:
714:
709:
682:
680:
679:
674:
666:
665:
593:Sign conventions
587:supercommutative
569:
567:
566:
561:
552:
551:
550:
542:
537:
529:
488:. Specifically,
473:identity element
455:
453:
452:
447:
445:
437:
429:
421:
413:
402:
356:
287:
285:
284:
279:
277:
276:
258:
257:
248:
247:
213:together with a
209:
207:
206:
201:
199:
198:
186:
185:
114:commutative ring
90:graded manifolds
59:commutative ring
21:
4242:
4241:
4237:
4236:
4235:
4233:
4232:
4231:
4212:
4211:
4210:
4205:
4099:
4016:
3958:
3902:
3888:ColemanâMandula
3876:
3837:Seiberg duality
3832:Konishi anomaly
3765:
3702:
3656:
3651:
3621:
3616:
3588:
3550:
3534:
3496:
3444:
3410:Poisson algebra
3386:
3268:
3261:
3219:
3115:Operator theory
3013:
2971:
2937:Tensor software
2913:
2862:Automata theory
2831:
2826:
2791:
2773:
2765:
2749:
2741:
2719:
2711:
2695:
2692:
2687:
2678:
2671:
2663:
2659:
2651:
2647:
2643:
2621:
2620:
2604:
2598:
2597:
2575:
2563:
2562:
2499:as a superring
2408:
2352:
2351:
2303:
2252:
2242:
2229:
2219:
2199:
2179:
2169:
2144:
2131:
2115:
2102:
2094:
2093:
2067:
1954:
1953:
1927:
1854:
1813:
1812:
1801:supercommutator
1796:
1761:
1709:
1704:
1703:
1682:
1651:
1638:
1618:
1617:
1577:
1548:
1547:
1530:
1519:
1432:
1431:
1404:
1391:
1379:
1374:
1364:
1353:
1352:
1344:
1325:
1318:
1299:even subalgebra
1284:
1265:
1263:Even subalgebra
1260:
1222:
1165:
1164:
1134:
1133:
1068:
1067:
1062:The set of all
1017:tensor algebras
999:
982:
939:
938:
903:
902:
869:
846:
845:
820:
819:
800:
799:
768:
767:
748:
747:
728:
727:
718:is the degree (
688:
687:
645:
622:
621:
603:
595:
520:
497:
496:
393:
392:
370:
363:
352:
339:are said to be
338:
315:
301:
262:
249:
239:
234:
233:
217:multiplication
190:
177:
166:
165:
106:
48:
28:
23:
22:
18:Even subalgebra
15:
12:
11:
5:
4240:
4238:
4230:
4229:
4224:
4214:
4213:
4207:
4206:
4204:
4203:
4198:
4193:
4188:
4183:
4178:
4173:
4168:
4163:
4158:
4153:
4148:
4143:
4138:
4133:
4128:
4123:
4118:
4113:
4107:
4105:
4101:
4100:
4098:
4097:
4092:
4087:
4082:
4077:
4072:
4067:
4062:
4057:
4052:
4047:
4042:
4037:
4032:
4026:
4024:
4018:
4017:
4015:
4014:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3968:
3966:
3960:
3959:
3957:
3956:
3951:
3946:
3941:
3936:
3931:
3926:
3921:
3916:
3910:
3908:
3907:Field theories
3904:
3903:
3901:
3900:
3895:
3890:
3884:
3882:
3878:
3877:
3875:
3874:
3869:
3864:
3859:
3854:
3849:
3844:
3839:
3834:
3829:
3824:
3819:
3814:
3809:
3804:
3802:Superpotential
3799:
3794:
3789:
3787:Supermultiplet
3784:
3779:
3773:
3771:
3767:
3766:
3764:
3763:
3758:
3753:
3748:
3743:
3738:
3733:
3728:
3723:
3718:
3712:
3710:
3704:
3703:
3701:
3700:
3695:
3690:
3685:
3680:
3675:
3670:
3664:
3662:
3661:General topics
3658:
3657:
3652:
3650:
3649:
3642:
3635:
3627:
3618:
3617:
3615:
3614:
3601:
3593:
3590:
3589:
3587:
3586:
3581:
3576:
3571:
3570:
3569:
3558:
3556:
3552:
3551:
3549:
3548:
3542:
3540:
3536:
3535:
3533:
3532:
3525:
3520:
3515:
3510:
3504:
3502:
3498:
3497:
3495:
3494:
3489:
3484:
3479:
3474:
3469:
3464:
3458:
3456:
3450:
3449:
3446:
3445:
3443:
3442:
3437:
3432:
3427:
3422:
3417:
3412:
3407:
3402:
3396:
3394:
3385:
3384:
3383:
3382:
3377:
3367:
3366:
3365:
3360:
3350:
3345:
3344:
3343:
3333:
3332:
3331:
3326:
3321:
3316:
3311:
3306:
3301:
3291:
3290:
3289:
3284:
3273:
3271:
3263:
3262:
3260:
3259:
3254:
3249:
3240:
3229:
3227:
3221:
3220:
3218:
3217:
3212:
3211:
3210:
3205:
3200:
3195:
3185:
3184:
3183:
3178:
3173:
3168:
3158:
3157:
3156:
3151:
3146:
3141:
3131:
3130:
3129:
3119:
3118:
3117:
3112:
3102:
3101:
3100:
3098:Control theory
3095:
3085:
3084:
3083:
3078:
3073:
3063:
3062:
3061:
3056:
3051:
3041:
3040:
3039:
3029:
3023:
3021:
3015:
3014:
3012:
3011:
3006:
3001:
2996:
2991:
2985:
2983:
2977:
2976:
2973:
2972:
2970:
2969:
2964:
2959:
2954:
2949:
2944:
2939:
2934:
2929:
2923:
2921:
2912:
2911:
2906:
2901:
2896:
2891:
2890:
2889:
2879:
2874:
2869:
2864:
2859:
2858:
2857:
2852:
2841:
2839:
2833:
2832:
2827:
2825:
2824:
2817:
2810:
2802:
2796:
2795:
2789:
2770:
2769:
2763:
2746:
2745:
2739:
2716:
2715:
2709:
2691:
2688:
2686:
2685:
2669:
2657:
2644:
2642:
2639:
2635:
2634:
2619:
2616:
2613:
2610:
2607:
2605:
2603:
2600:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2576:
2574:
2571:
2570:
2470:
2469:
2458:
2455:
2452:
2449:
2446:
2443:
2437:
2433:
2429:
2424:
2420:
2416:
2411:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2302:
2299:
2279:
2278:
2267:
2264:
2259:
2255:
2249:
2245:
2241:
2236:
2232:
2226:
2222:
2218:
2212:
2206:
2202:
2197:
2192:
2186:
2182:
2177:
2172:
2168:
2165:
2162:
2159:
2156:
2151:
2147:
2143:
2138:
2134:
2130:
2127:
2122:
2118:
2114:
2109:
2105:
2101:
2071:tensor product
2066:
2063:
2043:
2042:
2031:
2028:
2025:
2022:
2019:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1962:
1904:
1903:
1892:
1889:
1883:
1879:
1875:
1870:
1866:
1862:
1857:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1795:
1792:
1791:
1790:
1779:
1776:
1773:
1768:
1764:
1760:
1757:
1754:
1751:
1745:
1742:
1736:
1733:
1730:
1727:
1724:
1721:
1716:
1712:
1678:
1672:
1671:
1658:
1654:
1650:
1645:
1641:
1637:
1631:
1628:
1611:
1610:
1599:
1593:
1589:
1585:
1580:
1576:
1573:
1570:
1567:
1561:
1558:
1529:
1526:
1517:
1499:
1498:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1425:
1424:
1411:
1407:
1403:
1398:
1394:
1386:
1382:
1377:
1371:
1367:
1363:
1360:
1342:
1323:
1316:
1282:
1264:
1261:
1259:
1256:
1255:
1254:
1240:
1214:
1200:
1191:, composed of
1179:
1176:
1173:
1148:
1145:
1142:
1121:
1118:
1115:
1112:
1109:
1105:
1102:
1099:
1095:
1092:
1089:
1086:
1082:
1079:
1076:
1060:
1054:
1043:
1028:
1013:graded algebra
1001:
1000:to be trivial.
997:
981:
978:
975:Pierre Deligne
960:
956:
952:
949:
946:
924:
920:
916:
913:
910:
899:
898:
887:
884:
879:
876:
872:
868:
865:
862:
859:
856:
853:
830:
827:
807:
787:
784:
781:
778:
775:
755:
735:
707:
704:
701:
698:
695:
684:
683:
672:
669:
664:
661:
658:
655:
652:
648:
644:
641:
638:
635:
632:
629:
614:graded algebra
601:
594:
591:
571:
570:
558:
555:
549:
545:
541:
536:
532:
528:
523:
519:
516:
513:
510:
507:
504:
444:
440:
436:
432:
428:
424:
420:
416:
412:
408:
405:
401:
368:
361:
334:
313:
299:
289:
288:
275:
272:
269:
265:
261:
256:
252:
246:
242:
211:
210:
197:
193:
189:
184:
180:
176:
173:
161:decomposition
126:characteristic
105:
102:
94:supermanifolds
51:graded algebra
46:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4239:
4228:
4225:
4223:
4220:
4219:
4217:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4177:
4174:
4172:
4169:
4167:
4164:
4162:
4159:
4157:
4154:
4152:
4149:
4147:
4144:
4142:
4139:
4137:
4134:
4132:
4129:
4127:
4124:
4122:
4119:
4117:
4114:
4112:
4109:
4108:
4106:
4102:
4096:
4093:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4043:
4041:
4038:
4036:
4033:
4031:
4028:
4027:
4025:
4023:
4022:Superpartners
4019:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3969:
3967:
3965:
3961:
3955:
3952:
3950:
3947:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3915:
3912:
3911:
3909:
3905:
3899:
3896:
3894:
3891:
3889:
3886:
3885:
3883:
3879:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3778:
3775:
3774:
3772:
3768:
3762:
3761:Supermanifold
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3717:
3714:
3713:
3711:
3709:
3705:
3699:
3698:Supergeometry
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3668:Supersymmetry
3666:
3665:
3663:
3659:
3655:
3654:Supersymmetry
3648:
3643:
3641:
3636:
3634:
3629:
3628:
3625:
3613:
3609:
3605:
3602:
3600:
3599:
3595:
3594:
3591:
3585:
3582:
3580:
3577:
3575:
3572:
3568:
3565:
3564:
3563:
3560:
3559:
3557:
3555:Organizations
3553:
3547:
3544:
3543:
3541:
3537:
3530:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3505:
3503:
3499:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3459:
3457:
3455:
3451:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3415:Quantum group
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3397:
3395:
3393:
3389:
3381:
3378:
3376:
3373:
3372:
3371:
3370:Supersymmetry
3368:
3364:
3361:
3359:
3356:
3355:
3354:
3353:String theory
3351:
3349:
3346:
3342:
3339:
3338:
3337:
3334:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3296:
3295:
3292:
3288:
3285:
3283:
3280:
3279:
3278:
3275:
3274:
3272:
3270:
3264:
3258:
3255:
3253:
3252:Path integral
3250:
3248:
3244:
3241:
3238:
3234:
3233:Distributions
3231:
3230:
3228:
3226:
3222:
3216:
3213:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3190:
3189:
3186:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3163:
3162:
3159:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3136:
3135:
3132:
3128:
3125:
3124:
3123:
3120:
3116:
3113:
3111:
3108:
3107:
3106:
3103:
3099:
3096:
3094:
3091:
3090:
3089:
3086:
3082:
3079:
3077:
3074:
3072:
3069:
3068:
3067:
3064:
3060:
3057:
3055:
3052:
3050:
3047:
3046:
3045:
3042:
3038:
3035:
3034:
3033:
3030:
3028:
3025:
3024:
3022:
3020:
3016:
3010:
3007:
3005:
3002:
3000:
2999:Combinatorics
2997:
2995:
2992:
2990:
2987:
2986:
2984:
2982:
2978:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2924:
2922:
2920:
2916:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2888:
2885:
2884:
2883:
2880:
2878:
2875:
2873:
2872:Coding theory
2870:
2868:
2865:
2863:
2860:
2856:
2853:
2851:
2848:
2847:
2846:
2843:
2842:
2840:
2838:
2837:Computational
2834:
2830:
2823:
2818:
2816:
2811:
2809:
2804:
2803:
2800:
2792:
2786:
2782:
2781:
2776:
2772:
2771:
2766:
2764:3-540-61378-1
2760:
2756:
2752:
2748:
2747:
2742:
2736:
2732:
2731:
2726:
2722:
2718:
2717:
2712:
2710:0-8218-2012-5
2706:
2702:
2698:
2694:
2693:
2689:
2682:
2676:
2674:
2670:
2666:
2661:
2658:
2654:
2649:
2646:
2640:
2638:
2617:
2611:
2608:
2606:
2601:
2594:
2588:
2585:
2582:
2579:
2577:
2572:
2561:
2560:
2559:
2557:
2554:-supermodule
2553:
2549:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2520:categorically
2516:
2514:
2510:
2506:
2502:
2498:
2493:
2491:
2487:
2483:
2479:
2475:
2453:
2450:
2447:
2441:
2431:
2418:
2405:
2402:
2396:
2393:
2387:
2384:
2381:
2375:
2369:
2366:
2360:
2357:
2350:
2349:
2348:
2346:
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2334:
2330:
2327:
2325:
2320:
2316:
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2307:
2300:
2298:
2296:
2292:
2288:
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2265:
2257:
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2243:
2239:
2234:
2230:
2224:
2220:
2204:
2200:
2184:
2180:
2166:
2163:
2157:
2149:
2145:
2141:
2136:
2132:
2120:
2116:
2112:
2107:
2103:
2092:
2091:
2090:
2088:
2084:
2080:
2076:
2072:
2064:
2062:
2060:
2056:
2052:
2048:
2029:
2023:
2020:
2017:
2009:
2006:
2000:
1997:
1994:
1988:
1985:
1982:
1979:
1973:
1967:
1952:
1951:
1950:
1948:
1944:
1940:
1936:
1931:
1925:
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1877:
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1793:
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1597:
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1437:
1430:
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1396:
1392:
1384:
1380:
1375:
1369:
1365:
1361:
1358:
1351:
1350:
1349:
1348:
1347:bilinear form
1341:
1337:
1333:
1329:
1322:
1315:
1310:
1308:
1304:
1300:
1296:
1292:
1288:
1281:
1278:
1274:
1270:
1262:
1257:
1252:
1248:
1244:
1241:
1238:
1234:
1230:
1226:
1221:
1217:
1213:
1209:
1205:
1204:supermatrices
1201:
1198:
1194:
1163:
1116:
1113:
1110:
1093:
1087:
1065:
1064:endomorphisms
1061:
1058:
1055:
1052:
1048:
1044:
1041:
1037:
1033:
1029:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
996:
992:
988:
984:
983:
979:
977:
976:
958:
950:
947:
944:
922:
914:
911:
908:
885:
882:
877:
874:
866:
863:
854:
851:
844:
843:
842:
828:
825:
805:
785:
782:
779:
776:
773:
753:
733:
726:-grading) of
725:
721:
705:
702:
699:
696:
693:
670:
667:
662:
659:
656:
653:
650:
642:
639:
630:
627:
620:
619:
618:
615:
611:
607:
600:
592:
590:
588:
584:
580:
576:
556:
553:
543:
530:
517:
514:
508:
505:
502:
495:
494:
493:
491:
487:
486:commutativity
483:
482:
476:
474:
470:
466:
462:
457:
438:
430:
422:
414:
406:
403:
390:
386:
382:
378:
374:
367:
360:
355:
350:
346:
342:
337:
333:
328:
326:
323:
319:
312:
308:
303:
298:
294:
273:
270:
267:
263:
259:
254:
250:
244:
240:
232:
231:
230:
228:
224:
220:
216:
195:
191:
187:
182:
178:
174:
171:
164:
163:
162:
160:
156:
153:
151:
146:
142:
137:
135:
131:
127:
123:
119:
115:
111:
103:
101:
99:
95:
91:
87:
86:supergeometry
83:
79:
75:
74:supersymmetry
71:
66:
64:
60:
56:
52:
45:
41:
37:
33:
19:
3964:Supergravity
3857:Localization
3847:Witten index
3822:Moduli space
3716:Superalgebra
3715:
3683:Supergravity
3610: /
3606: /
3596:
3472:Optimization
3435:Superalgebra
3434:
3294:Field theory
3267:Mathematical
3245: /
3093:Chaos theory
3076:Gauge theory
3004:Graph theory
2899:Cryptography
2779:
2754:
2751:Manin, Y. I.
2729:
2725:Zelmanov, E.
2700:
2667:, p. 87
2660:
2648:
2636:
2555:
2551:
2547:
2539:
2535:
2527:
2517:
2512:
2508:
2504:
2500:
2496:
2494:
2489:
2485:
2481:
2477:
2473:
2471:
2344:
2340:
2336:
2332:
2328:
2326:-supermodule
2323:
2318:
2315:superalgebra
2314:
2310:
2308:
2304:
2294:
2290:
2286:
2282:
2280:
2086:
2082:
2078:
2074:
2068:
2058:
2054:
2046:
2044:
1946:
1942:
1938:
1934:
1932:
1924:supercommute
1923:
1922:are said to
1919:
1915:
1911:
1907:
1905:
1805:
1799:
1797:
1696:
1688:
1684:
1679:
1675:
1673:
1612:
1540:
1537:automorphism
1531:
1521:
1514:
1510:
1506:
1502:
1500:
1426:
1339:
1335:
1331:
1320:
1313:
1311:
1306:
1298:
1294:
1286:
1279:
1272:
1268:
1266:
1247:Lie algebras
1236:
1232:
1228:
1224:
1219:
1215:
1211:
1207:
1192:
1161:
1035:
1024:
1008:
1004:
994:
990:
986:
900:
723:
719:
685:
609:
605:
598:
596:
586:
582:
578:
574:
572:
489:
479:
477:
468:
460:
458:
388:
384:
380:
376:
372:
365:
358:
353:
348:
344:
340:
335:
331:
329:
324:
310:
306:
304:
296:
290:
226:
222:
218:
212:
154:
149:
144:
141:superalgebra
140:
138:
133:
129:
117:
109:
107:
98:superschemes
78:supermodules
69:
67:
43:
40:superalgebra
39:
29:
4104:Researchers
4090:Stop squark
4055:Graviscalar
4050:Graviphoton
3914:WessâZumino
3777:Supercharge
3612:topics list
3546:Mathematics
3462:Game theory
3363:Topological
3329:Topological
3324:Statistical
3287:Hamiltonian
2697:Deligne, P.
2655:, p. 3
2069:The graded
1935:supercenter
465:associative
341:homogeneous
318:graded ring
128:0, such as
68:The prefix
32:mathematics
4216:Categories
4151:Iliopoulos
4095:Superghost
4085:Sgoldstino
4070:Neutralino
3862:Mu problem
3782:R-symmetry
3746:Superspace
3741:Supergroup
3518:Psychology
3482:Statistics
3282:Lagrangian
2909:Statistics
2845:Algorithms
2721:Kac, V. G.
2690:References
2281:If either
1534:involutive
1427:such that
1291:subalgebra
229:such that
159:direct sum
4121:Batchelor
4045:Goldstino
3934:Super QCD
3812:FI D-term
3797:BPS state
3523:Sociology
3513:Chemistry
3309:Effective
3304:Conformal
3299:Classical
3171:Geometric
3144:Geometric
2615:→
2602:η
2592:→
2586:⊗
2573:μ
2451:⋅
2403:−
2385:⋅
2361:⋅
2240:⊗
2164:−
2142:⊗
2113:⊗
2021:∈
1983:∈
1849:−
1843:−
1756:−
1744:^
1729:∈
1693:2-torsion
1649:−
1630:^
1572:−
1560:^
1480:⊗
1471:μ
1468:⋅
1456:⋅
1447:⊗
1438:μ
1402:→
1376:⊗
1359:μ
1277:submodule
1094:≡
1066:(denoted
864:−
858:↦
783:
703:
640:−
634:↦
597:When the
515:−
307:superring
260:⊆
188:⊕
4222:Algebras
4156:Montonen
4080:Sfermion
4075:R-hadron
4060:Higgsino
4035:Chargino
3924:4D N = 1
3881:Theorems
3770:Concepts
3598:Category
3247:analysis
3166:Exterior
3139:Exterior
3019:Analysis
2981:Discrete
2855:analysis
2777:(2004).
2753:(1997).
2727:(2001).
2524:category
2507:→
2488:∈
2476:∈
2343:→
2085:⊗
1501:for all
1328:bimodule
1231:of rank
1162:internal
980:Examples
322:integers
225:→
215:bilinear
4171:Seiberg
4146:Golfand
4126:Berezin
4111:Affleck
4040:Gaugino
3608:outline
3539:Related
3508:Biology
3358:Bosonic
3319:Quantum
3269:physics
3235: (
2967:Solvers
2526:of all
2339:×
2331:with a
1691:has no
1345:with a
1338:equips
1303:algebra
221:×
157:with a
152:-module
57:over a
55:algebra
4201:Zumino
4196:Witten
4186:Rogers
4176:Siegel
4116:Bagger
3817:F-term
3807:D-term
3181:Vector
3176:Tensor
3154:Vector
3149:Tensor
2850:design
2787:
2761:
2737:
2707:
2544:monoid
2522:. The
2051:center
1674:where
1509:, and
1319:is an
1275:. The
686:where
467:and a
345:parity
343:. The
293:modulo
70:super-
4181:RoÄek
4166:Salam
4161:Olive
4141:Gates
4136:Fayet
4030:Axino
3944:NMSSM
3314:Gauge
2641:Notes
2321:is a
2317:over
1687:. If
1305:over
1034:over
1023:over
1007:- or
722:- or
608:- or
379:. If
309:, or
147:is a
143:over
122:field
120:is a
112:be a
63:field
42:is a
4191:Wess
4131:Dine
3939:MSSM
2785:ISBN
2759:ISBN
2735:ISBN
2705:ISBN
2679:See
2480:and
2309:Let
2293:and
2077:and
1933:The
1914:and
1798:The
1267:Let
1049:and
1045:The
1019:and
1003:Any
937:and
746:and
577:and
391:and
383:and
373:even
108:Let
96:and
38:, a
34:and
4065:LSP
2285:or
2053:of
1937:of
1928:= 0
1926:if
1918:of
1804:on
1513:in
1293:of
1193:all
955:mod
919:mod
780:deg
700:deg
581:of
459:An
377:odd
364:or
132:or
124:of
61:or
30:In
4218::
2672:^
2515:.
2492:.
2484:,
2061:.
1949::
1930:.
1699::
1524:.
1505:,
1309:.
478:A
456:.
389:xy
327:.
305:A
302:.
139:A
136:.
100:.
92:,
3646:e
3639:t
3632:v
3531:"
3527:"
3239:)
2821:e
2814:t
2807:v
2793:.
2767:.
2743:.
2713:.
2618:A
2612:R
2609::
2595:A
2589:A
2583:A
2580::
2556:A
2552:R
2548:R
2540:R
2536:R
2528:R
2513:A
2509:A
2505:R
2501:A
2497:R
2490:A
2486:y
2482:x
2478:R
2474:r
2457:)
2454:y
2448:r
2445:(
2442:x
2436:|
2432:x
2428:|
2423:|
2419:r
2415:|
2410:)
2406:1
2400:(
2397:=
2394:y
2391:)
2388:x
2382:r
2379:(
2376:=
2373:)
2370:y
2367:x
2364:(
2358:r
2345:A
2341:A
2337:A
2333:R
2329:A
2324:R
2319:R
2311:R
2295:B
2291:A
2287:B
2283:A
2266:.
2263:)
2258:2
2254:b
2248:1
2244:b
2235:2
2231:a
2225:1
2221:a
2217:(
2211:|
2205:2
2201:a
2196:|
2191:|
2185:1
2181:b
2176:|
2171:)
2167:1
2161:(
2158:=
2155:)
2150:2
2146:b
2137:2
2133:a
2129:(
2126:)
2121:1
2117:b
2108:1
2104:a
2100:(
2087:B
2083:A
2079:B
2075:A
2059:A
2055:A
2047:A
2030:.
2027:}
2024:A
2018:x
2010:0
2007:=
2004:]
2001:x
1998:,
1995:a
1992:[
1989::
1986:A
1980:a
1977:{
1974:=
1971:)
1968:A
1965:(
1961:Z
1947:A
1943:A
1939:A
1920:A
1916:y
1912:x
1908:A
1891:x
1888:y
1882:|
1878:y
1874:|
1869:|
1865:x
1861:|
1856:)
1852:1
1846:(
1840:y
1837:x
1834:=
1831:]
1828:y
1825:,
1822:x
1819:[
1806:A
1778:.
1775:}
1772:x
1767:i
1763:)
1759:1
1753:(
1750:=
1741:x
1735::
1732:A
1726:x
1723:{
1720:=
1715:i
1711:A
1697:A
1689:A
1685:x
1680:i
1676:x
1657:1
1653:x
1644:0
1640:x
1636:=
1627:x
1598:x
1592:|
1588:x
1584:|
1579:)
1575:1
1569:(
1566:=
1557:x
1522:A
1518:1
1515:A
1511:z
1507:y
1503:x
1486:)
1483:z
1477:y
1474:(
1465:x
1462:=
1459:z
1453:)
1450:y
1444:x
1441:(
1410:0
1406:A
1397:1
1393:A
1385:0
1381:A
1370:1
1366:A
1362::
1343:1
1340:A
1336:A
1332:A
1326:-
1324:0
1321:A
1317:1
1314:A
1307:K
1295:A
1287:A
1283:0
1280:A
1273:K
1269:A
1237:q
1235:|
1233:p
1229:K
1225:K
1223:(
1220:q
1218:|
1216:p
1212:M
1208:K
1178:m
1175:o
1172:H
1147:m
1144:o
1141:H
1120:)
1117:V
1114:,
1111:V
1108:(
1104:m
1101:o
1098:H
1091:)
1088:V
1085:(
1081:d
1078:n
1075:E
1042:.
1036:K
1027:.
1025:K
1011:-
1009:N
1005:Z
998:1
995:A
991:K
987:K
959:2
951:n
948:=
945:q
923:2
915:m
912:=
909:p
886:x
883:y
878:q
875:p
871:)
867:1
861:(
855:y
852:x
829:.
826:q
806:y
786:y
777:=
774:n
754:p
734:x
724:N
720:Z
706:x
697:=
694:m
671:x
668:y
663:q
660:p
657:+
654:n
651:m
647:)
643:1
637:(
631:y
628:x
612:-
610:N
606:Z
602:2
599:Z
583:A
579:y
575:x
557:y
554:x
548:|
544:y
540:|
535:|
531:x
527:|
522:)
518:1
512:(
509:=
506:x
503:y
490:A
443:|
439:y
435:|
431:+
427:|
423:x
419:|
415:=
411:|
407:y
404:x
400:|
385:y
381:x
369:1
366:A
362:0
359:A
354:x
349:x
336:i
332:A
325:Z
316:-
314:2
311:Z
300:2
297:Z
274:j
271:+
268:i
264:A
255:j
251:A
245:i
241:A
227:A
223:A
219:A
196:1
192:A
183:0
179:A
175:=
172:A
155:A
150:K
145:K
134:C
130:R
118:K
110:K
49:-
47:2
44:Z
20:)
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