1262:
920:
1968:). There is an important difference in the graded case, however. A homomorphism from one super vector space to another is, by definition, one that preserves the grading (i.e. maps even elements to even elements and odd elements to odd elements). The coordinate representation of such a transformation is always an
646:
Supermatrices of compatible dimensions can be added or multiplied just as for ordinary matrices. These operations are exactly the same as the ordinary ones with the restriction that they are defined only when the blocks have compatible dimensions. One can also multiply supermatrices by elements of
1794:
1552:
1257:{\displaystyle {\begin{bmatrix}X_{00}&X_{01}\\X_{10}&X_{11}\end{bmatrix}}{\begin{bmatrix}Y_{00}&Y_{01}\\Y_{10}&Y_{11}\end{bmatrix}}={\begin{bmatrix}X_{00}Y_{00}+X_{01}Y_{10}&X_{00}Y_{01}+X_{01}Y_{11}\\X_{10}Y_{00}+X_{11}Y_{10}&X_{10}Y_{01}+X_{11}Y_{11}\end{bmatrix}}.}
1972:
supermatrix. Odd supermatrices correspond to linear transformations that reverse the grading. General supermatrices represent an arbitrary ungraded linear transformation. Such transformations are still important in the graded case, although less so than the graded (even) transformations.
606:
505:
2776:
853:
to obtain a supermatrix of the same dimension. The addition can be performed blockwise since the blocks have compatible sizes. It is easy to see that the sum of two even supermatrices is even and the sum of two odd supermatrices is odd.
3744:
4043:
267:
2892:
1669:
1430:
3201:
3489:
3421:
3866:
1938:
3090:
2576:
4137:
2483:
2386:
1658:
3016:
4058:
The
Berezinian satisfies similar properties to the ordinary determinant. In particular, it is multiplicative and invariant under the supertranspose. It is related to the supertrace by the formula
1368:
3292:
3590:
3353:
520:
419:
1832:
2129:
2623:
1584:
3532:
2265:
3628:
3916:
185:
1789:{\displaystyle X\cdot \alpha ={\begin{bmatrix}X_{00}\,\alpha &X_{01}\,{\hat {\alpha }}\\X_{10}\,\alpha &X_{11}\,{\hat {\alpha }}\end{bmatrix}}.}
1547:{\displaystyle \alpha \cdot X={\begin{bmatrix}\alpha \,X_{00}&\alpha \,X_{01}\\{\hat {\alpha }}\,X_{10}&{\hat {\alpha }}\,X_{11}\end{bmatrix}}}
2803:
2493:
Many operations on ordinary matrices can be generalized to supermatrices, although the generalizations are not always obvious or straightforward.
3137:
638:
is 0 or 1 according to whether it is even or odd. Every supermatrix can be written uniquely as the sum of an even supermatrix and an odd one.
3427:
3359:
3762:
1860:
3036:
2789:. This can be extended to arbitrary supermatrices by linearity. Unlike the ordinary transpose, the supertranspose is not generally an
2522:
4181:
4162:
4064:
2409:
2292:
1596:
1401:|. That is, the product of two even or two odd supermatrices is even while the product of an even and odd supermatrix is odd.
2907:
4205:
1281:
2286:) supermatrix relative to the chosen bases. The components of the associated supermatrix are determined by the formula
3231:
651:(on the left or right), however, this operation differs from the ungraded case due to the presence of odd elements in
3538:
3298:
601:{\displaystyle {\begin{bmatrix}\mathrm {odd} &\mathrm {even} \\\mathrm {even} &\mathrm {odd} \end{bmatrix}}}
500:{\displaystyle {\begin{bmatrix}\mathrm {even} &\mathrm {odd} \\\mathrm {odd} &\mathrm {even} \end{bmatrix}}}
2790:
801:
63:
1802:
2071:
3619:
2771:{\displaystyle X^{st}={\begin{bmatrix}A^{t}&(-1)^{|X|}C^{t}\\-(-1)^{|X|}B^{t}&D^{t}\end{bmatrix}}}
3906:. The Berezinian is only well-defined on even, invertible supermatrices over a commutative superalgebra
1409:
895:
82:
4048:
where det denotes the ordinary determinant (of square matrices with entries in the commutative algebra
1960:). Likewise, supermatrices can be thought of as the coordinate representations of linear maps between
1560:
514:
is one for which the reverse holds: the diagonal blocks are odd and the off-diagonal blocks are even.
169:
78:
47:
17:
3500:
2018:
is supercommutative, the rank is independent of the choice of basis, just as in the ungraded case.
71:
32:
1961:
745:
86:
4172:
Deligne, Pierre; Morgan, John W. (1999). "Notes on
Supersymmetry (following Joseph Bernstein)".
2237:
4200:
4177:
4158:
67:
1412:
for supermatrices is different than the ungraded case due to the presence of odd elements in
3892:
1965:
126:
850:
3739:{\displaystyle \mathrm {str} (X)=\mathrm {tr} (X_{00})-(-1)^{|X|}\mathrm {tr} (X_{11})\,}
4038:{\displaystyle \mathrm {Ber} (X)=\det(X_{00}-X_{01}X_{11}^{-1}X_{10})\det(X_{11})^{-1}.}
616:
114:
4194:
353:
94:
615:
are purely even there are no nonzero odd elements, so the even supermatices are the
262:{\displaystyle X={\begin{bmatrix}X_{00}&X_{01}\\X_{10}&X_{11}\end{bmatrix}}}
1984:
1953:
177:
110:
55:
51:
21:
1834:
and both of these operations are the same as the ungraded versions. If α and
914:). The multiplication can be performed at the block level in the obvious manner:
3903:
1977:
1957:
704:
118:
90:
28:
2887:{\displaystyle (X^{st})^{st}={\begin{bmatrix}A&-B\\-C&D\end{bmatrix}}.}
3888:
3608:
1949:
318:). An ordinary (ungraded) matrix can be thought of as a supermatrix for which
2793:, but rather has order 4. Applying the supertranspose twice to a supermatrix
1557:
where the internal scalar multiplications are the ordinary ungraded ones and
2513:
793:
59:
1948:
Ordinary matrices can be thought of as the coordinate representations of
3196:{\displaystyle X^{\pi }={\begin{bmatrix}D&C\\B&A\end{bmatrix}}.}
2215:. Such a choice of bases is equivalent to a choice of isomorphisms from
711:
under supermatrix addition and scalar multiplication. In particular, if
3484:{\displaystyle (X\cdot \alpha )^{\pi }=X^{\pi }\cdot {\hat {\alpha }}}
3416:{\displaystyle (\alpha \cdot X)^{\pi }={\hat {\alpha }}\cdot X^{\pi }}
2142:
is just supermatrix multiplication (this action is not generally left
2025:
be the space of column supervectors—supermatrices of dimension (
20:. In other contexts, “supermatrix” is sometimes used as a synonym for
3030:
of a supermatrix is a new operation without an ungraded analog. Let
395:(i.e. homogeneous elements of parity 0) and the off-diagonal blocks (
3861:{\displaystyle \mathrm {str} (XY)=(-1)^{|X||Y|}\mathrm {str} (YX)\,}
804:, then supermatrix multiplication is a bilinear operation, so that
129:
as well (for essentially the same reasons as in the ungraded case).
1933:{\displaystyle \alpha \cdot X=(-1)^{|\alpha ||X|}X\cdot \alpha .}
3085:{\displaystyle X={\begin{bmatrix}A&B\\C&D\end{bmatrix}}}
2571:{\displaystyle X={\begin{bmatrix}A&B\\C&D\end{bmatrix}}}
1994:
if it has a free homogeneous basis. If such a basis consists of
2901:
is supercommutative, the supertranspose satisfies the identity
1420:
be a supermatrix. Left scalar multiplication by α ∈
796:
under supermatrix addition and multiplication. Furthermore, if
4176:. Vol. 1. American Mathematical Society. pp. 41–97.
4132:{\displaystyle \mathrm {Ber} (e^{X})=e^{\mathrm {str(X)} }.\,}
2478:{\displaystyle M=M_{0}\oplus M_{1}\qquad N=N_{0}\oplus N_{1}.}
1663:
Right scalar multiplication by α is defined analogously:
2381:{\displaystyle T(e_{i})=\sum _{k=1}^{r+s}f_{k}\,{T^{k}}_{i}.}
3622:. It is defined on homogeneous supermatrices by the formula
3756:
is supercommutative, the supertrace satisfies the identity
1846:⋅α are both homogeneous with parity |α| + |
62:). The most important examples are those with entries in a
619:
ones and the odd supermatrices are the off-diagonal ones.
1653:{\displaystyle {\hat {\alpha }}=(-1)^{|\alpha |}\alpha .}
81:
where they appear as the coordinate representations of a
349:). This means that not only is the unpartitioned matrix
74:(thought of as a purely even commutative superalgebra).
4174:
Quantum Fields and
Strings: A Course for Mathematicians
3159:
3051:
3011:{\displaystyle (XY)^{st}=(-1)^{|X||Y|}Y^{st}X^{st}.\,}
2844:
2648:
2537:
1690:
1451:
1060:
993:
929:
529:
428:
200:
4067:
3919:
3765:
3631:
3541:
3503:
3430:
3362:
3301:
3234:
3140:
3039:
2910:
2806:
2626:
2525:
2412:
2295:
2240:
2074:
1863:
1805:
1672:
1599:
1563:
1433:
1284:
923:
523:
422:
188:
3225:
The parity transpose operation obeys the identities
1363:{\displaystyle Z_{ij}=X_{i0}Y_{0j}+X_{i1}Y_{1j}.\,}
93:. They have important applications in the field of
4131:
4037:
3860:
3738:
3584:
3526:
3483:
3415:
3347:
3286:
3195:
3084:
3010:
2886:
2770:
2570:
2477:
2380:
2259:
2123:
1932:
1826:
1788:
1652:
1578:
1546:
1362:
1256:
600:
499:
261:
4151:Supersymmetry for Mathematicians: An Introduction
3214:) block of the transposed matrix is the (1−
772:) denote the set of all square supermatices over
4003:
3943:
3287:{\displaystyle (X+Y)^{\pi }=X^{\pi }+Y^{\pi }\,}
1267:Note that the blocks of the product supermatrix
862:One can multiply a supermatrix with dimensions (
3585:{\displaystyle \pi \circ st\circ \pi =(st)^{3}}
3348:{\displaystyle (XY)^{\pi }=X^{\pi }Y^{\pi }\,}
8:
50:. Specifically, a supermatrix is a 2×2
1590:. This is given on homogeneous elements by
683:) denote the set of all supermatrices over
3910:. In this case it is given by the formula
4128:
4103:
4102:
4086:
4068:
4066:
4023:
4013:
3994:
3981:
3976:
3966:
3953:
3920:
3918:
3857:
3834:
3827:
3819:
3814:
3806:
3805:
3766:
3764:
3735:
3726:
3711:
3704:
3696:
3695:
3670:
3655:
3632:
3630:
3576:
3540:
3523:
3508:
3502:
3470:
3469:
3460:
3447:
3429:
3407:
3389:
3388:
3379:
3361:
3344:
3338:
3328:
3315:
3300:
3283:
3277:
3264:
3251:
3233:
3154:
3145:
3139:
3046:
3038:
3007:
2995:
2982:
2971:
2963:
2958:
2950:
2949:
2924:
2909:
2839:
2827:
2814:
2805:
2754:
2742:
2731:
2723:
2722:
2696:
2685:
2677:
2676:
2655:
2643:
2631:
2625:
2532:
2524:
2466:
2453:
2436:
2423:
2411:
2391:The block decomposition of a supermatrix
2369:
2362:
2357:
2355:
2349:
2333:
2322:
2306:
2294:
2256:
2239:
2120:
2110:
2106:
2089:
2085:
2073:
1911:
1903:
1898:
1890:
1889:
1862:
1807:
1806:
1804:
1764:
1763:
1762:
1756:
1746:
1740:
1721:
1720:
1719:
1713:
1703:
1697:
1685:
1671:
1637:
1629:
1628:
1601:
1600:
1598:
1565:
1564:
1562:
1530:
1525:
1514:
1513:
1505:
1500:
1489:
1488:
1478:
1473:
1462:
1457:
1446:
1432:
1359:
1347:
1334:
1318:
1305:
1289:
1283:
1237:
1227:
1214:
1204:
1192:
1182:
1169:
1159:
1145:
1135:
1122:
1112:
1100:
1090:
1077:
1067:
1055:
1038:
1026:
1012:
1000:
988:
974:
962:
948:
936:
924:
922:
579:
563:
545:
532:
524:
522:
475:
462:
447:
431:
423:
421:
245:
233:
219:
207:
195:
187:
2033:)×(1|0). This is naturally a right
1827:{\displaystyle {\hat {\alpha }}=\alpha }
634:|, of a nonzero homogeneous supermatrix
4153:. Courant Lecture Notes in Mathematics
3111:) supermatrix. The parity transpose of
16:This article is about supermatrices in
3599:denotes the supertranspose operation.
2124:{\displaystyle T:R^{p|q}\to R^{r|s}\,}
377:is one for which the diagonal blocks (
3749:where tr denotes the ordinary trace.
2597:) supermatrix. The supertranspose of
391:) consist solely of even elements of
288:total columns (so that the submatrix
7:
2395:corresponds to the decomposition of
2061:) can then be thought of as a right
878:) by a supermatrix with dimensions (
409:) consist solely of odd elements of
176:that is partitioned into a 2×2
77:Supermatrices arise in the study of
4116:
4110:
4107:
4104:
4075:
4072:
4069:
3927:
3924:
3921:
3841:
3838:
3835:
3773:
3770:
3767:
3715:
3712:
3659:
3656:
3639:
3636:
3633:
2785:denotes the ordinary transpose of
1838:are homogeneous then α⋅
586:
583:
580:
573:
570:
567:
564:
555:
552:
549:
546:
539:
536:
533:
485:
482:
479:
476:
469:
466:
463:
454:
451:
448:
441:
438:
435:
432:
14:
4157:. American Mathematical Society.
3895:) of a square supermatrix is the
2146:-linear which is why we think of
1854:is supercommutative then one has
898:to obtain a matrix of dimension (
626:if it is either even or odd. The
3222:) block of the original matrix.
1586:denotes the grade involution in
1579:{\displaystyle {\hat {\alpha }}}
833:Two supermatrices of dimension (
3611:of a square supermatrix is the
2442:
1381:are homogeneous with parities |
715:is a superalgebra over a field
4119:
4113:
4092:
4079:
4020:
4006:
4000:
3946:
3937:
3931:
3871:for homogeneous supermatrices
3854:
3845:
3828:
3820:
3815:
3807:
3802:
3792:
3786:
3777:
3732:
3719:
3705:
3697:
3692:
3682:
3676:
3663:
3649:
3643:
3573:
3563:
3475:
3444:
3431:
3394:
3376:
3363:
3312:
3302:
3248:
3235:
2972:
2964:
2959:
2951:
2946:
2936:
2921:
2911:
2824:
2807:
2732:
2724:
2719:
2709:
2686:
2678:
2673:
2663:
2403:into even and odd submodules:
2312:
2299:
2250:
2111:
2099:
2090:
1912:
1904:
1899:
1891:
1886:
1876:
1812:
1769:
1726:
1638:
1630:
1625:
1615:
1606:
1570:
1519:
1494:
849:) can be added just as in the
333:supermatrix is one for which (
46:-graded analog of an ordinary
1:
3527:{\displaystyle \pi ^{2}=id\,}
2231:. Any (ungraded) linear map
1393:is homogeneous with parity |
821:) forms a superalgebra over
4149:Varadarajan, V. S. (2004).
148:be nonnegative integers. A
85:between finite-dimensional
4222:
2260:{\displaystyle T:M\to N\,}
356:, but the diagonal blocks
15:
2037:-supermodule, called the
1944:As linear transformations
2505:of a supermatrix is the
802:commutative superalgebra
101:Definitions and notation
64:commutative superalgebra
1799:If α is even then
4133:
4039:
3902:-graded analog of the
3862:
3740:
3618:-graded analog of the
3586:
3528:
3485:
3417:
3349:
3288:
3197:
3086:
3012:
2888:
2772:
2572:
2512:-graded analog of the
2479:
2382:
2344:
2261:
2211:) be a free basis for
2198:) be a free basis for
2125:
2039:right coordinate space
1934:
1828:
1790:
1654:
1580:
1548:
1364:
1258:
602:
501:
263:
121:). Often one requires
83:linear transformations
4134:
4040:
3863:
3741:
3587:
3529:
3486:
3418:
3350:
3289:
3198:
3087:
3013:
2889:
2773:
2573:
2480:
2383:
2318:
2270:can be written as a (
2262:
2181:-supermodule of rank
2165:-supermodule of rank
2126:
2006:is said to have rank
1935:
1829:
1791:
1655:
1581:
1549:
1410:Scalar multiplication
1405:Scalar multiplication
1365:
1259:
603:
502:
264:
4206:Super linear algebra
4065:
3917:
3763:
3629:
3539:
3501:
3428:
3360:
3299:
3232:
3138:
3037:
2908:
2804:
2624:
2523:
2410:
2293:
2238:
2134:where the action of
2072:
1861:
1803:
1670:
1597:
1561:
1431:
1282:
921:
792:). This set forms a
703:). This set forms a
521:
420:
186:
79:super linear algebra
18:super linear algebra
3989:
2002:odd elements, then
1962:super vector spaces
1850:|. Furthermore, if
642:Algebraic structure
87:super vector spaces
33:theoretical physics
4129:
4035:
3972:
3858:
3736:
3582:
3524:
3481:
3413:
3345:
3284:
3193:
3184:
3082:
3076:
3008:
2884:
2875:
2768:
2762:
2581:be a homogeneous (
2568:
2562:
2475:
2378:
2257:
2121:
1998:even elements and
1930:
1824:
1786:
1777:
1650:
1576:
1544:
1538:
1360:
1254:
1245:
1046:
982:
746:super vector space
598:
592:
497:
491:
259:
253:
54:with entries in a
3478:
3397:
1966:free supermodules
1815:
1772:
1729:
1609:
1573:
1522:
1497:
622:A supermatrix is
70:) or an ordinary
68:Grassmann algebra
4213:
4187:
4168:
4138:
4136:
4135:
4130:
4124:
4123:
4122:
4091:
4090:
4078:
4044:
4042:
4041:
4036:
4031:
4030:
4018:
4017:
3999:
3998:
3988:
3980:
3971:
3970:
3958:
3957:
3930:
3893:superdeterminant
3867:
3865:
3864:
3859:
3844:
3833:
3832:
3831:
3823:
3818:
3810:
3776:
3745:
3743:
3742:
3737:
3731:
3730:
3718:
3710:
3709:
3708:
3700:
3675:
3674:
3662:
3642:
3591:
3589:
3588:
3583:
3581:
3580:
3533:
3531:
3530:
3525:
3513:
3512:
3490:
3488:
3487:
3482:
3480:
3479:
3471:
3465:
3464:
3452:
3451:
3422:
3420:
3419:
3414:
3412:
3411:
3399:
3398:
3390:
3384:
3383:
3354:
3352:
3351:
3346:
3343:
3342:
3333:
3332:
3320:
3319:
3293:
3291:
3290:
3285:
3282:
3281:
3269:
3268:
3256:
3255:
3202:
3200:
3199:
3194:
3189:
3188:
3150:
3149:
3091:
3089:
3088:
3083:
3081:
3080:
3028:parity transpose
3022:Parity transpose
3017:
3015:
3014:
3009:
3003:
3002:
2990:
2989:
2977:
2976:
2975:
2967:
2962:
2954:
2932:
2931:
2893:
2891:
2890:
2885:
2880:
2879:
2835:
2834:
2822:
2821:
2777:
2775:
2774:
2769:
2767:
2766:
2759:
2758:
2747:
2746:
2737:
2736:
2735:
2727:
2701:
2700:
2691:
2690:
2689:
2681:
2660:
2659:
2639:
2638:
2577:
2575:
2574:
2569:
2567:
2566:
2484:
2482:
2481:
2476:
2471:
2470:
2458:
2457:
2441:
2440:
2428:
2427:
2387:
2385:
2384:
2379:
2374:
2373:
2368:
2367:
2366:
2354:
2353:
2343:
2332:
2311:
2310:
2266:
2264:
2263:
2258:
2177:be a free right
2130:
2128:
2127:
2122:
2119:
2118:
2114:
2098:
2097:
2093:
2041:. A supermatrix
1939:
1937:
1936:
1931:
1917:
1916:
1915:
1907:
1902:
1894:
1833:
1831:
1830:
1825:
1817:
1816:
1808:
1795:
1793:
1792:
1787:
1782:
1781:
1774:
1773:
1765:
1761:
1760:
1745:
1744:
1731:
1730:
1722:
1718:
1717:
1702:
1701:
1659:
1657:
1656:
1651:
1643:
1642:
1641:
1633:
1611:
1610:
1602:
1585:
1583:
1582:
1577:
1575:
1574:
1566:
1553:
1551:
1550:
1545:
1543:
1542:
1535:
1534:
1524:
1523:
1515:
1510:
1509:
1499:
1498:
1490:
1483:
1482:
1467:
1466:
1369:
1367:
1366:
1361:
1355:
1354:
1342:
1341:
1326:
1325:
1313:
1312:
1297:
1296:
1263:
1261:
1260:
1255:
1250:
1249:
1242:
1241:
1232:
1231:
1219:
1218:
1209:
1208:
1197:
1196:
1187:
1186:
1174:
1173:
1164:
1163:
1150:
1149:
1140:
1139:
1127:
1126:
1117:
1116:
1105:
1104:
1095:
1094:
1082:
1081:
1072:
1071:
1051:
1050:
1043:
1042:
1031:
1030:
1017:
1016:
1005:
1004:
987:
986:
979:
978:
967:
966:
953:
952:
941:
940:
776:with dimension (
687:with dimension (
607:
605:
604:
599:
597:
596:
589:
576:
558:
542:
506:
504:
503:
498:
496:
495:
488:
472:
457:
444:
375:even supermatrix
268:
266:
265:
260:
258:
257:
250:
249:
238:
237:
224:
223:
212:
211:
172:with entries in
127:supercommutative
4221:
4220:
4216:
4215:
4214:
4212:
4211:
4210:
4191:
4190:
4184:
4171:
4165:
4148:
4145:
4098:
4082:
4063:
4062:
4054:
4019:
4009:
3990:
3962:
3949:
3915:
3914:
3901:
3885:
3801:
3761:
3760:
3722:
3691:
3666:
3627:
3626:
3617:
3605:
3572:
3537:
3536:
3504:
3499:
3498:
3456:
3443:
3426:
3425:
3403:
3375:
3358:
3357:
3334:
3324:
3311:
3297:
3296:
3273:
3260:
3247:
3230:
3229:
3183:
3182:
3177:
3171:
3170:
3165:
3155:
3141:
3136:
3135:
3075:
3074:
3069:
3063:
3062:
3057:
3047:
3035:
3034:
3024:
2991:
2978:
2945:
2920:
2906:
2905:
2874:
2873:
2868:
2859:
2858:
2850:
2840:
2823:
2810:
2802:
2801:
2761:
2760:
2750:
2748:
2738:
2718:
2703:
2702:
2692:
2672:
2661:
2651:
2644:
2627:
2622:
2621:
2561:
2560:
2555:
2549:
2548:
2543:
2533:
2521:
2520:
2511:
2499:
2491:
2462:
2449:
2432:
2419:
2408:
2407:
2358:
2356:
2345:
2302:
2291:
2290:
2236:
2235:
2210:
2197:
2102:
2081:
2070:
2069:
1946:
1885:
1859:
1858:
1801:
1800:
1776:
1775:
1752:
1750:
1736:
1733:
1732:
1709:
1707:
1693:
1686:
1668:
1667:
1624:
1595:
1594:
1559:
1558:
1537:
1536:
1526:
1511:
1501:
1485:
1484:
1474:
1468:
1458:
1447:
1429:
1428:
1407:
1343:
1330:
1314:
1301:
1285:
1280:
1279:
1244:
1243:
1233:
1223:
1210:
1200:
1198:
1188:
1178:
1165:
1155:
1152:
1151:
1141:
1131:
1118:
1108:
1106:
1096:
1086:
1073:
1063:
1056:
1045:
1044:
1034:
1032:
1022:
1019:
1018:
1008:
1006:
996:
989:
981:
980:
970:
968:
958:
955:
954:
944:
942:
932:
925:
919:
918:
860:
831:
816:
767:
739:
678:
644:
611:If the scalars
591:
590:
577:
560:
559:
543:
525:
519:
518:
512:odd supermatrix
490:
489:
473:
459:
458:
445:
424:
418:
417:
408:
401:
390:
383:
369:
362:
326:are both zero.
310:has dimensions
309:
295:has dimensions
294:
280:total rows and
252:
251:
241:
239:
229:
226:
225:
215:
213:
203:
196:
184:
183:
178:block structure
113:(assumed to be
103:
45:
25:
12:
11:
5:
4219:
4217:
4209:
4208:
4203:
4193:
4192:
4189:
4188:
4182:
4169:
4163:
4144:
4141:
4140:
4139:
4127:
4121:
4118:
4115:
4112:
4109:
4106:
4101:
4097:
4094:
4089:
4085:
4081:
4077:
4074:
4071:
4052:
4046:
4045:
4034:
4029:
4026:
4022:
4016:
4012:
4008:
4005:
4002:
3997:
3993:
3987:
3984:
3979:
3975:
3969:
3965:
3961:
3956:
3952:
3948:
3945:
3942:
3939:
3936:
3933:
3929:
3926:
3923:
3899:
3884:
3881:
3869:
3868:
3856:
3853:
3850:
3847:
3843:
3840:
3837:
3830:
3826:
3822:
3817:
3813:
3809:
3804:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3775:
3772:
3769:
3747:
3746:
3734:
3729:
3725:
3721:
3717:
3714:
3707:
3703:
3699:
3694:
3690:
3687:
3684:
3681:
3678:
3673:
3669:
3665:
3661:
3658:
3654:
3651:
3648:
3645:
3641:
3638:
3635:
3615:
3604:
3601:
3593:
3592:
3579:
3575:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3534:
3522:
3519:
3516:
3511:
3507:
3492:
3491:
3477:
3474:
3468:
3463:
3459:
3455:
3450:
3446:
3442:
3439:
3436:
3433:
3423:
3410:
3406:
3402:
3396:
3393:
3387:
3382:
3378:
3374:
3371:
3368:
3365:
3355:
3341:
3337:
3331:
3327:
3323:
3318:
3314:
3310:
3307:
3304:
3294:
3280:
3276:
3272:
3267:
3263:
3259:
3254:
3250:
3246:
3243:
3240:
3237:
3206:That is, the (
3204:
3203:
3192:
3187:
3181:
3178:
3176:
3173:
3172:
3169:
3166:
3164:
3161:
3160:
3158:
3153:
3148:
3144:
3131:) supermatrix
3093:
3092:
3079:
3073:
3070:
3068:
3065:
3064:
3061:
3058:
3056:
3053:
3052:
3050:
3045:
3042:
3023:
3020:
3019:
3018:
3006:
3001:
2998:
2994:
2988:
2985:
2981:
2974:
2970:
2966:
2961:
2957:
2953:
2948:
2944:
2941:
2938:
2935:
2930:
2927:
2923:
2919:
2916:
2913:
2895:
2894:
2883:
2878:
2872:
2869:
2867:
2864:
2861:
2860:
2857:
2854:
2851:
2849:
2846:
2845:
2843:
2838:
2833:
2830:
2826:
2820:
2817:
2813:
2809:
2779:
2778:
2765:
2757:
2753:
2749:
2745:
2741:
2734:
2730:
2726:
2721:
2717:
2714:
2711:
2708:
2705:
2704:
2699:
2695:
2688:
2684:
2680:
2675:
2671:
2668:
2665:
2662:
2658:
2654:
2650:
2649:
2647:
2642:
2637:
2634:
2630:
2617:) supermatrix
2579:
2578:
2565:
2559:
2556:
2554:
2551:
2550:
2547:
2544:
2542:
2539:
2538:
2536:
2531:
2528:
2509:
2503:supertranspose
2498:
2497:Supertranspose
2495:
2490:
2487:
2486:
2485:
2474:
2469:
2465:
2461:
2456:
2452:
2448:
2445:
2439:
2435:
2431:
2426:
2422:
2418:
2415:
2389:
2388:
2377:
2372:
2365:
2361:
2352:
2348:
2342:
2339:
2336:
2331:
2328:
2325:
2321:
2317:
2314:
2309:
2305:
2301:
2298:
2268:
2267:
2255:
2252:
2249:
2246:
2243:
2206:
2193:
2161:be free right
2154:supermodule).
2132:
2131:
2117:
2113:
2109:
2105:
2101:
2096:
2092:
2088:
2084:
2080:
2077:
2045:of dimension (
1945:
1942:
1941:
1940:
1929:
1926:
1923:
1920:
1914:
1910:
1906:
1901:
1897:
1893:
1888:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1823:
1820:
1814:
1811:
1797:
1796:
1785:
1780:
1771:
1768:
1759:
1755:
1751:
1749:
1743:
1739:
1735:
1734:
1728:
1725:
1716:
1712:
1708:
1706:
1700:
1696:
1692:
1691:
1689:
1684:
1681:
1678:
1675:
1661:
1660:
1649:
1646:
1640:
1636:
1632:
1627:
1623:
1620:
1617:
1614:
1608:
1605:
1572:
1569:
1555:
1554:
1541:
1533:
1529:
1521:
1518:
1512:
1508:
1504:
1496:
1493:
1487:
1486:
1481:
1477:
1472:
1469:
1465:
1461:
1456:
1453:
1452:
1450:
1445:
1442:
1439:
1436:
1424:is defined by
1406:
1403:
1371:
1370:
1358:
1353:
1350:
1346:
1340:
1337:
1333:
1329:
1324:
1321:
1317:
1311:
1308:
1304:
1300:
1295:
1292:
1288:
1265:
1264:
1253:
1248:
1240:
1236:
1230:
1226:
1222:
1217:
1213:
1207:
1203:
1199:
1195:
1191:
1185:
1181:
1177:
1172:
1168:
1162:
1158:
1154:
1153:
1148:
1144:
1138:
1134:
1130:
1125:
1121:
1115:
1111:
1107:
1103:
1099:
1093:
1089:
1085:
1080:
1076:
1070:
1066:
1062:
1061:
1059:
1054:
1049:
1041:
1037:
1033:
1029:
1025:
1021:
1020:
1015:
1011:
1007:
1003:
999:
995:
994:
992:
985:
977:
973:
969:
965:
961:
957:
956:
951:
947:
943:
939:
935:
931:
930:
928:
859:
858:Multiplication
856:
830:
827:
808:
759:
723:
662:
643:
640:
617:block diagonal
609:
608:
595:
588:
585:
582:
578:
575:
572:
569:
566:
562:
561:
557:
554:
551:
548:
544:
541:
538:
535:
531:
530:
528:
508:
507:
494:
487:
484:
481:
478:
474:
471:
468:
465:
461:
460:
456:
453:
450:
446:
443:
440:
437:
434:
430:
429:
427:
406:
399:
388:
381:
367:
360:
307:
292:
270:
269:
256:
248:
244:
240:
236:
232:
228:
227:
222:
218:
214:
210:
206:
202:
201:
199:
194:
191:
152:of dimension (
102:
99:
43:
13:
10:
9:
6:
4:
3:
2:
4218:
4207:
4204:
4202:
4199:
4198:
4196:
4185:
4183:0-8218-2012-5
4179:
4175:
4170:
4166:
4164:0-8218-3574-2
4160:
4156:
4152:
4147:
4146:
4142:
4125:
4099:
4095:
4087:
4083:
4061:
4060:
4059:
4056:
4051:
4032:
4027:
4024:
4014:
4010:
3995:
3991:
3985:
3982:
3977:
3973:
3967:
3963:
3959:
3954:
3950:
3940:
3934:
3913:
3912:
3911:
3909:
3905:
3898:
3894:
3890:
3882:
3880:
3878:
3874:
3851:
3848:
3824:
3811:
3798:
3795:
3789:
3783:
3780:
3759:
3758:
3757:
3755:
3750:
3727:
3723:
3701:
3688:
3685:
3679:
3671:
3667:
3652:
3646:
3625:
3624:
3623:
3621:
3614:
3610:
3602:
3600:
3598:
3577:
3569:
3566:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3535:
3520:
3517:
3514:
3509:
3505:
3497:
3496:
3495:
3472:
3466:
3461:
3457:
3453:
3448:
3440:
3437:
3434:
3424:
3408:
3404:
3400:
3391:
3385:
3380:
3372:
3369:
3366:
3356:
3339:
3335:
3329:
3325:
3321:
3316:
3308:
3305:
3295:
3278:
3274:
3270:
3265:
3261:
3257:
3252:
3244:
3241:
3238:
3228:
3227:
3226:
3223:
3221:
3217:
3213:
3209:
3190:
3185:
3179:
3174:
3167:
3162:
3156:
3151:
3146:
3142:
3134:
3133:
3132:
3130:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3098:
3077:
3071:
3066:
3059:
3054:
3048:
3043:
3040:
3033:
3032:
3031:
3029:
3021:
3004:
2999:
2996:
2992:
2986:
2983:
2979:
2968:
2955:
2942:
2939:
2933:
2928:
2925:
2917:
2914:
2904:
2903:
2902:
2900:
2881:
2876:
2870:
2865:
2862:
2855:
2852:
2847:
2841:
2836:
2831:
2828:
2818:
2815:
2811:
2800:
2799:
2798:
2796:
2792:
2788:
2784:
2763:
2755:
2751:
2743:
2739:
2728:
2715:
2712:
2706:
2697:
2693:
2682:
2669:
2666:
2656:
2652:
2645:
2640:
2635:
2632:
2628:
2620:
2619:
2618:
2616:
2612:
2608:
2604:
2600:
2596:
2592:
2588:
2584:
2563:
2557:
2552:
2545:
2540:
2534:
2529:
2526:
2519:
2518:
2517:
2515:
2508:
2504:
2496:
2494:
2488:
2472:
2467:
2463:
2459:
2454:
2450:
2446:
2443:
2437:
2433:
2429:
2424:
2420:
2416:
2413:
2406:
2405:
2404:
2402:
2398:
2394:
2375:
2370:
2363:
2359:
2350:
2346:
2340:
2337:
2334:
2329:
2326:
2323:
2319:
2315:
2307:
2303:
2296:
2289:
2288:
2287:
2285:
2281:
2277:
2273:
2253:
2247:
2244:
2241:
2234:
2233:
2232:
2230:
2226:
2222:
2218:
2214:
2209:
2205:
2201:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2164:
2160:
2155:
2153:
2149:
2145:
2141:
2137:
2115:
2107:
2103:
2094:
2086:
2082:
2078:
2075:
2068:
2067:
2066:
2064:
2060:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2019:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1986:
1982:
1979:
1974:
1971:
1967:
1963:
1959:
1955:
1954:vector spaces
1951:
1943:
1927:
1924:
1921:
1918:
1908:
1895:
1882:
1879:
1873:
1870:
1867:
1864:
1857:
1856:
1855:
1853:
1849:
1845:
1841:
1837:
1821:
1818:
1809:
1783:
1778:
1766:
1757:
1753:
1747:
1741:
1737:
1723:
1714:
1710:
1704:
1698:
1694:
1687:
1682:
1679:
1676:
1673:
1666:
1665:
1664:
1647:
1644:
1634:
1621:
1618:
1612:
1603:
1593:
1592:
1591:
1589:
1567:
1539:
1531:
1527:
1516:
1506:
1502:
1491:
1479:
1475:
1470:
1463:
1459:
1454:
1448:
1443:
1440:
1437:
1434:
1427:
1426:
1425:
1423:
1419:
1415:
1411:
1404:
1402:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1356:
1351:
1348:
1344:
1338:
1335:
1331:
1327:
1322:
1319:
1315:
1309:
1306:
1302:
1298:
1293:
1290:
1286:
1278:
1277:
1276:
1275:are given by
1274:
1270:
1251:
1246:
1238:
1234:
1228:
1224:
1220:
1215:
1211:
1205:
1201:
1193:
1189:
1183:
1179:
1175:
1170:
1166:
1160:
1156:
1146:
1142:
1136:
1132:
1128:
1123:
1119:
1113:
1109:
1101:
1097:
1091:
1087:
1083:
1078:
1074:
1068:
1064:
1057:
1052:
1047:
1039:
1035:
1027:
1023:
1013:
1009:
1001:
997:
990:
983:
975:
971:
963:
959:
949:
945:
937:
933:
926:
917:
916:
915:
913:
909:
905:
901:
897:
896:ungraded case
893:
889:
885:
881:
877:
873:
869:
865:
857:
855:
852:
851:ungraded case
848:
844:
840:
836:
828:
826:
824:
820:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
766:
762:
758:
753:
751:
747:
743:
738:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
690:
686:
682:
677:
673:
669:
665:
661:
656:
654:
650:
641:
639:
637:
633:
629:
625:
620:
618:
614:
593:
526:
517:
516:
515:
513:
492:
425:
416:
415:
414:
412:
405:
398:
394:
387:
380:
376:
371:
370:are as well.
366:
359:
355:
352:
348:
344:
340:
336:
332:
327:
325:
321:
317:
313:
306:
302:
298:
291:
287:
283:
279:
275:
254:
246:
242:
234:
230:
220:
216:
208:
204:
197:
192:
189:
182:
181:
180:
179:
175:
171:
167:
163:
159:
155:
151:
147:
143:
139:
135:
130:
128:
124:
120:
116:
112:
108:
100:
98:
96:
95:supersymmetry
92:
88:
84:
80:
75:
73:
69:
65:
61:
57:
53:
49:
42:
38:
34:
30:
23:
19:
4173:
4154:
4150:
4057:
4049:
4047:
3907:
3896:
3886:
3876:
3872:
3870:
3753:
3751:
3748:
3612:
3606:
3596:
3594:
3493:
3224:
3219:
3215:
3211:
3207:
3205:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3094:
3027:
3025:
2898:
2896:
2794:
2786:
2782:
2780:
2614:
2610:
2606:
2602:
2598:
2594:
2590:
2586:
2582:
2580:
2506:
2502:
2500:
2492:
2400:
2396:
2392:
2390:
2283:
2279:
2275:
2271:
2269:
2228:
2224:
2220:
2216:
2212:
2207:
2203:
2199:
2194:
2190:
2186:
2182:
2178:
2174:
2170:
2166:
2162:
2158:
2156:
2151:
2147:
2143:
2139:
2135:
2133:
2065:-linear map
2062:
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2020:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1985:superalgebra
1980:
1975:
1969:
1958:free modules
1947:
1851:
1847:
1843:
1839:
1835:
1798:
1662:
1587:
1556:
1421:
1417:
1413:
1408:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1372:
1272:
1268:
1266:
911:
907:
903:
899:
894:) as in the
891:
887:
883:
879:
875:
871:
867:
863:
861:
846:
842:
838:
834:
832:
822:
818:
813:
809:
805:
797:
789:
785:
781:
777:
773:
769:
764:
760:
756:
754:
749:
741:
736:
732:
728:
724:
720:
716:
712:
708:
700:
696:
692:
688:
684:
680:
675:
671:
667:
663:
659:
657:
652:
648:
645:
635:
631:
627:
623:
621:
612:
610:
511:
509:
410:
403:
396:
392:
385:
378:
374:
372:
364:
357:
350:
346:
342:
338:
334:
330:
328:
323:
319:
315:
311:
304:
300:
296:
289:
285:
281:
277:
273:
271:
173:
165:
161:
157:
153:
149:
145:
141:
137:
133:
131:
122:
111:superalgebra
106:
104:
91:supermodules
76:
56:superalgebra
52:block matrix
40:
36:
26:
22:block matrix
3904:determinant
3494:as well as
1978:supermodule
1950:linear maps
705:supermodule
624:homogeneous
150:supermatrix
119:associative
109:be a fixed
66:(such as a
37:supermatrix
29:mathematics
4195:Categories
4143:References
3889:Berezinian
3883:Berezinian
3609:supertrace
3603:Supertrace
2791:involution
2489:Operations
744:) forms a
4025:−
3983:−
3960:−
3796:−
3686:−
3680:−
3558:π
3555:∘
3546:∘
3543:π
3506:π
3476:^
3473:α
3467:⋅
3462:π
3449:π
3441:α
3438:⋅
3409:π
3401:⋅
3395:^
3392:α
3381:π
3370:⋅
3367:α
3340:π
3330:π
3317:π
3279:π
3266:π
3253:π
3218:,1−
3147:π
2940:−
2863:−
2853:−
2713:−
2707:−
2667:−
2514:transpose
2460:⊕
2430:⊕
2320:∑
2251:→
2223:and from
2202:and let (
2100:→
1925:α
1922:⋅
1896:α
1880:−
1868:⋅
1865:α
1822:α
1813:^
1810:α
1770:^
1767:α
1748:α
1727:^
1724:α
1705:α
1680:α
1677:⋅
1645:α
1635:α
1619:−
1607:^
1604:α
1571:^
1568:α
1520:^
1517:α
1495:^
1492:α
1471:α
1455:α
1438:⋅
1435:α
794:superring
60:superring
4201:Matrices
3123:)×(
3115:is the (
3103:)×(
2609:)×(
2601:is the (
2589:)×(
2278:)×(
2173:and let
2053:)×(
1952:between
906:)×(
886:)×(
870:)×(
841:)×(
829:Addition
160:)×(
89:or free
2189:. Let (
1983:over a
1389:| then
1385:| and |
168:) is a
4180:
4161:
3595:where
3095:be a (
2797:gives
2781:where
2516:. Let
1416:. Let
719:then
628:parity
354:square
331:square
314:×
299:×
170:matrix
144:, and
115:unital
48:matrix
3620:trace
2152:right
2150:as a
2014:. If
1397:| + |
800:is a
748:over
707:over
341:) = (
272:with
72:field
39:is a
4178:ISBN
4159:ISBN
3891:(or
3887:The
3875:and
3607:The
3026:The
2501:The
2399:and
2157:Let
2021:Let
1992:free
1970:even
1964:(or
1956:(or
1842:and
1377:and
755:Let
658:Let
402:and
384:and
363:and
322:and
303:and
132:Let
117:and
105:Let
58:(or
35:, a
31:and
4055:).
4004:det
3944:det
3752:If
2897:If
2227:to
2219:to
2138:on
1990:is
1373:If
784:)Ă—(
695:)Ă—(
630:, |
510:An
373:An
125:be
27:In
4197::
4155:11
4015:11
3996:10
3978:11
3968:01
3955:00
3879:.
3728:11
3672:00
3597:st
1976:A
1758:11
1742:10
1715:01
1699:00
1532:11
1507:10
1480:01
1464:00
1391:XY
1273:XY
1271:=
1239:11
1229:11
1216:01
1206:10
1194:10
1184:11
1171:00
1161:10
1147:11
1137:01
1124:01
1114:00
1102:10
1092:01
1079:00
1069:00
1040:11
1028:10
1014:01
1002:00
976:11
964:10
950:01
938:00
825:.
752:.
655:.
413:.
407:10
400:01
389:11
382:00
368:11
361:00
329:A
308:11
293:00
247:11
235:10
221:01
209:00
140:,
136:,
97:.
4186:.
4167:.
4126:.
4120:)
4117:X
4114:(
4111:r
4108:t
4105:s
4100:e
4096:=
4093:)
4088:X
4084:e
4080:(
4076:r
4073:e
4070:B
4053:0
4050:R
4033:.
4028:1
4021:)
4011:X
4007:(
4001:)
3992:X
3986:1
3974:X
3964:X
3951:X
3947:(
3941:=
3938:)
3935:X
3932:(
3928:r
3925:e
3922:B
3908:R
3900:2
3897:Z
3877:Y
3873:X
3855:)
3852:X
3849:Y
3846:(
3842:r
3839:t
3836:s
3829:|
3825:Y
3821:|
3816:|
3812:X
3808:|
3803:)
3799:1
3793:(
3790:=
3787:)
3784:Y
3781:X
3778:(
3774:r
3771:t
3768:s
3754:R
3733:)
3724:X
3720:(
3716:r
3713:t
3706:|
3702:X
3698:|
3693:)
3689:1
3683:(
3677:)
3668:X
3664:(
3660:r
3657:t
3653:=
3650:)
3647:X
3644:(
3640:r
3637:t
3634:s
3616:2
3613:Z
3578:3
3574:)
3570:t
3567:s
3564:(
3561:=
3552:t
3549:s
3521:d
3518:i
3515:=
3510:2
3458:X
3454:=
3445:)
3435:X
3432:(
3405:X
3386:=
3377:)
3373:X
3364:(
3336:Y
3326:X
3322:=
3313:)
3309:Y
3306:X
3303:(
3275:Y
3271:+
3262:X
3258:=
3249:)
3245:Y
3242:+
3239:X
3236:(
3220:j
3216:i
3212:j
3210:,
3208:i
3191:.
3186:]
3180:A
3175:B
3168:C
3163:D
3157:[
3152:=
3143:X
3129:p
3127:|
3125:q
3121:r
3119:|
3117:s
3113:X
3109:q
3107:|
3105:p
3101:s
3099:|
3097:r
3078:]
3072:D
3067:C
3060:B
3055:A
3049:[
3044:=
3041:X
3005:.
3000:t
2997:s
2993:X
2987:t
2984:s
2980:Y
2973:|
2969:Y
2965:|
2960:|
2956:X
2952:|
2947:)
2943:1
2937:(
2934:=
2929:t
2926:s
2922:)
2918:Y
2915:X
2912:(
2899:R
2882:.
2877:]
2871:D
2866:C
2856:B
2848:A
2842:[
2837:=
2832:t
2829:s
2825:)
2819:t
2816:s
2812:X
2808:(
2795:X
2787:A
2783:A
2764:]
2756:t
2752:D
2744:t
2740:B
2733:|
2729:X
2725:|
2720:)
2716:1
2710:(
2698:t
2694:C
2687:|
2683:X
2679:|
2674:)
2670:1
2664:(
2657:t
2653:A
2646:[
2641:=
2636:t
2633:s
2629:X
2615:s
2613:|
2611:r
2607:q
2605:|
2603:p
2599:X
2595:q
2593:|
2591:p
2587:s
2585:|
2583:r
2564:]
2558:D
2553:C
2546:B
2541:A
2535:[
2530:=
2527:X
2510:2
2507:Z
2473:.
2468:1
2464:N
2455:0
2451:N
2447:=
2444:N
2438:1
2434:M
2425:0
2421:M
2417:=
2414:M
2401:N
2397:M
2393:T
2376:.
2371:i
2364:k
2360:T
2351:k
2347:f
2341:s
2338:+
2335:r
2330:1
2327:=
2324:k
2316:=
2313:)
2308:i
2304:e
2300:(
2297:T
2284:q
2282:|
2280:p
2276:s
2274:|
2272:r
2254:N
2248:M
2245::
2242:T
2229:R
2225:N
2221:R
2217:M
2213:N
2208:k
2204:f
2200:M
2195:i
2191:e
2187:s
2185:|
2183:r
2179:R
2175:N
2171:q
2169:|
2167:p
2163:R
2159:M
2148:R
2144:R
2140:R
2136:T
2116:s
2112:|
2108:r
2104:R
2095:q
2091:|
2087:p
2083:R
2079::
2076:T
2063:R
2059:q
2057:|
2055:p
2051:s
2049:|
2047:r
2043:T
2035:R
2031:q
2029:|
2027:p
2023:R
2016:R
2012:q
2010:|
2008:p
2004:M
2000:q
1996:p
1988:R
1981:M
1928:.
1919:X
1913:|
1909:X
1905:|
1900:|
1892:|
1887:)
1883:1
1877:(
1874:=
1871:X
1852:R
1848:X
1844:X
1840:X
1836:X
1819:=
1784:.
1779:]
1754:X
1738:X
1711:X
1695:X
1688:[
1683:=
1674:X
1648:.
1639:|
1631:|
1626:)
1622:1
1616:(
1613:=
1588:R
1540:]
1528:X
1503:X
1476:X
1460:X
1449:[
1444:=
1441:X
1422:R
1418:X
1414:R
1399:Y
1395:X
1387:Y
1383:X
1379:Y
1375:X
1357:.
1352:j
1349:1
1345:Y
1339:1
1336:i
1332:X
1328:+
1323:j
1320:0
1316:Y
1310:0
1307:i
1303:X
1299:=
1294:j
1291:i
1287:Z
1269:Z
1252:.
1247:]
1235:Y
1225:X
1221:+
1212:Y
1202:X
1190:Y
1180:X
1176:+
1167:Y
1157:X
1143:Y
1133:X
1129:+
1120:Y
1110:X
1098:Y
1088:X
1084:+
1075:Y
1065:X
1058:[
1053:=
1048:]
1036:Y
1024:Y
1010:Y
998:Y
991:[
984:]
972:X
960:X
946:X
934:X
927:[
912:l
910:|
908:k
904:s
902:|
900:r
892:l
890:|
888:k
884:q
882:|
880:p
876:q
874:|
872:p
868:s
866:|
864:r
847:q
845:|
843:p
839:s
837:|
835:r
823:R
819:R
817:(
814:q
812:|
810:p
806:M
798:R
790:q
788:|
786:p
782:q
780:|
778:p
774:R
770:R
768:(
765:q
763:|
761:p
757:M
750:K
742:R
740:(
737:q
735:|
733:p
731:Ă—
729:s
727:|
725:r
721:M
717:K
713:R
709:R
701:q
699:|
697:p
693:s
691:|
689:r
685:R
681:R
679:(
676:q
674:|
672:p
670:Ă—
668:s
666:|
664:r
660:M
653:R
649:R
636:X
632:X
613:R
594:]
587:d
584:d
581:o
574:n
571:e
568:v
565:e
556:n
553:e
550:v
547:e
540:d
537:d
534:o
527:[
493:]
486:n
483:e
480:v
477:e
470:d
467:d
464:o
455:d
452:d
449:o
442:n
439:e
436:v
433:e
426:[
411:R
404:X
397:X
393:R
386:X
379:X
365:X
358:X
351:X
347:q
345:|
343:p
339:s
337:|
335:r
324:s
320:q
316:q
312:s
305:X
301:p
297:r
290:X
286:q
284:+
282:p
278:s
276:+
274:r
255:]
243:X
231:X
217:X
205:X
198:[
193:=
190:X
174:R
166:q
164:|
162:p
158:s
156:|
154:r
146:s
142:r
138:q
134:p
123:R
107:R
44:2
41:Z
24:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.