5645:
5133:
5640:{\displaystyle {\begin{aligned}&\chi _{{\mathcal {V}}_{c,h_{r,s}}/({\mathcal {V}}_{c,h_{r,s}+rs}+{\mathcal {V}}_{c,h_{r,s}+(p-r)(p'-s)})}\\&=\sum _{k\in \mathbb {Z} }\left(\chi _{{\mathcal {V}}_{c,{\frac {1}{4pp'}}\left((p'r-ps+2kpp')^{2}-(p-p')^{2}\right)}}-\chi _{{\mathcal {V}}_{c,{\frac {1}{4pp'}}\left((p'r+ps+2kpp')^{2}-(p-p')^{2}\right)}}\right).\end{aligned}}}
41:
4238:
4842:
3592:
5850:
obeys the commutation relations of (two copies of) the
Virasoro algebra. This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones. Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes. This is known
5126:
has infinitely many singular vectors, and is therefore reducible with infinitely many submodules. This Verma module has an irreducible quotient by its largest nontrivial submodule. (The spectrums of minimal models are built from such irreducible representations.) The character of the irreducible
3961:
2260:
4577:
3413:
3913:
5957:. On a higher-genus compact Riemann surface, the Lie algebra of meromorphic vector fields with two poles also has a central extension, which is a generalization of the Virasoro algebra. This can be further generalized to supermanifolds.
2908:
3300:
4233:{\displaystyle \chi _{{\mathcal {V}}_{c,h}}(q)={\frac {q^{h-{\frac {c}{24}}}}{\prod _{n=1}^{\infty }(1-q^{n})}}={\frac {q^{h-{\frac {c-1}{24}}}}{\eta (q)}}=q^{h-{\frac {c}{24}}}\left(1+q+2q^{2}+3q^{3}+5q^{4}+\cdots \right),}
3736:
6013:, though he did not find the central extension. The central extension giving the Virasoro algebra was rediscovered in physics shortly after by J. H. Weis, according to Brower and Thorn (1971, footnote on page 167).
1063:
2057:
2488:
5795:
4837:{\displaystyle \chi _{{\mathcal {V}}_{c,h_{r,s}}/{\mathcal {V}}_{c,h_{r,s}+rs}}=\chi _{{\mathcal {V}}_{c,h_{r,s}}}-\chi _{{\mathcal {V}}_{c,h_{r,s}+rs}}=(1-q^{rs})\chi _{{\mathcal {V}}_{c,h_{r,s}}}.}
1763:
1456:
1825:
1518:
5138:
1234:
5707:
4469:
2790:
5124:
4569:
4519:
4387:
2558:
4337:
3134:
1598:
5038:
3953:
2348:
1926:
1669:
3587:{\displaystyle c\in \left\{1-{\frac {6}{m(m+1)}}\right\}_{m=2,3,4,\ldots }=\left\{0,{\frac {1}{2}},{\frac {7}{10}},{\frac {4}{5}},{\frac {6}{7}},{\frac {25}{28}},\ldots \right\}}
2675:
4926:
4296:
4995:
4889:
3821:
3025:. It can be useful to use non-integer Kac indices for parametrizing the conformal dimensions of Verma modules that do not have singular vectors, for example in the critical
2944:
2723:
2311:
1994:
486:
461:
424:
3829:
6379:
3019:
4957:
6533:
2381:
1851:
1544:
1343:
5070:
4261:
3342:
3086:
1869:, called degenerate representations, which are cosets of the Verma module. In particular, the unique irreducible highest weight representation with these values of
1373:
1291:
2049:
1094:
2977:
2584:
2407:
2020:
1317:
788:
4410:
2813:
920:
6484:
L. D. Faddeev (ed.) A. A. Mal'tsev (ed.), Topology. Proc. Internat. Topol. Conf. Leningrad 1982, Lect. notes in math., 1060, Springer (1984) pp. 230–245
3382:
3362:
3165:
3055:
2627:
2607:
1946:
1618:
1261:
1160:
2821:
6781:
3177:
3607:
3140:
if that
Hermitian form is positive definite. Since any singular vector has zero norm, all unitary highest weight representations are irreducible.
6738:
Dobrev, V. K. (1986). "Multiplet classification of the indecomposable highest weight modules over the Neveu-Schwarz and Ramond superalgebras".
6558:
5924:
5819:
826:
346:
6629:
928:
2255:{\displaystyle h_{r,s}(c)={\frac {1}{4}}{\Big (}(b+b^{-1})^{2}-(br+b^{-1}s)^{2}{\Big )}\ ,\quad {\text{where}}\quad c=1+6(b+b^{-1})^{2}\ .}
810:
296:
3314:
781:
291:
6731:
6222:
2412:
5989:
in about the 1930s. The central extension of the Witt algebra that gives the
Virasoro algebra was first found (in characteristic
5973:
counterparts, which basically come from arranging all the basis elements into generating series and working with single objects.
6235:
Krichever, I. M.; Novikov, S.P. (1987). "Algebras of
Virasoro type, Riemann surfaces and structures of the theory of solitons".
6834:
1139:
707:
5712:
1685:
1378:
6779:
V. K. Dobrev, "Characters of the irreducible highest weight modules over the
Virasoro and super-Virasoro algebras", Suppl.
5903:
1768:
1461:
6844:
6706:
774:
6488:
Friedan, D., Qiu, Z. and
Shenker, S. (1984). "Conformal invariance, unitarity and critical exponents in two dimensions".
3787:(identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine
6331:
5953:
The
Virasoro algebra is a central extension of the Lie algebra of meromorphic vector fields with two poles on a genus 0
5860:
6701:
391:
205:
5941:
The
Virasoro algebra is a subalgebra of the universal enveloping algebra of any affine Lie algebra, as shown by the
1888:
A singular vector or null vector of a highest weight representation is a state that is both descendant and primary.
1861:
it is also irreducible. When it is reducible, there exist other highest weight representations with these values of
1168:
6810:(2010). "Direct proofs of the Feigin-Fuchs character formula for unitary representations of the Virasoro algebra".
5653:
4415:
6028:
5891:
3768:
5843:
5079:
4524:
4474:
4342:
5827:
2493:
589:
323:
200:
88:
6548:
Funct. Anal. Appl., 2 (1968) pp. 342–343 Funkts. Anal. i
Prilozh., 2 : 4 (1968) pp. 92–93
4301:
3091:
6490:
6335:
5966:
5923:
W-algebras are associative algebras which contain the
Virasoro algebra, and which play an important role in
5073:
2947:
1549:
1108:
5830:, special functions that include and generalize the characters of representations of the Virasoro algebra.
6717:
6048:
6023:
5877:
3921:
2316:
1894:
1637:
739:
529:
3788:
1096:
is merely a matter of convention. For a derivation of the algebra as the unique central extension of the
6839:
6527:
6038:
5970:
5942:
4264:
1101:
818:
613:
4273:
2728:
6696:
6005:(1968). Virasoro (1970) wrote down some operators generating the Virasoro algebra (later known as the
5000:
6747:
6667:
6567:
6499:
6425:
6353:
6289:
6177:
6134:
6095:
6010:
5859:, cannot be applied to all the states in the theory, but rather only on the physical states (compare
4962:
3026:
553:
541:
159:
93:
3908:{\displaystyle \chi _{\mathcal {R}}(q)=\operatorname {Tr} _{\mathcal {R}}q^{L_{0}-{\frac {c}{24}}}.}
3802:
5852:
5823:
4894:
128:
23:
2917:
2680:
2268:
1951:
469:
444:
407:
6811:
6793:
6763:
6683:
6657:
6591:
6515:
6398:
6305:
6279:
6252:
6193:
6150:
6033:
5936:
3780:
113:
85:
4850:
2982:
1853:
form a basis of the Verma module. The Verma module is indecomposable, and for generic values of
2632:
6807:
6789:
6727:
6648:
6625:
6416:
6344:
6327:
6218:
6086:
5856:
684:
518:
361:
255:
2815:. This type of singular vectors can however only exist if the central charge is of the type
2353:
1830:
1523:
1322:
6755:
6675:
6617:
6599:
6575:
6507:
6469:
6459:
6433:
6388:
6361:
6297:
6244:
6185:
6142:
6103:
6043:
5899:
5883:
3784:
3144:
669:
661:
653:
645:
637:
625:
565:
505:
495:
337:
279:
154:
123:
6639:
6587:
5043:
4246:
3320:
3064:
1351:
1269:
6635:
6603:
6583:
6473:
5954:
4931:
3764:
2025:
1071:
753:
746:
732:
689:
577:
500:
330:
244:
184:
64:
2956:
2563:
2386:
1999:
1296:
6751:
6671:
6571:
6503:
6429:
6357:
6293:
6181:
6138:
6099:
5981:
The Witt algebra (the Virasoro algebra without the central extension) was discovered by
4392:
2903:{\displaystyle c=1-6{\frac {(p-q)^{2}}{pq}}\quad {\text{with}}\quad p,q\in \mathbb {Z} }
2795:
880:
5927:. Among W-algebras, the Virasoro algebra has the particularity of being a Lie algebra.
5895:
3760:
3367:
3347:
3150:
3058:
3040:
2612:
2592:
1931:
1603:
1246:
1145:
870:
760:
696:
386:
366:
303:
268:
189:
179:
164:
149:
103:
80:
6393:
6374:
5902:. There are further extensions of these algebras with more supersymmetry, such as the
4412:. This singular vector generates a submodule, which is isomorphic to the Verma module
6828:
6767:
6679:
6541:
6519:
6437:
6365:
6301:
6197:
6154:
5998:
5847:
5811:
830:
679:
601:
435:
308:
174:
6687:
6595:
6309:
6256:
6084:
M. A. Virasoro (1970). "Subsidiary conditions and ghosts in dual-resonance models".
3295:{\displaystyle A_{N}\prod _{1\leq r,s\leq N}{\big (}h-h_{r,s}(c){\big )}^{p(N-rs)},}
6058:
6002:
5815:
2792:. This singular vector is now a descendant of another singular vector at the level
1632:
1142:
of the Virasoro algebra is a representation generated by a primary state: a vector
1097:
822:
534:
233:
222:
169:
144:
139:
98:
69:
32:
6712:
V. G. Kac, "Highest weight representations of infinite dimensional Lie algebras",
5982:
5838:
Since the Virasoro algebra comprises the generators of the conformal group of the
6553:
6122:
6511:
5994:
3776:
3772:
846:
814:
802:
3731:{\displaystyle h=h_{r,s}(c)={\frac {{\big (}(m+1)r-ms{\big )}^{2}-1}{4m(m+1)}}}
6621:
5839:
701:
429:
5945:. In this sense, affine Lie algebras are extensions of the Virasoro algebra.
6107:
6063:
6053:
5918:
522:
6616:, Springer Monographs in Mathematics, London: Springer-Verlag London Ltd.,
3388:(1978), and its first published proof was given by Feigin and Fuks (1984).
6168:
Uretsky, J. L. (1989). "Redundancy of conditions for a Virasoro algebra".
40:
5997:(1966, page 381) and independently rediscovered (in characteristic 0) by
5898:. Their theory is similar to that of the Virasoro algebra, now involving
5797:
have a nontrivial intersection, which is itself a complicated submodule.
2609:
is not necessary. In particular, there is a singular vector at the level
1880:
A Verma module is irreducible if and only if it has no singular vectors.
1671:
is the largest possible highest weight representation. (The same letter
59:
6759:
6662:
6579:
6464:
6447:
6402:
6284:
6248:
6189:
6146:
5986:
3136:
and the norm of the primary state is one. The representation is called
401:
315:
1058:{\displaystyle =(m-n)L_{m+n}+{\frac {c}{12}}(m^{3}-m)\delta _{m+n,0}.}
6554:"Unitary representations of the Virasoro and super-Virasoro algebras"
6340:"Infinite conformal symmetry in two-dimensional quantum field theory"
3385:
2490:, and the corresponding reducible Verma module has a singular vector
6411:
6339:
1600:. Any state whose level is not zero is called a descendant state of
2589:
This condition for the existence of a singular vector at the level
6816:
6798:
6375:"On the Mills–Seligman axioms for Lie algebras of classical type"
1679:
of the Virasoro algebra and its eigenvalue in a representation.)
4389:
is reducible due to the existence of a singular vector at level
6546:
The cohomology of the Lie algebra of vector fields in a circle
2483:{\displaystyle h_{2,1}(c)=-{\frac {1}{2}}-{\frac {3}{4}}b^{2}}
1877:
is the quotient of the Verma module by its maximal submodule.
1348:
More precisely, a highest weight representation is spanned by
6123:"A presentation for the Virasoro and super-Virasoro algebras"
1266:
A highest weight representation is spanned by eigenstates of
6792:(2010). "Lecture notes on Kac-Moody and Virasoro algebras".
6646:
A. Kent (1991). "Singular vectors of the Virasoro algebra".
5719:
5660:
5493:
5350:
5236:
5191:
5150:
5086:
4799:
4722:
4679:
4627:
4589:
4571:
does not have other singular vectors, and its character is
4531:
4481:
4422:
4349:
3973:
3928:
3863:
3839:
3808:
2323:
1901:
1644:
6448:"Les groupes de transformations continus, infinis, simples"
6412:"Eliminating spurious states from the dual resonance model"
3391:
The irreducible highest weight representation with values
5650:
This expression is an infinite sum because the submodules
5985:(1909). Its analogues over finite fields were studied by
1243:
is called the conformal dimension or conformal weight of
3767:(1984) showed that these conditions are necessary, and
5790:{\displaystyle {\mathcal {V}}_{c,h_{r,s}+(p-r)(p'-s)}}
1758:{\displaystyle L_{-n_{1}}L_{-n_{2}}\cdots L_{-n_{k}}v}
1451:{\displaystyle L_{-n_{1}}L_{-n_{2}}\cdots L_{-n_{k}}v}
5715:
5656:
5136:
5082:
5046:
5003:
4965:
4934:
4897:
4853:
4580:
4527:
4477:
4418:
4395:
4345:
4304:
4276:
4249:
3964:
3924:
3832:
3805:
3610:
3416:
3370:
3350:
3323:
3180:
3153:
3094:
3067:
3043:
3037:
A highest weight representation with a real value of
2985:
2959:
2920:
2824:
2798:
2731:
2683:
2635:
2615:
2595:
2566:
2496:
2415:
2389:
2356:
2319:
2271:
2060:
2028:
2002:
1954:
1934:
1897:
1833:
1820:{\displaystyle 0<n_{1}\leq n_{2}\leq \cdots n_{k}}
1771:
1688:
1640:
1606:
1552:
1526:
1513:{\displaystyle 0<n_{1}\leq n_{2}\leq \cdots n_{k}}
1464:
1381:
1354:
1345:
is called the level of the corresponding eigenstate.
1325:
1299:
1272:
1249:
1171:
1148:
1074:
931:
883:
472:
447:
410:
6452:
Annales Scientifiques de l'École Normale Supérieure
5789:
5701:
5639:
5118:
5064:
5032:
4989:
4951:
4920:
4883:
4836:
4563:
4513:
4463:
4404:
4381:
4331:
4290:
4255:
4232:
3947:
3907:
3815:
3730:
3586:
3376:
3356:
3336:
3294:
3159:
3128:
3080:
3049:
3013:
2971:
2938:
2902:
2807:
2784:
2717:
2669:
2621:
2601:
2578:
2552:
2482:
2401:
2375:
2342:
2305:
2254:
2043:
2014:
1988:
1940:
1920:
1845:
1819:
1757:
1663:
1612:
1592:
1538:
1512:
1450:
1367:
1337:
1311:
1285:
1255:
1228:
1154:
1088:
1057:
914:
480:
455:
418:
6724:Bombay lectures on highest weight representations
6380:Transactions of the American Mathematical Society
6121:Fairlie, D. B.; Nuyts, J.; Zachos, C. K. (1988).
2181:
2101:
3344:is a positive constant that does not depend on
6270:Rabin, J. M. (1995). "Super elliptic curves".
6213:P. Di Francesco, P. Mathieu, and D. Sénéchal,
2946:coprime, these are the central charges of the
1229:{\displaystyle L_{n>0}v=0,\quad L_{0}v=hv,}
6614:Representation theory of the Virasoro algebra
5949:Meromorphic vector fields on Riemann surfaces
5702:{\displaystyle {\mathcal {V}}_{c,h_{r,s}+rs}}
4464:{\displaystyle {\mathcal {V}}_{c,h_{r,s}+rs}}
3685:
3650:
3384:. The Kac determinant formula was stated by
3260:
3221:
782:
8:
6782:Rendiconti del Circolo Matematico di Palermo
6532:: CS1 maint: multiple names: authors list (
1891:A sufficient condition for the Verma module
6552:P. Goddard, A. Kent & D. Olive (1986).
5826:approach to two-dimensional CFT relies on
5822:is the Virasoro algebra. Technically, the
5818:. It follows that the symmetry algebra of
5119:{\displaystyle {\mathcal {V}}_{c,h_{r,s}}}
4564:{\displaystyle {\mathcal {V}}_{c,h_{r,s}}}
4514:{\displaystyle {\mathcal {V}}_{c,h_{r,s}}}
4382:{\displaystyle {\mathcal {V}}_{c,h_{r,s}}}
789:
775:
227:
53:
18:
6815:
6797:
6661:
6463:
6392:
6283:
5735:
5724:
5718:
5717:
5714:
5676:
5665:
5659:
5658:
5655:
5610:
5580:
5505:
5498:
5492:
5491:
5489:
5467:
5437:
5362:
5355:
5349:
5348:
5346:
5331:
5330:
5323:
5252:
5241:
5235:
5234:
5207:
5196:
5190:
5189:
5180:
5166:
5155:
5149:
5148:
5146:
5137:
5135:
5102:
5091:
5085:
5084:
5081:
5072:is in the Kac table of the corresponding
5045:
5002:
4964:
4933:
4896:
4864:
4852:
4815:
4804:
4798:
4797:
4795:
4779:
4738:
4727:
4721:
4720:
4718:
4695:
4684:
4678:
4677:
4675:
4643:
4632:
4626:
4625:
4619:
4605:
4594:
4588:
4587:
4585:
4579:
4547:
4536:
4530:
4529:
4526:
4497:
4486:
4480:
4479:
4476:
4438:
4427:
4421:
4420:
4417:
4394:
4365:
4354:
4348:
4347:
4344:
4323:
4319:
4318:
4303:
4284:
4283:
4275:
4248:
4210:
4194:
4178:
4142:
4135:
4092:
4085:
4079:
4064:
4045:
4034:
4017:
4010:
4004:
3978:
3972:
3971:
3969:
3963:
3933:
3927:
3926:
3923:
3890:
3881:
3876:
3862:
3861:
3838:
3837:
3831:
3807:
3806:
3804:
3690:
3684:
3683:
3649:
3648:
3645:
3621:
3609:
3563:
3550:
3537:
3524:
3511:
3467:
3435:
3415:
3369:
3349:
3328:
3322:
3265:
3259:
3258:
3236:
3220:
3219:
3195:
3185:
3179:
3152:
3117:
3104:
3099:
3093:
3072:
3066:
3042:
2990:
2984:
2958:
2919:
2896:
2895:
2877:
2859:
2840:
2823:
2797:
2751:
2730:
2694:
2682:
2634:
2614:
2594:
2565:
2553:{\displaystyle (L_{-1}^{2}+b^{2}L_{-2})v}
2535:
2525:
2512:
2504:
2495:
2474:
2460:
2447:
2420:
2414:
2388:
2361:
2355:
2328:
2322:
2321:
2318:
2276:
2270:
2240:
2227:
2193:
2180:
2179:
2173:
2157:
2132:
2119:
2100:
2099:
2089:
2065:
2059:
2027:
2001:
1965:
1953:
1933:
1906:
1900:
1899:
1896:
1832:
1811:
1795:
1782:
1770:
1744:
1736:
1721:
1713:
1701:
1693:
1687:
1649:
1643:
1642:
1639:
1605:
1584:
1574:
1563:
1551:
1525:
1504:
1488:
1475:
1463:
1437:
1429:
1414:
1406:
1394:
1386:
1380:
1359:
1353:
1324:
1298:
1277:
1271:
1248:
1205:
1176:
1170:
1147:
1078:
1073:
1034:
1015:
998:
983:
952:
939:
930:
897:
882:
474:
473:
471:
449:
448:
446:
412:
411:
409:
5810:In two dimensions, the algebra of local
3823:of the Virasoro algebra is the function
16:Algebra describing 2D conformal symmetry
6714:Proc. Internat. Congress Mathematicians
6612:Iohara, Kenji; Koga, Yoshiyuki (2011),
6482:Verma modules over the Virasoro algebra
6076:
4332:{\displaystyle r,s\in \mathbb {N} ^{*}}
3129:{\displaystyle L_{n}^{\dagger }=L_{-n}}
1928:to have a singular vector at the level
345:
111:
21:
6559:Communications in Mathematical Physics
6525:
6170:Communications in Mathematical Physics
6127:Communications in Mathematical Physics
5961:Vertex algebras and conformal algebras
5925:two-dimensional conformal field theory
5820:two-dimensional conformal field theory
1593:{\displaystyle N=\sum _{i=1}^{k}n_{i}}
827:two-dimensional conformal field theory
347:Classification of finite simple groups
6209:
6207:
7:
5890:of the Virasoro algebra, called the
4521:by this submodule is irreducible if
3948:{\displaystyle {\mathcal {V}}_{c,h}}
3791:) to show that they are sufficient.
2343:{\displaystyle {\mathcal {V}}_{c,0}}
1921:{\displaystyle {\mathcal {V}}_{c,h}}
1664:{\displaystyle {\mathcal {V}}_{c,h}}
6785:, Serie II, Numero 14 (1987) 25-42.
3061:such that the Hermitian adjoint of
6410:R. C. Brower; C. B. Thorn (1971).
4046:
3918:The character of the Verma module
3799:The character of a representation
3399:is unitary if and only if either
1102:derivation of the Virasoro algebra
14:
6394:10.1090/S0002-9947-1966-0188356-3
4291:{\displaystyle c\in \mathbb {C} }
2785:{\displaystyle h+rs=h_{r',s'}(c)}
2313:, and the reducible Verma module
1111:in terms of two generators (e.g.
5033:{\displaystyle 1\leq s\leq p'-1}
1623:For any pair of complex numbers
1293:. The eigenvalues take the form
39:
6272:Journal of Geometry and Physics
4990:{\displaystyle 1\leq r\leq p-1}
2882:
2876:
2198:
2192:
1200:
5965:The Virasoro algebra also has
5782:
5765:
5762:
5750:
5607:
5589:
5577:
5533:
5464:
5446:
5434:
5390:
5304:
5299:
5282:
5279:
5267:
5185:
5059:
5047:
4788:
4766:
4122:
4116:
4070:
4051:
3998:
3992:
3851:
3845:
3816:{\displaystyle {\mathcal {R}}}
3722:
3710:
3667:
3655:
3639:
3633:
3456:
3444:
3284:
3269:
3254:
3248:
3008:
3002:
2856:
2843:
2779:
2773:
2712:
2706:
2544:
2497:
2438:
2432:
2294:
2288:
2237:
2214:
2170:
2141:
2129:
2106:
2083:
2077:
1983:
1977:
1134:Highest weight representations
1027:
1008:
976:
964:
958:
932:
903:
884:
708:Infinite dimensional Lie group
1:
5814:is made of two copies of the
4921:{\displaystyle 2\leq p<p'}
1675:is used for both the element
1140:highest weight representation
6680:10.1016/0370-2693(91)90553-3
6438:10.1016/0550-3213(71)90452-4
6366:10.1016/0550-3213(84)90052-X
6302:10.1016/0393-0440(94)00012-S
3033:Hermitian form and unitarity
2939:{\displaystyle p>q\geq 2}
2718:{\displaystyle h=h_{r,s}(c)}
2306:{\displaystyle h_{1,1}(c)=0}
1989:{\displaystyle h=h_{r,s}(c)}
481:{\displaystyle \mathbb {Z} }
456:{\displaystyle \mathbb {Z} }
419:{\displaystyle \mathbb {Z} }
6702:Encyclopedia of Mathematics
6512:10.1103/PhysRevLett.52.1575
6480:B. L. Feigin, D. B. Fuchs,
1996:for some positive integers
1107:The Virasoro algebra has a
877:. These generators satisfy
809:(named after the physicist
206:List of group theory topics
6861:
5934:
5916:
5907:= 2 superconformal algebra
5875:
4884:{\displaystyle c=c_{p,p'}}
3014:{\displaystyle h_{r,s}(c)}
6622:10.1007/978-0-85729-160-8
5888: = 1 extensions
5828:Virasoro conformal blocks
5812:conformal transformations
2670:{\displaystyle N=rs+r's'}
1375:-eigenstates of the type
6772:& correction: ibid.
6722:V. G. Kac, A. K. Raina,
3147:of a basis of the level
324:Elementary abelian group
201:Glossary of group theory
6491:Physical Review Letters
6336:Alexander Zamolodchikov
6108:10.1103/PhysRevD.1.2933
5872:Super Virasoro algebras
5861:Gupta–Bleuler formalism
3169:Kac determinant formula
2376:{\displaystyle L_{-1}v}
1846:{\displaystyle k\geq 0}
1539:{\displaystyle k\geq 0}
1338:{\displaystyle N\geq 0}
825:. It is widely used in
6835:Conformal field theory
6215:Conformal Field Theory
6049:Super Virasoro algebra
6024:Conformal field theory
5878:Super Virasoro algebra
5806:Conformal field theory
5791:
5703:
5641:
5120:
5066:
5034:
4991:
4953:
4922:
4885:
4838:
4565:
4515:
4465:
4406:
4383:
4333:
4292:
4257:
4234:
4050:
3949:
3909:
3817:
3732:
3601:is one of the values
3588:
3378:
3358:
3338:
3296:
3161:
3130:
3082:
3051:
3015:
2973:
2940:
2904:
2809:
2786:
2719:
2671:
2623:
2603:
2580:
2554:
2484:
2403:
2377:
2350:has a singular vector
2344:
2307:
2256:
2045:
2016:
1990:
1942:
1922:
1847:
1821:
1759:
1665:
1614:
1594:
1579:
1540:
1514:
1452:
1369:
1339:
1313:
1287:
1257:
1230:
1156:
1090:
1059:
916:
740:Linear algebraic group
482:
457:
420:
6726:, World Sci. (1987)
6039:Lie conformal algebra
6029:Goddard–Thorn theorem
6011:dual resonance models
5943:Sugawara construction
5892:Neveu–Schwarz algebra
5792:
5704:
5642:
5121:
5067:
5065:{\displaystyle (r,s)}
5035:
4992:
4954:
4923:
4886:
4839:
4566:
4516:
4466:
4407:
4384:
4334:
4293:
4265:Dedekind eta function
4258:
4256:{\displaystyle \eta }
4235:
4030:
3950:
3910:
3818:
3733:
3589:
3379:
3359:
3339:
3337:{\displaystyle A_{N}}
3297:
3162:
3131:
3083:
3081:{\displaystyle L_{n}}
3052:
3016:
2974:
2941:
2905:
2810:
2787:
2720:
2672:
2624:
2604:
2581:
2555:
2485:
2404:
2378:
2345:
2308:
2257:
2046:
2017:
1991:
1943:
1923:
1848:
1822:
1760:
1666:
1615:
1595:
1559:
1541:
1515:
1453:
1370:
1368:{\displaystyle L_{0}}
1340:
1314:
1288:
1286:{\displaystyle L_{0}}
1258:
1231:
1157:
1129:Representation theory
1125:) and six relations.
1091:
1060:
917:
811:Miguel Ángel Virasoro
483:
458:
421:
6845:Mathematical physics
6695:Victor Kac (2001) ,
6373:R. E. Block (1966).
5713:
5654:
5134:
5080:
5076:). The Verma module
5044:
5001:
4963:
4952:{\displaystyle p,p'}
4932:
4895:
4851:
4578:
4525:
4475:
4416:
4393:
4343:
4302:
4274:
4247:
3962:
3922:
3830:
3803:
3608:
3414:
3407: ≥ 0, or
3368:
3348:
3321:
3178:
3151:
3092:
3065:
3041:
3027:random cluster model
2983:
2957:
2918:
2822:
2796:
2729:
2681:
2633:
2613:
2593:
2564:
2494:
2413:
2387:
2354:
2317:
2269:
2058:
2044:{\displaystyle N=rs}
2026:
2000:
1952:
1932:
1895:
1831:
1769:
1686:
1638:
1604:
1550:
1524:
1462:
1379:
1352:
1323:
1319:, where the integer
1297:
1270:
1247:
1169:
1146:
1089:{\displaystyle 1/12}
1072:
929:
881:
470:
445:
408:
6752:1986LMaPh..11..225D
6672:1991PhLB..273...56K
6572:1986CMaPh.103..105G
6504:1984PhRvL..52.1575F
6430:1971NuPhB..31..163B
6358:1984NuPhB.241..333B
6294:1995JGP....15..252R
6182:1989CMaPh.122..171U
6139:1988CMaPh.117..595F
6100:1970PhRvD...1.2933V
5971:conformal algebraic
5931:Affine Lie algebras
5853:Virasoro constraint
5824:conformal bootstrap
4339:, the Verma module
3749: − 1 and
3403: ≥ 1 and
3305:where the function
3109:
2972:{\displaystyle r,s}
2579:{\displaystyle N=2}
2517:
2402:{\displaystyle N=1}
2015:{\displaystyle r,s}
1546:, whose levels are
1312:{\displaystyle h+N}
114:Group homomorphisms
24:Algebraic structure
6760:10.1007/bf00400220
6716:(Helsinki, 1978),
6697:"Virasoro algebra"
6580:10.1007/BF01464283
6465:10.24033/asens.603
6446:E. Cartan (1909).
6332:Alexander Polyakov
6249:10.1007/BF01078026
6237:Funkts. Anal. Appl
6190:10.1007/BF01221412
6147:10.1007/BF01218387
6034:Heisenberg algebra
6007:Virasoro operators
5937:affine Lie algebra
5787:
5699:
5637:
5635:
5336:
5116:
5062:
5030:
4987:
4949:
4918:
4881:
4834:
4561:
4511:
4471:. The quotient of
4461:
4405:{\displaystyle rs}
4402:
4379:
4329:
4288:
4253:
4230:
3945:
3905:
3813:
3789:Kac–Moody algebras
3781:coset construction
3763:, Zongan Qiu, and
3728:
3584:
3374:
3354:
3334:
3315:partition function
3292:
3218:
3157:
3126:
3095:
3078:
3047:
3011:
2969:
2936:
2900:
2808:{\displaystyle rs}
2805:
2782:
2715:
2667:
2619:
2599:
2576:
2550:
2500:
2480:
2399:
2373:
2340:
2303:
2252:
2041:
2012:
1986:
1938:
1918:
1843:
1817:
1755:
1661:
1610:
1590:
1536:
1510:
1448:
1365:
1335:
1309:
1283:
1253:
1226:
1152:
1086:
1055:
915:{\displaystyle =0}
912:
590:Special orthogonal
478:
453:
416:
297:Lagrange's theorem
6808:Antony Wassermann
6790:Antony Wassermann
6649:Physics Letters B
6631:978-0-85729-159-2
6544:, D. B. Fuchs,
6498:(18): 1575–1578.
6417:Nuclear Physics B
6345:Nuclear Physics B
6328:Alexander Belavin
6094:(10): 2933–2936.
6087:Physical Review D
6009:) while studying
5900:Grassmann numbers
5526:
5383:
5319:
4150:
4126:
4108:
4074:
4025:
3898:
3726:
3571:
3558:
3545:
3532:
3519:
3460:
3377:{\displaystyle c}
3357:{\displaystyle h}
3191:
3160:{\displaystyle N}
3050:{\displaystyle c}
2880:
2874:
2622:{\displaystyle N}
2602:{\displaystyle N}
2468:
2455:
2248:
2196:
2188:
2097:
1941:{\displaystyle N}
1613:{\displaystyle v}
1256:{\displaystyle v}
1239:where the number
1155:{\displaystyle v}
1006:
819:central extension
799:
798:
374:
373:
256:Alternating group
213:
212:
6852:
6821:
6819:
6803:
6801:
6771:
6740:Lett. Math. Phys
6709:
6691:
6665:
6642:
6607:
6537:
6531:
6523:
6477:
6467:
6441:
6406:
6396:
6369:
6314:
6313:
6287:
6267:
6261:
6260:
6232:
6226:
6211:
6202:
6201:
6165:
6159:
6158:
6118:
6112:
6111:
6081:
6044:Pohlmeyer charge
5967:vertex algebraic
5796:
5794:
5793:
5788:
5786:
5785:
5775:
5746:
5745:
5723:
5722:
5708:
5706:
5705:
5700:
5698:
5697:
5687:
5686:
5664:
5663:
5646:
5644:
5643:
5638:
5636:
5629:
5625:
5624:
5623:
5622:
5621:
5620:
5616:
5615:
5614:
5605:
5585:
5584:
5575:
5543:
5527:
5525:
5524:
5506:
5497:
5496:
5481:
5480:
5479:
5478:
5477:
5473:
5472:
5471:
5462:
5442:
5441:
5432:
5400:
5384:
5382:
5381:
5363:
5354:
5353:
5335:
5334:
5312:
5308:
5307:
5303:
5302:
5292:
5263:
5262:
5240:
5239:
5229:
5228:
5218:
5217:
5195:
5194:
5184:
5179:
5178:
5177:
5176:
5154:
5153:
5140:
5125:
5123:
5122:
5117:
5115:
5114:
5113:
5112:
5090:
5089:
5071:
5069:
5068:
5063:
5039:
5037:
5036:
5031:
5023:
4996:
4994:
4993:
4988:
4958:
4956:
4955:
4950:
4948:
4927:
4925:
4924:
4919:
4917:
4890:
4888:
4887:
4882:
4880:
4879:
4878:
4843:
4841:
4840:
4835:
4830:
4829:
4828:
4827:
4826:
4825:
4803:
4802:
4787:
4786:
4762:
4761:
4760:
4759:
4749:
4748:
4726:
4725:
4710:
4709:
4708:
4707:
4706:
4705:
4683:
4682:
4667:
4666:
4665:
4664:
4654:
4653:
4631:
4630:
4623:
4618:
4617:
4616:
4615:
4593:
4592:
4570:
4568:
4567:
4562:
4560:
4559:
4558:
4557:
4535:
4534:
4520:
4518:
4517:
4512:
4510:
4509:
4508:
4507:
4485:
4484:
4470:
4468:
4467:
4462:
4460:
4459:
4449:
4448:
4426:
4425:
4411:
4409:
4408:
4403:
4388:
4386:
4385:
4380:
4378:
4377:
4376:
4375:
4353:
4352:
4338:
4336:
4335:
4330:
4328:
4327:
4322:
4297:
4295:
4294:
4289:
4287:
4262:
4260:
4259:
4254:
4239:
4237:
4236:
4231:
4226:
4222:
4215:
4214:
4199:
4198:
4183:
4182:
4153:
4152:
4151:
4143:
4127:
4125:
4111:
4110:
4109:
4104:
4093:
4080:
4075:
4073:
4069:
4068:
4049:
4044:
4028:
4027:
4026:
4018:
4005:
3991:
3990:
3989:
3988:
3977:
3976:
3954:
3952:
3951:
3946:
3944:
3943:
3932:
3931:
3914:
3912:
3911:
3906:
3901:
3900:
3899:
3891:
3886:
3885:
3868:
3867:
3866:
3844:
3843:
3842:
3822:
3820:
3819:
3814:
3812:
3811:
3785:GKO construction
3779:(1986) used the
3753:= 1, 2, 3, ...,
3745:= 1, 2, 3, ...,
3737:
3735:
3734:
3729:
3727:
3725:
3702:
3695:
3694:
3689:
3688:
3654:
3653:
3646:
3632:
3631:
3593:
3591:
3590:
3585:
3583:
3579:
3572:
3564:
3559:
3551:
3546:
3538:
3533:
3525:
3520:
3512:
3496:
3495:
3466:
3462:
3461:
3459:
3436:
3406:
3402:
3398:
3394:
3383:
3381:
3380:
3375:
3363:
3361:
3360:
3355:
3343:
3341:
3340:
3335:
3333:
3332:
3301:
3299:
3298:
3293:
3288:
3287:
3264:
3263:
3247:
3246:
3225:
3224:
3217:
3190:
3189:
3167:is given by the
3166:
3164:
3163:
3158:
3145:Gram determinant
3135:
3133:
3132:
3127:
3125:
3124:
3108:
3103:
3087:
3085:
3084:
3079:
3077:
3076:
3056:
3054:
3053:
3048:
3020:
3018:
3017:
3012:
3001:
3000:
2978:
2976:
2975:
2970:
2945:
2943:
2942:
2937:
2909:
2907:
2906:
2901:
2899:
2881:
2878:
2875:
2873:
2865:
2864:
2863:
2841:
2814:
2812:
2811:
2806:
2791:
2789:
2788:
2783:
2772:
2771:
2770:
2759:
2724:
2722:
2721:
2716:
2705:
2704:
2676:
2674:
2673:
2668:
2666:
2658:
2628:
2626:
2625:
2620:
2608:
2606:
2605:
2600:
2585:
2583:
2582:
2577:
2559:
2557:
2556:
2551:
2543:
2542:
2530:
2529:
2516:
2511:
2489:
2487:
2486:
2481:
2479:
2478:
2469:
2461:
2456:
2448:
2431:
2430:
2408:
2406:
2405:
2400:
2382:
2380:
2379:
2374:
2369:
2368:
2349:
2347:
2346:
2341:
2339:
2338:
2327:
2326:
2312:
2310:
2309:
2304:
2287:
2286:
2261:
2259:
2258:
2253:
2246:
2245:
2244:
2235:
2234:
2197:
2194:
2186:
2185:
2184:
2178:
2177:
2165:
2164:
2137:
2136:
2127:
2126:
2105:
2104:
2098:
2090:
2076:
2075:
2050:
2048:
2047:
2042:
2021:
2019:
2018:
2013:
1995:
1993:
1992:
1987:
1976:
1975:
1947:
1945:
1944:
1939:
1927:
1925:
1924:
1919:
1917:
1916:
1905:
1904:
1884:Singular vectors
1876:
1872:
1868:
1864:
1860:
1856:
1852:
1850:
1849:
1844:
1826:
1824:
1823:
1818:
1816:
1815:
1800:
1799:
1787:
1786:
1764:
1762:
1761:
1756:
1751:
1750:
1749:
1748:
1728:
1727:
1726:
1725:
1708:
1707:
1706:
1705:
1678:
1674:
1670:
1668:
1667:
1662:
1660:
1659:
1648:
1647:
1630:
1626:
1619:
1617:
1616:
1611:
1599:
1597:
1596:
1591:
1589:
1588:
1578:
1573:
1545:
1543:
1542:
1537:
1519:
1517:
1516:
1511:
1509:
1508:
1493:
1492:
1480:
1479:
1457:
1455:
1454:
1449:
1444:
1443:
1442:
1441:
1421:
1420:
1419:
1418:
1401:
1400:
1399:
1398:
1374:
1372:
1371:
1366:
1364:
1363:
1344:
1342:
1341:
1336:
1318:
1316:
1315:
1310:
1292:
1290:
1289:
1284:
1282:
1281:
1262:
1260:
1259:
1254:
1242:
1235:
1233:
1232:
1227:
1210:
1209:
1187:
1186:
1161:
1159:
1158:
1153:
1121:
1114:
1095:
1093:
1092:
1087:
1082:
1064:
1062:
1061:
1056:
1051:
1050:
1020:
1019:
1007:
999:
994:
993:
957:
956:
944:
943:
921:
919:
918:
913:
902:
901:
876:
867:
860:
843:Virasoro algebra
807:Virasoro algebra
791:
784:
777:
733:Algebraic groups
506:Hyperbolic group
496:Arithmetic group
487:
485:
484:
479:
477:
462:
460:
459:
454:
452:
425:
423:
422:
417:
415:
338:Schur multiplier
292:Cauchy's theorem
280:Quaternion group
228:
54:
43:
30:
19:
6860:
6859:
6855:
6854:
6853:
6851:
6850:
6849:
6825:
6824:
6806:
6788:
6737:
6694:
6645:
6632:
6611:
6551:
6524:
6487:
6445:
6409:
6372:
6326:
6323:
6318:
6317:
6269:
6268:
6264:
6234:
6233:
6229:
6212:
6205:
6167:
6166:
6162:
6120:
6119:
6115:
6083:
6082:
6078:
6073:
6068:
6019:
5979:
5963:
5955:Riemann surface
5951:
5939:
5933:
5921:
5915:
5884:supersymmetric
5880:
5874:
5869:
5867:Generalizations
5836:
5808:
5803:
5768:
5731:
5716:
5711:
5710:
5672:
5657:
5652:
5651:
5634:
5633:
5606:
5598:
5576:
5568:
5536:
5532:
5528:
5517:
5510:
5490:
5485:
5463:
5455:
5433:
5425:
5393:
5389:
5385:
5374:
5367:
5347:
5342:
5341:
5337:
5310:
5309:
5285:
5248:
5233:
5203:
5188:
5162:
5147:
5142:
5132:
5131:
5098:
5083:
5078:
5077:
5042:
5041:
5016:
4999:
4998:
4961:
4960:
4941:
4930:
4929:
4910:
4893:
4892:
4871:
4860:
4849:
4848:
4811:
4796:
4791:
4775:
4734:
4719:
4714:
4691:
4676:
4671:
4639:
4624:
4601:
4586:
4581:
4576:
4575:
4543:
4528:
4523:
4522:
4493:
4478:
4473:
4472:
4434:
4419:
4414:
4413:
4391:
4390:
4361:
4346:
4341:
4340:
4317:
4300:
4299:
4272:
4271:
4245:
4244:
4206:
4190:
4174:
4158:
4154:
4131:
4112:
4094:
4081:
4060:
4029:
4006:
3970:
3965:
3960:
3959:
3925:
3920:
3919:
3877:
3872:
3857:
3833:
3828:
3827:
3801:
3800:
3797:
3765:Stephen Shenker
3703:
3682:
3647:
3617:
3606:
3605:
3504:
3500:
3440:
3428:
3424:
3423:
3412:
3411:
3404:
3400:
3396:
3392:
3366:
3365:
3346:
3345:
3324:
3319:
3318:
3257:
3232:
3181:
3176:
3175:
3149:
3148:
3113:
3090:
3089:
3068:
3063:
3062:
3039:
3038:
3035:
2986:
2981:
2980:
2979:that appear in
2955:
2954:
2916:
2915:
2866:
2855:
2842:
2820:
2819:
2794:
2793:
2763:
2752:
2747:
2727:
2726:
2690:
2679:
2678:
2659:
2651:
2631:
2630:
2611:
2610:
2591:
2590:
2562:
2561:
2531:
2521:
2492:
2491:
2470:
2416:
2411:
2410:
2385:
2384:
2357:
2352:
2351:
2320:
2315:
2314:
2272:
2267:
2266:
2265:In particular,
2236:
2223:
2169:
2153:
2128:
2115:
2061:
2056:
2055:
2024:
2023:
1998:
1997:
1961:
1950:
1949:
1930:
1929:
1898:
1893:
1892:
1886:
1874:
1870:
1866:
1862:
1858:
1854:
1829:
1828:
1807:
1791:
1778:
1767:
1766:
1740:
1732:
1717:
1709:
1697:
1689:
1684:
1683:
1676:
1672:
1641:
1636:
1635:
1628:
1624:
1602:
1601:
1580:
1548:
1547:
1522:
1521:
1500:
1484:
1471:
1460:
1459:
1433:
1425:
1410:
1402:
1390:
1382:
1377:
1376:
1355:
1350:
1349:
1321:
1320:
1295:
1294:
1273:
1268:
1267:
1245:
1244:
1240:
1201:
1172:
1167:
1166:
1144:
1143:
1136:
1131:
1124:
1119:
1117:
1112:
1070:
1069:
1066:
1030:
1011:
979:
948:
935:
927:
926:
893:
879:
878:
874:
862:
858:
853:
839:
817:and the unique
813:) is a complex
795:
766:
765:
754:Abelian variety
747:Reductive group
735:
725:
724:
723:
722:
673:
665:
657:
649:
641:
614:Special unitary
525:
511:
510:
492:
491:
468:
467:
443:
442:
406:
405:
397:
396:
387:Discrete groups
376:
375:
331:Frobenius group
276:
263:
252:
245:Symmetric group
241:
225:
215:
214:
65:Normal subgroup
51:
31:
22:
17:
12:
11:
5:
6858:
6856:
6848:
6847:
6842:
6837:
6827:
6826:
6823:
6822:
6804:
6786:
6777:
6746:(3): 225–234.
6735:
6720:
6710:
6692:
6663:hep-th/9204097
6656:(1–2): 56–62.
6643:
6630:
6609:
6566:(1): 105–119.
6549:
6539:
6485:
6478:
6443:
6424:(1): 163–182.
6407:
6387:(2): 378–392.
6370:
6352:(2): 333–380.
6322:
6319:
6316:
6315:
6285:hep-th/9302105
6278:(3): 252–280.
6262:
6227:
6203:
6176:(1): 171–173.
6160:
6113:
6075:
6074:
6072:
6069:
6067:
6066:
6061:
6056:
6051:
6046:
6041:
6036:
6031:
6026:
6020:
6018:
6015:
5978:
5975:
5962:
5959:
5950:
5947:
5935:Main article:
5932:
5929:
5917:Main article:
5914:
5911:
5896:Ramond algebra
5882:There are two
5876:Main article:
5873:
5870:
5868:
5865:
5857:quantum theory
5835:
5832:
5807:
5804:
5802:
5799:
5784:
5781:
5778:
5774:
5771:
5767:
5764:
5761:
5758:
5755:
5752:
5749:
5744:
5741:
5738:
5734:
5730:
5727:
5721:
5696:
5693:
5690:
5685:
5682:
5679:
5675:
5671:
5668:
5662:
5648:
5647:
5632:
5628:
5619:
5613:
5609:
5604:
5601:
5597:
5594:
5591:
5588:
5583:
5579:
5574:
5571:
5567:
5564:
5561:
5558:
5555:
5552:
5549:
5546:
5542:
5539:
5535:
5531:
5523:
5520:
5516:
5513:
5509:
5504:
5501:
5495:
5488:
5484:
5476:
5470:
5466:
5461:
5458:
5454:
5451:
5448:
5445:
5440:
5436:
5431:
5428:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5399:
5396:
5392:
5388:
5380:
5377:
5373:
5370:
5366:
5361:
5358:
5352:
5345:
5340:
5333:
5329:
5326:
5322:
5318:
5315:
5313:
5311:
5306:
5301:
5298:
5295:
5291:
5288:
5284:
5281:
5278:
5275:
5272:
5269:
5266:
5261:
5258:
5255:
5251:
5247:
5244:
5238:
5232:
5227:
5224:
5221:
5216:
5213:
5210:
5206:
5202:
5199:
5193:
5187:
5183:
5175:
5172:
5169:
5165:
5161:
5158:
5152:
5145:
5141:
5139:
5111:
5108:
5105:
5101:
5097:
5094:
5088:
5061:
5058:
5055:
5052:
5049:
5029:
5026:
5022:
5019:
5015:
5012:
5009:
5006:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4947:
4944:
4940:
4937:
4916:
4913:
4909:
4906:
4903:
4900:
4877:
4874:
4870:
4867:
4863:
4859:
4856:
4845:
4844:
4833:
4824:
4821:
4818:
4814:
4810:
4807:
4801:
4794:
4790:
4785:
4782:
4778:
4774:
4771:
4768:
4765:
4758:
4755:
4752:
4747:
4744:
4741:
4737:
4733:
4730:
4724:
4717:
4713:
4704:
4701:
4698:
4694:
4690:
4687:
4681:
4674:
4670:
4663:
4660:
4657:
4652:
4649:
4646:
4642:
4638:
4635:
4629:
4622:
4614:
4611:
4608:
4604:
4600:
4597:
4591:
4584:
4556:
4553:
4550:
4546:
4542:
4539:
4533:
4506:
4503:
4500:
4496:
4492:
4489:
4483:
4458:
4455:
4452:
4447:
4444:
4441:
4437:
4433:
4430:
4424:
4401:
4398:
4374:
4371:
4368:
4364:
4360:
4357:
4351:
4326:
4321:
4316:
4313:
4310:
4307:
4286:
4282:
4279:
4252:
4241:
4240:
4229:
4225:
4221:
4218:
4213:
4209:
4205:
4202:
4197:
4193:
4189:
4186:
4181:
4177:
4173:
4170:
4167:
4164:
4161:
4157:
4149:
4146:
4141:
4138:
4134:
4130:
4124:
4121:
4118:
4115:
4107:
4103:
4100:
4097:
4091:
4088:
4084:
4078:
4072:
4067:
4063:
4059:
4056:
4053:
4048:
4043:
4040:
4037:
4033:
4024:
4021:
4016:
4013:
4009:
4003:
4000:
3997:
3994:
3987:
3984:
3981:
3975:
3968:
3942:
3939:
3936:
3930:
3916:
3915:
3904:
3897:
3894:
3889:
3884:
3880:
3875:
3871:
3865:
3860:
3856:
3853:
3850:
3847:
3841:
3836:
3810:
3796:
3793:
3761:Daniel Friedan
3739:
3738:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3701:
3698:
3693:
3687:
3681:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3652:
3644:
3641:
3638:
3635:
3630:
3627:
3624:
3620:
3616:
3613:
3595:
3594:
3582:
3578:
3575:
3570:
3567:
3562:
3557:
3554:
3549:
3544:
3541:
3536:
3531:
3528:
3523:
3518:
3515:
3510:
3507:
3503:
3499:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3465:
3458:
3455:
3452:
3449:
3446:
3443:
3439:
3434:
3431:
3427:
3422:
3419:
3373:
3353:
3331:
3327:
3303:
3302:
3291:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3262:
3256:
3253:
3250:
3245:
3242:
3239:
3235:
3231:
3228:
3223:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3194:
3188:
3184:
3156:
3123:
3120:
3116:
3112:
3107:
3102:
3098:
3075:
3071:
3059:Hermitian form
3046:
3034:
3031:
3010:
3007:
3004:
2999:
2996:
2993:
2989:
2968:
2965:
2962:
2948:minimal models
2935:
2932:
2929:
2926:
2923:
2912:
2911:
2898:
2894:
2891:
2888:
2885:
2872:
2869:
2862:
2858:
2854:
2851:
2848:
2845:
2839:
2836:
2833:
2830:
2827:
2804:
2801:
2781:
2778:
2775:
2769:
2766:
2762:
2758:
2755:
2750:
2746:
2743:
2740:
2737:
2734:
2714:
2711:
2708:
2703:
2700:
2697:
2693:
2689:
2686:
2665:
2662:
2657:
2654:
2650:
2647:
2644:
2641:
2638:
2618:
2598:
2575:
2572:
2569:
2549:
2546:
2541:
2538:
2534:
2528:
2524:
2520:
2515:
2510:
2507:
2503:
2499:
2477:
2473:
2467:
2464:
2459:
2454:
2451:
2446:
2443:
2440:
2437:
2434:
2429:
2426:
2423:
2419:
2398:
2395:
2392:
2372:
2367:
2364:
2360:
2337:
2334:
2331:
2325:
2302:
2299:
2296:
2293:
2290:
2285:
2282:
2279:
2275:
2263:
2262:
2251:
2243:
2239:
2233:
2230:
2226:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2191:
2183:
2176:
2172:
2168:
2163:
2160:
2156:
2152:
2149:
2146:
2143:
2140:
2135:
2131:
2125:
2122:
2118:
2114:
2111:
2108:
2103:
2096:
2093:
2088:
2085:
2082:
2079:
2074:
2071:
2068:
2064:
2040:
2037:
2034:
2031:
2011:
2008:
2005:
1985:
1982:
1979:
1974:
1971:
1968:
1964:
1960:
1957:
1937:
1915:
1912:
1909:
1903:
1885:
1882:
1842:
1839:
1836:
1814:
1810:
1806:
1803:
1798:
1794:
1790:
1785:
1781:
1777:
1774:
1754:
1747:
1743:
1739:
1735:
1731:
1724:
1720:
1716:
1712:
1704:
1700:
1696:
1692:
1658:
1655:
1652:
1646:
1609:
1587:
1583:
1577:
1572:
1569:
1566:
1562:
1558:
1555:
1535:
1532:
1529:
1507:
1503:
1499:
1496:
1491:
1487:
1483:
1478:
1474:
1470:
1467:
1447:
1440:
1436:
1432:
1428:
1424:
1417:
1413:
1409:
1405:
1397:
1393:
1389:
1385:
1362:
1358:
1334:
1331:
1328:
1308:
1305:
1302:
1280:
1276:
1252:
1237:
1236:
1225:
1222:
1219:
1216:
1213:
1208:
1204:
1199:
1196:
1193:
1190:
1185:
1182:
1179:
1175:
1151:
1135:
1132:
1130:
1127:
1122:
1115:
1085:
1081:
1077:
1068:The factor of
1054:
1049:
1046:
1043:
1040:
1037:
1033:
1029:
1026:
1023:
1018:
1014:
1010:
1005:
1002:
997:
992:
989:
986:
982:
978:
975:
972:
969:
966:
963:
960:
955:
951:
947:
942:
938:
934:
924:
911:
908:
905:
900:
896:
892:
889:
886:
871:central charge
856:
838:
835:
797:
796:
794:
793:
786:
779:
771:
768:
767:
764:
763:
761:Elliptic curve
757:
756:
750:
749:
743:
742:
736:
731:
730:
727:
726:
721:
720:
717:
714:
710:
706:
705:
704:
699:
697:Diffeomorphism
693:
692:
687:
682:
676:
675:
671:
667:
663:
659:
655:
651:
647:
643:
639:
634:
633:
622:
621:
610:
609:
598:
597:
586:
585:
574:
573:
562:
561:
554:Special linear
550:
549:
542:General linear
538:
537:
532:
526:
517:
516:
513:
512:
509:
508:
503:
498:
490:
489:
476:
464:
451:
438:
436:Modular groups
434:
433:
432:
427:
414:
398:
395:
394:
389:
383:
382:
381:
378:
377:
372:
371:
370:
369:
364:
359:
356:
350:
349:
343:
342:
341:
340:
334:
333:
327:
326:
321:
312:
311:
309:Hall's theorem
306:
304:Sylow theorems
300:
299:
294:
286:
285:
284:
283:
277:
272:
269:Dihedral group
265:
264:
259:
253:
248:
242:
237:
226:
221:
220:
217:
216:
211:
210:
209:
208:
203:
195:
194:
193:
192:
187:
182:
177:
172:
167:
162:
160:multiplicative
157:
152:
147:
142:
134:
133:
132:
131:
126:
118:
117:
109:
108:
107:
106:
104:Wreath product
101:
96:
91:
89:direct product
83:
81:Quotient group
75:
74:
73:
72:
67:
62:
52:
49:
48:
45:
44:
36:
35:
15:
13:
10:
9:
6:
4:
3:
2:
6857:
6846:
6843:
6841:
6838:
6836:
6833:
6832:
6830:
6818:
6813:
6809:
6805:
6800:
6795:
6791:
6787:
6784:
6783:
6778:
6775:
6769:
6765:
6761:
6757:
6753:
6749:
6745:
6741:
6736:
6733:
6732:9971-5-0395-6
6729:
6725:
6721:
6719:
6715:
6711:
6708:
6704:
6703:
6698:
6693:
6689:
6685:
6681:
6677:
6673:
6669:
6664:
6659:
6655:
6651:
6650:
6644:
6641:
6637:
6633:
6627:
6623:
6619:
6615:
6610:
6605:
6601:
6597:
6593:
6589:
6585:
6581:
6577:
6573:
6569:
6565:
6561:
6560:
6555:
6550:
6547:
6543:
6542:I.M. Gel'fand
6540:
6535:
6529:
6521:
6517:
6513:
6509:
6505:
6501:
6497:
6493:
6492:
6486:
6483:
6479:
6475:
6471:
6466:
6461:
6457:
6453:
6449:
6444:
6439:
6435:
6431:
6427:
6423:
6419:
6418:
6413:
6408:
6404:
6400:
6395:
6390:
6386:
6382:
6381:
6376:
6371:
6367:
6363:
6359:
6355:
6351:
6347:
6346:
6341:
6337:
6333:
6329:
6325:
6324:
6320:
6311:
6307:
6303:
6299:
6295:
6291:
6286:
6281:
6277:
6273:
6266:
6263:
6258:
6254:
6250:
6246:
6242:
6238:
6231:
6228:
6224:
6223:0-387-94785-X
6220:
6216:
6210:
6208:
6204:
6199:
6195:
6191:
6187:
6183:
6179:
6175:
6171:
6164:
6161:
6156:
6152:
6148:
6144:
6140:
6136:
6132:
6128:
6124:
6117:
6114:
6109:
6105:
6101:
6097:
6093:
6089:
6088:
6080:
6077:
6070:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6045:
6042:
6040:
6037:
6035:
6032:
6030:
6027:
6025:
6022:
6021:
6016:
6014:
6012:
6008:
6004:
6000:
5999:I. M. Gelfand
5996:
5992:
5988:
5984:
5976:
5974:
5972:
5968:
5960:
5958:
5956:
5948:
5946:
5944:
5938:
5930:
5928:
5926:
5920:
5912:
5910:
5908:
5906:
5901:
5897:
5893:
5889:
5887:
5879:
5871:
5866:
5864:
5862:
5858:
5855:, and in the
5854:
5849:
5848:string theory
5845:
5844:stress tensor
5841:
5834:String theory
5833:
5831:
5829:
5825:
5821:
5817:
5813:
5805:
5800:
5798:
5779:
5776:
5772:
5769:
5759:
5756:
5753:
5747:
5742:
5739:
5736:
5732:
5728:
5725:
5694:
5691:
5688:
5683:
5680:
5677:
5673:
5669:
5666:
5630:
5626:
5617:
5611:
5602:
5599:
5595:
5592:
5586:
5581:
5572:
5569:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5544:
5540:
5537:
5529:
5521:
5518:
5514:
5511:
5507:
5502:
5499:
5486:
5482:
5474:
5468:
5459:
5456:
5452:
5449:
5443:
5438:
5429:
5426:
5422:
5419:
5416:
5413:
5410:
5407:
5404:
5401:
5397:
5394:
5386:
5378:
5375:
5371:
5368:
5364:
5359:
5356:
5343:
5338:
5327:
5324:
5320:
5316:
5314:
5296:
5293:
5289:
5286:
5276:
5273:
5270:
5264:
5259:
5256:
5253:
5249:
5245:
5242:
5230:
5225:
5222:
5219:
5214:
5211:
5208:
5204:
5200:
5197:
5181:
5173:
5170:
5167:
5163:
5159:
5156:
5143:
5130:
5129:
5128:
5127:quotient is
5109:
5106:
5103:
5099:
5095:
5092:
5075:
5074:minimal model
5056:
5053:
5050:
5027:
5024:
5020:
5017:
5013:
5010:
5007:
5004:
4984:
4981:
4978:
4975:
4972:
4969:
4966:
4959:coprime, and
4945:
4942:
4938:
4935:
4914:
4911:
4907:
4904:
4901:
4898:
4875:
4872:
4868:
4865:
4861:
4857:
4854:
4831:
4822:
4819:
4816:
4812:
4808:
4805:
4792:
4783:
4780:
4776:
4772:
4769:
4763:
4756:
4753:
4750:
4745:
4742:
4739:
4735:
4731:
4728:
4715:
4711:
4702:
4699:
4696:
4692:
4688:
4685:
4672:
4668:
4661:
4658:
4655:
4650:
4647:
4644:
4640:
4636:
4633:
4620:
4612:
4609:
4606:
4602:
4598:
4595:
4582:
4574:
4573:
4572:
4554:
4551:
4548:
4544:
4540:
4537:
4504:
4501:
4498:
4494:
4490:
4487:
4456:
4453:
4450:
4445:
4442:
4439:
4435:
4431:
4428:
4399:
4396:
4372:
4369:
4366:
4362:
4358:
4355:
4324:
4314:
4311:
4308:
4305:
4280:
4277:
4268:
4266:
4250:
4227:
4223:
4219:
4216:
4211:
4207:
4203:
4200:
4195:
4191:
4187:
4184:
4179:
4175:
4171:
4168:
4165:
4162:
4159:
4155:
4147:
4144:
4139:
4136:
4132:
4128:
4119:
4113:
4105:
4101:
4098:
4095:
4089:
4086:
4082:
4076:
4065:
4061:
4057:
4054:
4041:
4038:
4035:
4031:
4022:
4019:
4014:
4011:
4007:
4001:
3995:
3985:
3982:
3979:
3966:
3958:
3957:
3956:
3940:
3937:
3934:
3902:
3895:
3892:
3887:
3882:
3878:
3873:
3869:
3858:
3854:
3848:
3834:
3826:
3825:
3824:
3794:
3792:
3790:
3786:
3782:
3778:
3774:
3770:
3769:Peter Goddard
3766:
3762:
3758:
3756:
3752:
3748:
3744:
3719:
3716:
3713:
3707:
3704:
3699:
3696:
3691:
3679:
3676:
3673:
3670:
3664:
3661:
3658:
3642:
3636:
3628:
3625:
3622:
3618:
3614:
3611:
3604:
3603:
3602:
3600:
3580:
3576:
3573:
3568:
3565:
3560:
3555:
3552:
3547:
3542:
3539:
3534:
3529:
3526:
3521:
3516:
3513:
3508:
3505:
3501:
3497:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3463:
3453:
3450:
3447:
3441:
3437:
3432:
3429:
3425:
3420:
3417:
3410:
3409:
3408:
3389:
3387:
3371:
3351:
3329:
3325:
3316:
3312:
3308:
3289:
3281:
3278:
3275:
3272:
3266:
3251:
3243:
3240:
3237:
3233:
3229:
3226:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3192:
3186:
3182:
3174:
3173:
3172:
3170:
3154:
3146:
3141:
3139:
3121:
3118:
3114:
3110:
3105:
3100:
3096:
3073:
3069:
3060:
3057:has a unique
3044:
3032:
3030:
3028:
3024:
3005:
2997:
2994:
2991:
2987:
2966:
2963:
2960:
2953:The integers
2951:
2949:
2933:
2930:
2927:
2924:
2921:
2892:
2889:
2886:
2883:
2870:
2867:
2860:
2852:
2849:
2846:
2837:
2834:
2831:
2828:
2825:
2818:
2817:
2816:
2802:
2799:
2776:
2767:
2764:
2760:
2756:
2753:
2748:
2744:
2741:
2738:
2735:
2732:
2709:
2701:
2698:
2695:
2691:
2687:
2684:
2663:
2660:
2655:
2652:
2648:
2645:
2642:
2639:
2636:
2616:
2596:
2587:
2573:
2570:
2567:
2560:at the level
2547:
2539:
2536:
2532:
2526:
2522:
2518:
2513:
2508:
2505:
2501:
2475:
2471:
2465:
2462:
2457:
2452:
2449:
2444:
2441:
2435:
2427:
2424:
2421:
2417:
2396:
2393:
2390:
2383:at the level
2370:
2365:
2362:
2358:
2335:
2332:
2329:
2300:
2297:
2291:
2283:
2280:
2277:
2273:
2249:
2241:
2231:
2228:
2224:
2220:
2217:
2211:
2208:
2205:
2202:
2199:
2189:
2174:
2166:
2161:
2158:
2154:
2150:
2147:
2144:
2138:
2133:
2123:
2120:
2116:
2112:
2109:
2094:
2091:
2086:
2080:
2072:
2069:
2066:
2062:
2054:
2053:
2052:
2038:
2035:
2032:
2029:
2009:
2006:
2003:
1980:
1972:
1969:
1966:
1962:
1958:
1955:
1935:
1913:
1910:
1907:
1889:
1883:
1881:
1878:
1840:
1837:
1834:
1812:
1808:
1804:
1801:
1796:
1792:
1788:
1783:
1779:
1775:
1772:
1752:
1745:
1741:
1737:
1733:
1729:
1722:
1718:
1714:
1710:
1702:
1698:
1694:
1690:
1680:
1656:
1653:
1650:
1634:
1621:
1607:
1585:
1581:
1575:
1570:
1567:
1564:
1560:
1556:
1553:
1533:
1530:
1527:
1505:
1501:
1497:
1494:
1489:
1485:
1481:
1476:
1472:
1468:
1465:
1445:
1438:
1434:
1430:
1426:
1422:
1415:
1411:
1407:
1403:
1395:
1391:
1387:
1383:
1360:
1356:
1346:
1332:
1329:
1326:
1306:
1303:
1300:
1278:
1274:
1264:
1250:
1223:
1220:
1217:
1214:
1211:
1206:
1202:
1197:
1194:
1191:
1188:
1183:
1180:
1177:
1173:
1165:
1164:
1163:
1149:
1141:
1133:
1128:
1126:
1110:
1105:
1103:
1099:
1083:
1079:
1075:
1065:
1052:
1047:
1044:
1041:
1038:
1035:
1031:
1024:
1021:
1016:
1012:
1003:
1000:
995:
990:
987:
984:
980:
973:
970:
967:
961:
953:
949:
945:
940:
936:
923:
909:
906:
898:
894:
890:
887:
873:
872:
865:
859:
852:
848:
844:
836:
834:
832:
831:string theory
828:
824:
820:
816:
812:
808:
804:
792:
787:
785:
780:
778:
773:
772:
770:
769:
762:
759:
758:
755:
752:
751:
748:
745:
744:
741:
738:
737:
734:
729:
728:
718:
715:
712:
711:
709:
703:
700:
698:
695:
694:
691:
688:
686:
683:
681:
678:
677:
674:
668:
666:
660:
658:
652:
650:
644:
642:
636:
635:
631:
627:
624:
623:
619:
615:
612:
611:
607:
603:
600:
599:
595:
591:
588:
587:
583:
579:
576:
575:
571:
567:
564:
563:
559:
555:
552:
551:
547:
543:
540:
539:
536:
533:
531:
528:
527:
524:
520:
515:
514:
507:
504:
502:
499:
497:
494:
493:
465:
440:
439:
437:
431:
428:
403:
400:
399:
393:
390:
388:
385:
384:
380:
379:
368:
365:
363:
360:
357:
354:
353:
352:
351:
348:
344:
339:
336:
335:
332:
329:
328:
325:
322:
320:
318:
314:
313:
310:
307:
305:
302:
301:
298:
295:
293:
290:
289:
288:
287:
281:
278:
275:
270:
267:
266:
262:
257:
254:
251:
246:
243:
240:
235:
232:
231:
230:
229:
224:
223:Finite groups
219:
218:
207:
204:
202:
199:
198:
197:
196:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
137:
136:
135:
130:
127:
125:
122:
121:
120:
119:
116:
115:
110:
105:
102:
100:
97:
95:
92:
90:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
57:
56:
55:
50:Basic notions
47:
46:
42:
38:
37:
34:
29:
25:
20:
6840:Lie algebras
6780:
6773:
6743:
6739:
6723:
6713:
6700:
6653:
6647:
6613:
6563:
6557:
6545:
6528:cite journal
6495:
6489:
6481:
6455:
6451:
6421:
6415:
6384:
6378:
6349:
6343:
6275:
6271:
6265:
6243:(2): 46–63.
6240:
6236:
6230:
6214:
6173:
6169:
6163:
6130:
6126:
6116:
6091:
6085:
6079:
6059:Witt algebra
6006:
6003:Dmitry Fuchs
5990:
5980:
5964:
5952:
5940:
5922:
5904:
5885:
5881:
5837:
5816:Witt algebra
5809:
5801:Applications
5649:
4846:
4269:
4242:
3917:
3798:
3759:
3754:
3750:
3746:
3742:
3740:
3598:
3596:
3390:
3310:
3306:
3304:
3168:
3142:
3137:
3036:
3022:
2952:
2913:
2588:
2264:
1890:
1887:
1879:
1681:
1633:Verma module
1622:
1347:
1265:
1238:
1137:
1109:presentation
1106:
1098:Witt algebra
1067:
925:
869:
863:
854:
850:
842:
840:
823:Witt algebra
806:
800:
629:
617:
605:
593:
581:
569:
557:
545:
316:
273:
260:
249:
238:
234:Cyclic group
112:
99:Free product
70:Group action
33:Group theory
28:Group theory
27:
6776:(1987) 260.
5995:R. E. Block
5993:> 0) by
3777:David Olive
3773:Adrian Kent
3023:Kac indices
3021:are called
1682:The states
1162:such that
815:Lie algebra
803:mathematics
519:Topological
358:alternating
6829:Categories
6718:pp.299-304
6604:0588.17014
6474:40.0193.02
6458:: 93–161.
6321:References
6133:(4): 595.
5913:W-algebras
5840:worldsheet
3795:Characters
2022:such that
851:generators
837:Definition
626:Symplectic
566:Orthogonal
523:Lie groups
430:Free group
155:continuous
94:Direct sum
6817:1012.6003
6799:1004.1287
6768:122201087
6707:EMS Press
6520:122320349
6198:119887710
6155:119811901
6064:WZW model
6054:W-algebra
5983:É. Cartan
5919:W-algebra
5777:−
5757:−
5596:−
5587:−
5487:χ
5483:−
5453:−
5444:−
5405:−
5344:χ
5328:∈
5321:∑
5294:−
5274:−
5144:χ
5025:−
5014:≤
5008:≤
4982:−
4976:≤
4970:≤
4902:≤
4793:χ
4773:−
4716:χ
4712:−
4673:χ
4583:χ
4325:∗
4315:∈
4281:∈
4251:η
4220:⋯
4140:−
4114:η
4099:−
4090:−
4058:−
4047:∞
4032:∏
4015:−
3967:χ
3888:−
3870:
3835:χ
3697:−
3674:−
3577:…
3493:…
3433:−
3421:∈
3313:) is the
3276:−
3230:−
3212:≤
3200:≤
3193:∏
3119:−
3106:†
2931:≥
2893:∈
2850:−
2835:−
2537:−
2506:−
2458:−
2445:−
2363:−
2229:−
2159:−
2139:−
2121:−
1838:≥
1805:⋯
1802:≤
1789:≤
1738:−
1730:⋯
1715:−
1695:−
1561:∑
1531:≥
1498:⋯
1495:≤
1482:≤
1431:−
1423:⋯
1408:−
1388:−
1330:≥
1032:δ
1022:−
971:−
690:Conformal
578:Euclidean
185:nilpotent
6688:15105921
6596:91181508
6338:(1984).
6310:10921054
6257:55989582
6217:, 1997,
6017:See also
5894:and the
5773:′
5603:′
5573:′
5541:′
5522:′
5460:′
5430:′
5398:′
5379:′
5290:′
5040:. (Then
5021:′
4946:′
4915:′
4876:′
4298:and for
4270:For any
2768:′
2757:′
2664:′
2656:′
868:and the
685:Poincaré
530:Solenoid
402:Integers
392:Lattices
367:sporadic
362:Lie type
190:solvable
180:dihedral
165:additive
150:infinite
60:Subgroup
6748:Bibcode
6668:Bibcode
6640:2744610
6588:0826859
6568:Bibcode
6500:Bibcode
6426:Bibcode
6403:1994485
6354:Bibcode
6290:Bibcode
6178:Bibcode
6135:Bibcode
6096:Bibcode
5987:E. Witt
5977:History
5851:as the
4263:is the
3138:unitary
2409:. Then
2051:, with
847:spanned
829:and in
821:of the
680:Lorentz
602:Unitary
501:Lattice
441:PSL(2,
175:abelian
86:(Semi-)
6766:
6730:
6686:
6638:
6628:
6602:
6594:
6586:
6518:
6472:
6401:
6308:
6255:
6221:
6196:
6153:
5842:, the
4243:where
3775:, and
3386:V. Kac
3317:, and
2247:
2187:
1631:, the
1100:, see
805:, the
535:Circle
466:SL(2,
355:cyclic
319:-group
170:cyclic
145:finite
140:simple
124:kernel
6812:arXiv
6794:arXiv
6764:S2CID
6684:S2CID
6658:arXiv
6592:S2CID
6516:S2CID
6399:JSTOR
6306:S2CID
6280:arXiv
6253:S2CID
6194:S2CID
6151:S2CID
6071:Notes
4891:with
2914:(For
2677:with
2195:where
1765:with
1458:with
922:and
719:Sp(∞)
716:SU(∞)
129:image
6728:ISBN
6626:ISBN
6534:link
6334:and
6219:ISBN
6001:and
5969:and
5709:and
4997:and
4928:and
4908:<
4847:Let
3955:is
3741:for
3597:and
3395:and
3143:The
2925:>
2879:with
2725:and
1873:and
1865:and
1857:and
1827:and
1776:<
1627:and
1520:and
1469:<
1181:>
1118:and
861:for
841:The
713:O(∞)
702:Loop
521:and
6756:doi
6676:doi
6654:273
6618:doi
6600:Zbl
6576:doi
6564:103
6508:doi
6470:JFM
6460:doi
6434:doi
6389:doi
6385:121
6362:doi
6350:241
6298:doi
6245:doi
6186:doi
6174:122
6143:doi
6131:117
6104:doi
5863:).
5846:in
3783:or
3364:or
3171:,
3088:is
2950:.)
2629:if
2586:.
1948:is
866:∈ ℤ
849:by
845:is
801:In
628:Sp(
616:SU(
592:SO(
556:SL(
544:GL(
6831::
6774:13
6762:.
6754:.
6744:11
6742:.
6705:,
6699:,
6682:.
6674:.
6666:.
6652:.
6636:MR
6634:,
6624:,
6598:.
6590:.
6584:MR
6582:.
6574:.
6562:.
6556:.
6530:}}
6526:{{
6514:.
6506:.
6496:52
6494:.
6468:.
6456:26
6454:.
6450:.
6432:.
6422:31
6420:.
6414:.
6397:.
6383:.
6377:.
6360:.
6348:.
6342:.
6330:,
6304:.
6296:.
6288:.
6276:15
6274:.
6251:.
6241:21
6239:.
6206:^
6192:.
6184:.
6172:.
6149:.
6141:.
6129:.
6125:.
6102:.
6090:.
5909:.
4267:.
4148:24
4106:24
4023:24
3896:24
3859:Tr
3771:,
3757:.
3569:28
3566:25
3530:10
3029:.
1620:.
1263:.
1138:A
1123:−2
1104:.
1084:12
1004:12
833:.
604:U(
580:E(
568:O(
26:→
6820:.
6814::
6802:.
6796::
6770:.
6758::
6750::
6734:.
6690:.
6678::
6670::
6660::
6620::
6608:.
6606:.
6578::
6570::
6538:.
6536:)
6522:.
6510::
6502::
6476:.
6462::
6442:.
6440:.
6436::
6428::
6405:.
6391::
6368:.
6364::
6356::
6312:.
6300::
6292::
6282::
6259:.
6247::
6225:.
6200:.
6188::
6180::
6157:.
6145::
6137::
6110:.
6106::
6098::
6092:1
5991:p
5905:N
5886:N
5783:)
5780:s
5770:p
5766:(
5763:)
5760:r
5754:p
5751:(
5748:+
5743:s
5740:,
5737:r
5733:h
5729:,
5726:c
5720:V
5695:s
5692:r
5689:+
5684:s
5681:,
5678:r
5674:h
5670:,
5667:c
5661:V
5631:.
5627:)
5618:)
5612:2
5608:)
5600:p
5593:p
5590:(
5582:2
5578:)
5570:p
5566:p
5563:k
5560:2
5557:+
5554:s
5551:p
5548:+
5545:r
5538:p
5534:(
5530:(
5519:p
5515:p
5512:4
5508:1
5503:,
5500:c
5494:V
5475:)
5469:2
5465:)
5457:p
5450:p
5447:(
5439:2
5435:)
5427:p
5423:p
5420:k
5417:2
5414:+
5411:s
5408:p
5402:r
5395:p
5391:(
5387:(
5376:p
5372:p
5369:4
5365:1
5360:,
5357:c
5351:V
5339:(
5332:Z
5325:k
5317:=
5305:)
5300:)
5297:s
5287:p
5283:(
5280:)
5277:r
5271:p
5268:(
5265:+
5260:s
5257:,
5254:r
5250:h
5246:,
5243:c
5237:V
5231:+
5226:s
5223:r
5220:+
5215:s
5212:,
5209:r
5205:h
5201:,
5198:c
5192:V
5186:(
5182:/
5174:s
5171:,
5168:r
5164:h
5160:,
5157:c
5151:V
5110:s
5107:,
5104:r
5100:h
5096:,
5093:c
5087:V
5060:)
5057:s
5054:,
5051:r
5048:(
5028:1
5018:p
5011:s
5005:1
4985:1
4979:p
4973:r
4967:1
4943:p
4939:,
4936:p
4912:p
4905:p
4899:2
4873:p
4869:,
4866:p
4862:c
4858:=
4855:c
4832:.
4823:s
4820:,
4817:r
4813:h
4809:,
4806:c
4800:V
4789:)
4784:s
4781:r
4777:q
4770:1
4767:(
4764:=
4757:s
4754:r
4751:+
4746:s
4743:,
4740:r
4736:h
4732:,
4729:c
4723:V
4703:s
4700:,
4697:r
4693:h
4689:,
4686:c
4680:V
4669:=
4662:s
4659:r
4656:+
4651:s
4648:,
4645:r
4641:h
4637:,
4634:c
4628:V
4621:/
4613:s
4610:,
4607:r
4603:h
4599:,
4596:c
4590:V
4555:s
4552:,
4549:r
4545:h
4541:,
4538:c
4532:V
4505:s
4502:,
4499:r
4495:h
4491:,
4488:c
4482:V
4457:s
4454:r
4451:+
4446:s
4443:,
4440:r
4436:h
4432:,
4429:c
4423:V
4400:s
4397:r
4373:s
4370:,
4367:r
4363:h
4359:,
4356:c
4350:V
4320:N
4312:s
4309:,
4306:r
4285:C
4278:c
4228:,
4224:)
4217:+
4212:4
4208:q
4204:5
4201:+
4196:3
4192:q
4188:3
4185:+
4180:2
4176:q
4172:2
4169:+
4166:q
4163:+
4160:1
4156:(
4145:c
4137:h
4133:q
4129:=
4123:)
4120:q
4117:(
4102:1
4096:c
4087:h
4083:q
4077:=
4071:)
4066:n
4062:q
4055:1
4052:(
4042:1
4039:=
4036:n
4020:c
4012:h
4008:q
4002:=
3999:)
3996:q
3993:(
3986:h
3983:,
3980:c
3974:V
3941:h
3938:,
3935:c
3929:V
3903:.
3893:c
3883:0
3879:L
3874:q
3864:R
3855:=
3852:)
3849:q
3846:(
3840:R
3809:R
3755:r
3751:s
3747:m
3743:r
3723:)
3720:1
3717:+
3714:m
3711:(
3708:m
3705:4
3700:1
3692:2
3686:)
3680:s
3677:m
3671:r
3668:)
3665:1
3662:+
3659:m
3656:(
3651:(
3643:=
3640:)
3637:c
3634:(
3629:s
3626:,
3623:r
3619:h
3615:=
3612:h
3599:h
3581:}
3574:,
3561:,
3556:7
3553:6
3548:,
3543:5
3540:4
3535:,
3527:7
3522:,
3517:2
3514:1
3509:,
3506:0
3502:{
3498:=
3490:,
3487:4
3484:,
3481:3
3478:,
3475:2
3472:=
3469:m
3464:}
3457:)
3454:1
3451:+
3448:m
3445:(
3442:m
3438:6
3430:1
3426:{
3418:c
3405:h
3401:c
3397:c
3393:h
3372:c
3352:h
3330:N
3326:A
3311:N
3309:(
3307:p
3290:,
3285:)
3282:s
3279:r
3273:N
3270:(
3267:p
3261:)
3255:)
3252:c
3249:(
3244:s
3241:,
3238:r
3234:h
3227:h
3222:(
3215:N
3209:s
3206:,
3203:r
3197:1
3187:N
3183:A
3155:N
3122:n
3115:L
3111:=
3101:n
3097:L
3074:n
3070:L
3045:c
3009:)
3006:c
3003:(
2998:s
2995:,
2992:r
2988:h
2967:s
2964:,
2961:r
2934:2
2928:q
2922:p
2910:.
2897:Z
2890:q
2887:,
2884:p
2871:q
2868:p
2861:2
2857:)
2853:q
2847:p
2844:(
2838:6
2832:1
2829:=
2826:c
2803:s
2800:r
2780:)
2777:c
2774:(
2765:s
2761:,
2754:r
2749:h
2745:=
2742:s
2739:r
2736:+
2733:h
2713:)
2710:c
2707:(
2702:s
2699:,
2696:r
2692:h
2688:=
2685:h
2661:s
2653:r
2649:+
2646:s
2643:r
2640:=
2637:N
2617:N
2597:N
2574:2
2571:=
2568:N
2548:v
2545:)
2540:2
2533:L
2527:2
2523:b
2519:+
2514:2
2509:1
2502:L
2498:(
2476:2
2472:b
2466:4
2463:3
2453:2
2450:1
2442:=
2439:)
2436:c
2433:(
2428:1
2425:,
2422:2
2418:h
2397:1
2394:=
2391:N
2371:v
2366:1
2359:L
2336:0
2333:,
2330:c
2324:V
2301:0
2298:=
2295:)
2292:c
2289:(
2284:1
2281:,
2278:1
2274:h
2250:.
2242:2
2238:)
2232:1
2225:b
2221:+
2218:b
2215:(
2212:6
2209:+
2206:1
2203:=
2200:c
2190:,
2182:)
2175:2
2171:)
2167:s
2162:1
2155:b
2151:+
2148:r
2145:b
2142:(
2134:2
2130:)
2124:1
2117:b
2113:+
2110:b
2107:(
2102:(
2095:4
2092:1
2087:=
2084:)
2081:c
2078:(
2073:s
2070:,
2067:r
2063:h
2039:s
2036:r
2033:=
2030:N
2010:s
2007:,
2004:r
1984:)
1981:c
1978:(
1973:s
1970:,
1967:r
1963:h
1959:=
1956:h
1936:N
1914:h
1911:,
1908:c
1902:V
1875:c
1871:h
1867:c
1863:h
1859:c
1855:h
1841:0
1835:k
1813:k
1809:n
1797:2
1793:n
1784:1
1780:n
1773:0
1753:v
1746:k
1742:n
1734:L
1723:2
1719:n
1711:L
1703:1
1699:n
1691:L
1677:c
1673:c
1657:h
1654:,
1651:c
1645:V
1629:c
1625:h
1608:v
1586:i
1582:n
1576:k
1571:1
1568:=
1565:i
1557:=
1554:N
1534:0
1528:k
1506:k
1502:n
1490:2
1486:n
1477:1
1473:n
1466:0
1446:v
1439:k
1435:n
1427:L
1416:2
1412:n
1404:L
1396:1
1392:n
1384:L
1361:0
1357:L
1333:0
1327:N
1307:N
1304:+
1301:h
1279:0
1275:L
1251:v
1241:h
1224:,
1221:v
1218:h
1215:=
1212:v
1207:0
1203:L
1198:,
1195:0
1192:=
1189:v
1184:0
1178:n
1174:L
1150:v
1120:L
1116:3
1113:L
1080:/
1076:1
1053:.
1048:0
1045:,
1042:n
1039:+
1036:m
1028:)
1025:m
1017:3
1013:m
1009:(
1001:c
996:+
991:n
988:+
985:m
981:L
977:)
974:n
968:m
965:(
962:=
959:]
954:n
950:L
946:,
941:m
937:L
933:[
910:0
907:=
904:]
899:n
895:L
891:,
888:c
885:[
875:c
864:n
857:n
855:L
790:e
783:t
776:v
672:8
670:E
664:7
662:E
656:6
654:E
648:4
646:F
640:2
638:G
632:)
630:n
620:)
618:n
608:)
606:n
596:)
594:n
584:)
582:n
572:)
570:n
560:)
558:n
548:)
546:n
488:)
475:Z
463:)
450:Z
426:)
413:Z
404:(
317:p
282:Q
274:n
271:D
261:n
258:A
250:n
247:S
239:n
236:Z
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
