3170:, and then a very delicate check that the resulting algebra-coalgebra structure is a Hopf algebra (see for example the article from 2001 below); the method of proof is considerably simplified by the 2003 article cited below (albeit packaged into the definition of Hopf algebroid). The Hopf algebroid structure on the endomorphism ring
1407:). Since a group algebra is a Hopf algebra, the first example above illustrates the back implication of the theorem. Other examples come from the fact that finite Hopf-Galois extensions are depth two in a strong sense (the split epimorphism in the definition may be replaced by a bimodule isomorphism).
1375:> 2 makes sense via the right endomorphism ring extension iterated to generate a tower of rings (a technical procedure beyond the scope of this survey, although the first step, the endomorphism ring theorem, is described in the section on Frobenius extension under
3161:
The proof of this theorem is a reconstruction theorem, requiring the construction of a Hopf algebra as a minimum, but in most papers done by construction of a nondegenerate pairing of two algebras in the iterated endomorphism algebra tower above
727:
556:
1333:
3116:
1214:
2165:
3246:
2862:
3027:> 2 Frobenius extension since such a depth n extension embeds in a depth two extension in a tower of iterated endomorphism rings. For example, given a depth three Frobenius extension of ring
2094:
608:
2922:
423:
162:
1089:
226:
782:
2035:
coefficients and combinatorics of skew tableaux to be (up to permutation) the 2 by 3 matrix with top row 1,1,0 and bottom row 0,1,1, which has depth three after applying the definition.
2978:
2727:
1836:
963:
3133:: in particular, somewhat related to A. Ocneanu's definition of depth, his theory of paragroups, and the articles by W. Szymanski, Nikshych-Vainerman, R. Longo and others.
369:
2800:
1720:
877:
2480:
2038:
In a 2011 article in the
Journal of Algebra by R. Boltje, S. Danz and B. Kuelshammer, they provide a simplified and extended definition of the depth of any unital subring
104:
2525:
1913:
280:.) Equivalently, the condition for left or right depth two may be given in terms of a split monomorphism of bimodules where the domains and codomains above are reversed.
2624:
2333:
996:
2582:
2013:
1977:
1781:
2276:
816:
1751:
428:
2642:
Main classes of examples of depth two extensions are Galois extensions of algebras being acted upon by groups, Hopf algebras, weak Hopf algebras or
613:
3038:
1110:
56:. A more recent definition of depth of any unital subring in any associative ring is proposed (see below) in a paper studying the depth of a
3248:
is one-dimensional. The action of an endomorphism on its space of definition is shown to be a Hopf-Galois action. The dual Hopf algebra
3430:
2103:
1219:
3310:
1371:
group algebras of a subgroup pair of finite index) the two one-sided conditions of depth two are equivalent, and a notion of depth
3284:
is not as important to the proof as the depth two hypothesis and might be avoided by imposing a progenerator module condition on
3181:
2813:
48:, whereas the notion of depth greater than two measures the defect, or distance, from being depth two in a tower of iterated
2366:
thereby showing that combinatorial depth is finite. In more detail, one defines an ascending chain of sets of subgroups of
3121:
The main theorem in this subject is the following based on algebraic arguments in two of the articles below, published in
2936:
extension, the right and left endomorphism rings are anti-isomorphic, which restricts to an antipode on the bialgebroid
2053:
561:
3001:
2879:
1587:
1474:
is isomorphic to the permutation module on the right cosets. The 2013 paper referenced below proves that the depth of
3276:
is precisely the invariant subalgebra of the Hopf-Galois action (and not just contained within). The condition that
382:
121:
3252:
introduced above as well in the Hopf algebroid context and the dual left action becomes a right coaction that makes
1021:
2032:
186:
732:
37:
1633:
3316:
Boltje, R.; Külshammer, B. (2010), "On the depth two condition for group algebra and Hopf algebra extensions",
2939:
2688:
3406:
3366:
3178:-bimodule A (discussed above) becomes a Hopf algebra in the presence of the hypothesis that the centralizer
3122:
2031:. The inclusion matrix may be computed in at least three ways via idempotents, via character tables or via
1786:
929:
328:
3364:
Kadison, L.; Nikshych, D. (2001), "Hopf algebra actions of strongly separable extensions of depth two",
1527:
3340:
Boltje, R.; Danz, S.; Külshammer, B. (2011), "On the depth of subgroups and group algebra extensions",
2924:(often called a theory of duality of actions, which dates back in operator algebras to the 1970s). If
2354:-set homomorphisms instead of modules and module homomorphisms. They characterize combinatorial depth
3477:
2772:
1683:
849:
2985:
1571:
315:
2437:
83:
3457:
3439:
3393:
3375:
3342:
3318:
3013:
2485:
1873:
1617:
2587:
2296:
968:
3404:
Kadison, L.; Szlachanyi, K. (2003), "Bialgebroid actions on depth two extensions and duality",
3306:
2933:
2667:
2546:
2220:
1982:
1946:
1583:
1567:
1376:
1352:
49:
21:
3428:
Kadison, L. (2014), "Hopf subalgebras and tensor powers of generalized permutation modules",
1760:
3449:
3415:
3385:
3351:
3327:
3126:
2246:
1726:(and each corresponding entry). Denoting the left-hand side of this inequality by the power
1661:
791:
69:
3020:; see Brzezinski-Wisbauer for the definition of the Amitsur cochain complex with product).
1729:
1652:
Depth in relation to finite-dimensional semisimple algebras and subgroups of finite groups
308:
551:{\displaystyle p((a_{1},\cdots ,a_{n}))=\sum _{i=1}^{n}a_{i}g_{i}^{-1}\otimes _{B}g_{i}}
2981:
2643:
41:
3420:
2761:
similar to that of Hopf-Galois theory. There is a right bialgebroid structure on the
3471:
3461:
2758:
895:
3397:
3145:
having depth 2, a surjective
Frobenius homomorphism and one-dimensional centralizer
3356:
3332:
2434:
to be minimum, and a technical definition of odd combinatorial depth. For example,
2215:-bimodules). (This definition is equivalent to an earlier notion of depth in case
1384:
296:
61:
45:
891:
25:
17:
3453:
722:{\displaystyle q(a\otimes _{B}a')=(a\gamma _{1}(a'),\cdots ,a\gamma _{n}(a'))}
300:
65:
3111:{\displaystyle \lambda :B\rightarrow {\mbox{End}}\,A_{B},\ \lambda (b)(a)=ba}
3305:. London Math. Soc. Lect. Note Ser., 309. Cambridge University Press, 2003.
3130:
1546:) (not counting multiplicities, an entirely similar definition for depth of
1209:{\displaystyle p(a_{1},\cdots ,a_{n})=\sum _{i=1}^{n}x_{i}\otimes _{B}a_{i}}
3389:
2235:
are complex semisimple algebras.) Again notice that a subring having depth
57:
3023:
The Galois theory of a depth two extension is not irrelevant to a depth
2278:
denote the minimal depth. They then apply this to the group algebras of
1554:-module with closely related results). As a consequence, the depth of
165:
53:
3380:
2682:
2626:, which in turn is bounded by twice the index of the normalizer of
2864:, i.e. isomorphic as rings to the smash product of the bialgebroid
3444:
2344:
2160:{\displaystyle \oplus _{i=1}^{m}A\otimes _{B}\cdots \otimes _{B}A}
1606:
1498:-module is defined in that paper to be the least positive integer
1335:
as the reader may verify. A similar argument naturally shows that
1328:{\displaystyle q(a\otimes _{B}a')=(f_{1}(a)a',\cdots ,f_{n}(a)a')}
2810:; certain endomorphism rings decompose as smash product, such as
890:
As another example (perhaps more elementary than the first; see
1624:
is the length of the descending chain of annihilator ideals in
2207:-bimodules (or equivalently for free Frobenius extensions, as
2015:, the order 2 and order 6 permutation groups on three letter
1403:(i.e. invariant under the left and right adjoint actions of
1562:
is finite if and only if its "generalized quotient module"
3241:{\displaystyle R=\{r\in A:br=rb{\text{ for all }}b\in B\}}
1414:
be a Hopf subalgebra of a finite-dimensional Hopf algebra
3035:, one can show that the left multiplication monomorphism
2638:
Galois theory for depth two extensions and a Main
Theorem
2343:
mimicking the definition of depth of a subring but using
1632:, which stabilize on the maximal Hopf ideal within the
1482:
is determined to the nearest even value by the depth of
2857:{\displaystyle {\mbox{End}}\,A_{B}\cong A\otimes _{R}S}
3055:
2944:
2884:
2818:
2693:
2430:. The minimum combinatorial depth follows from taking
1664:) of finite-dimensional semisimple (complex) algebras
36:. The notion of depth two is important in a certain
3264:. The condition that the Frobenius homomorphism map
3184:
3041:
2942:
2882:
2816:
2775:
2691:
2590:
2549:
2488:
2440:
2299:
2249:
2106:
2056:
1985:
1949:
1876:
1789:
1763:
1753:
and similarly for all powers of the inclusion matrix
1732:
1686:
1222:
1113:
1024:
971:
932:
852:
794:
735:
616:
564:
431:
385:
331:
189:
124:
86:
2227:
with surjective
Frobenius homomorphism, for example
2199:
if the same condition is satisfied more strongly as
1660:
is the inclusion matrix (or incidence matrix of the
1943:. As another example, consider the group algebras
264:; the theory works as well for a ring homomorphism
179:-bimodules, there is a corresponding definition of
3240:
3110:
2972:
2916:
2856:
2794:
2721:
2618:
2576:
2519:
2474:
2327:
2270:
2159:
2100:+1 times A) is isomorphic to a direct summand in
2088:
2007:
1971:
1907:
1830:
1783:on the subalgebra pair of semisimple algebras is:
1775:
1745:
1714:
1327:
1208:
1083:
990:
957:
871:
810:
776:
721:
602:
550:
417:
363:
236:is a split epimorphism if there is a homomorphism
220:
156:
98:
2670:extension (briefly called Frobenius extensions).
2089:{\displaystyle A\otimes _{B}\cdots \otimes _{B}A}
1850:+1 condition.) For example, a depth one subgroup
603:{\displaystyle q:A\otimes _{B}A\rightarrow A^{n}}
2984:. There is the following relation with relative
2917:{\displaystyle {\mbox{End}}\,A\otimes _{B}A_{A}}
2370:starting with the zeroth stage singleton set of
2666:-Galois, explained in detail in the article on
418:{\displaystyle A^{n}\rightarrow A\otimes _{B}A}
157:{\displaystyle A^{n}\rightarrow A\otimes _{B}A}
2358:as a condition on the number of conjugates of
1084:{\displaystyle \sum _{i=1}^{n}x_{i}f_{i}(a)=a}
1870:, satisfies the condition on the centralizer
1597:is a permutation module over a group algebra
1383:is a Hopf subalgebra of a finite-dimensional
221:{\displaystyle A^{n}=A\times \ldots \times A}
8:
3235:
3191:
1672:, the depth two condition on the subalgebra
777:{\displaystyle \gamma _{i}(g)=\delta _{ij}g}
2741:is a left bialgebroid over the centralizer
2658:is a depth two extension of its subalgebra
110:if there is a split epimorphism of natural
2872:it acts on. Something similar is true for
2382:th stage is to intersect all subgroups of
2293:They define a minimum combinatorial depth
1422:° denote the maximal ideal of elements of
3443:
3419:
3379:
3355:
3331:
3221:
3183:
3141:is a Frobenius extension of a subalgebra
3066:
3061:
3054:
3040:
2973:{\displaystyle {\mbox{End}}\,{}_{B}A_{B}}
2964:
2954:
2952:
2950:
2943:
2941:
2908:
2898:
2890:
2883:
2881:
2845:
2829:
2824:
2817:
2815:
2783:
2774:
2722:{\displaystyle {\mbox{End}}\,{}_{B}A_{B}}
2713:
2703:
2701:
2699:
2692:
2690:
2595:
2589:
2548:
2502:
2487:
2445:
2439:
2304:
2298:
2248:
2148:
2135:
2122:
2111:
2105:
2077:
2064:
2055:
1999:
1984:
1963:
1948:
1890:
1875:
1816:
1794:
1788:
1762:
1737:
1731:
1694:
1685:
1648:= 0 } (using a 1967 theorem of Rieffel).
1526:from the right) has the same constituent
1299:
1263:
1236:
1221:
1200:
1190:
1180:
1170:
1159:
1143:
1124:
1112:
1060:
1050:
1040:
1029:
1023:
976:
970:
937:
931:
898:for some of the elementary notions), let
860:
851:
799:
793:
762:
740:
734:
696:
660:
630:
615:
594:
578:
563:
542:
532:
519:
514:
504:
494:
483:
464:
445:
430:
406:
390:
384:
355:
336:
330:
194:
188:
145:
129:
123:
85:
2378:by all its conjugate subgroups, and the
1490:-module (by restriction). The depth of
1846:condition, then it satisfies the depth
287:be the group algebra of a finite group
3125:, that are inspired from the field of
2733:: denoting this endomorphism ring by
2685:theory based on the natural action of
2646:; for example, suppose a finite group
902:be an algebra over a commutative ring
2988:: the relative Hochschild complex of
2183:-bimodules for some positive integer
303:for the elementary definitions). Let
256:is referred to as the ring extension
232:times) as well as the common notion,
7:
3301:Tomasz Brzezinski; Robert Wisbauer,
3000:, and cup product, is isomorphic as
2932:is in addition to being depth two a
2673:Conversely, any depth two extension
2650:acts by automorphisms on an algebra
1831:{\displaystyle M^{m+1}\leq nM^{m-1}}
1470:, and one shows as an exercise that
1458:-module coalgebra. For example, if
958:{\displaystyle f_{i}:A\rightarrow B}
818:(and extended linearly to a mapping
3431:Journal of Pure and Applied Algebra
2414:−1)'st stage subset. For example,
1578:. This is the case for example if
364:{\displaystyle g_{1},\cdots ,g_{n}}
240:in the reverse direction such that
3137:Main Theorem: Suppose an algebra
2394:. Then the combinatorial depth of
2390:−1)'st stage by all conjugates of
1347:Depth in relation to Hopf algebras
14:
2543:. In general, the minimum depth
2410:th stage subset is equal to the (
183:. Here we use the usual notation
2531:and its centralizer subgroup in
1757:, the condition of being depth
1434:is a right ideal and coideal in
910:is taken to be in the center of
291:(over any commutative base ring
2662:of invariants if the action is
2527:(i.e., G equals the product of
2426:has combinatorial depth two in
2374:, the first stage intersecting
1628:of increasing tensor powers of
307:be the group (sub)algebra of a
272:, which induces right and left
44:in place of the more classical
3357:10.1016/j.jalgebra.2011.03.019
3333:10.1016/j.jalgebra.2009.11.043
3280:be a Frobenius extension over
3096:
3090:
3087:
3081:
3051:
3004:to the Amitsur complex of the
2795:{\displaystyle A\otimes _{B}A}
2613:
2601:
2571:
2553:
2514:
2508:
2463:
2451:
2322:
2310:
2265:
2253:
1902:
1896:
1715:{\displaystyle MM^{t}M\leq nM}
1322:
1311:
1305:
1275:
1269:
1256:
1250:
1226:
1149:
1117:
1072:
1066:
949:
872:{\displaystyle A\otimes _{B}A}
752:
746:
716:
713:
702:
677:
666:
650:
644:
620:
587:
558:. It is split by the mapping
473:
470:
438:
435:
396:
135:
34:depth of a Frobenius extension
1:
3421:10.1016/s0001-8708(02)00028-2
1522:-modules, diagonal action of
1426:having counit value 0. Then
76:Definition and first examples
3153:is Hopf-Galois extension of
3002:differential graded algebras
2868:(or its dual) with the ring
2475:{\displaystyle d_{c}(H,G)=1}
1927:a subgroup in the center of
1616:is a Hopf algebra that is a
842:): the splitting condition
99:{\displaystyle B\subseteq A}
38:noncommutative Galois theory
2520:{\displaystyle G=HC_{G}(H)}
1908:{\displaystyle G=HC_{G}(X)}
1858:, viewed as group algebras
834:-module homomorphism since
325:with coset representatives
244:= identity on the image of
3494:
3454:10.1016/j.jpaa.2013.06.008
2619:{\displaystyle d_{c}(H,G)}
2584:is shown to be bounded by
2328:{\displaystyle d_{c}(H,G)}
2286:over any commutative ring
2239:implies that it has depth
2033:Littlewood-Richardson rule
1722:for some positive integer
1680:is given by an inequality
1438:, and the quotient module
991:{\displaystyle x_{i}\in A}
171:; by switching to natural
2027:where the subgroup fixes
1915:for each cyclic subgroup
1866:over the complex numbers
1466:is a subgroup algebra of
1462:is a group algebra, then
248:. (Sometimes the subring
2577:{\displaystyle d(RH,RG)}
2418:is a normal subgroup of
2008:{\displaystyle A=CS_{3}}
1972:{\displaystyle B=CS_{2}}
3407:Advances in Mathematics
3367:Advances in Mathematics
3123:Advances in Mathematics
2980:satisfying axioms of a
1776:{\displaystyle m\geq 1}
918:is a finite projective
276:-modules structures on
28:, there is a notion of
3390:10.1006/aima.2001.2003
3242:
3112:
2974:
2918:
2858:
2796:
2765:-centralized elements
2723:
2620:
2578:
2521:
2476:
2329:
2272:
2271:{\displaystyle d(B,A)}
2161:
2090:
2009:
1973:
1923:(whence normal); e.g.
1909:
1832:
1777:
1747:
1716:
1605:has a basis that is a
1528:indecomposable modules
1339:is right depth two in
1329:
1210:
1175:
1085:
1045:
992:
959:
922:-module, so there are
883:is right depth two in
873:
812:
811:{\displaystyle g_{j}H}
778:
723:
604:
552:
499:
419:
365:
295:; see the articles on
222:
158:
100:
3303:Corings and Comodules
3272:is used to show that
3260:-Galois extension of
3243:
3113:
2996:with coefficients in
2975:
2919:
2859:
2797:
2724:
2621:
2579:
2522:
2477:
2330:
2273:
2162:
2091:
2010:
1974:
1910:
1833:
1778:
1748:
1746:{\displaystyle M^{3}}
1717:
1330:
1211:
1155:
1103:is left depth two in
1086:
1025:
993:
960:
874:
813:
779:
724:
605:
553:
479:
420:
366:
223:
159:
101:
24:extensions, areas of
3182:
3039:
2940:
2880:
2814:
2773:
2689:
2588:
2547:
2486:
2438:
2297:
2247:
2104:
2054:
2042:of associative ring
1983:
1947:
1874:
1842:satisfies the depth
1787:
1761:
1730:
1684:
1518:, tensor product of
1220:
1111:
1022:
969:
930:
879:is satisfied. Thus
850:
792:
733:
614:
562:
429:
383:
329:
187:
122:
84:
3223: for all
2986:homological algebra
2749:commuting with all
2127:
1574:(or Green ring) of
1572:representation ring
1379:). For example, if
1216:with splitting map
1099:. It follows that
527:
3343:Journal of Algebra
3319:Journal of Algebra
3238:
3108:
3059:
3014:group-like element
2970:
2948:
2914:
2888:
2854:
2822:
2792:
2719:
2697:
2616:
2574:
2535:); in particular,
2517:
2472:
2339:of a finite group
2325:
2268:
2157:
2107:
2086:
2005:
1969:
1905:
1854:of a finite group
1838:. (Notice that if
1828:
1773:
1743:
1712:
1618:semisimple algebra
1325:
1206:
1081:
988:
955:
869:
846:= the identity on
808:
774:
719:
600:
548:
510:
415:
361:
218:
164:for some positive
154:
96:
50:endomorphism rings
40:, which generates
30:depth two subring
3224:
3127:operator algebras
3077:
3058:
2947:
2934:Frobenius algebra
2887:
2821:
2696:
2668:Frobenius algebra
2221:Frobenius algebra
1636:ideal, Ann
1584:projective module
1568:algebraic element
1391:has depth two in
1377:Frobenius algebra
1353:Frobenius algebra
371:. Define a split
283:For example, let
80:A unital subring
22:Frobenius algebra
3485:
3464:
3447:
3424:
3423:
3400:
3383:
3360:
3359:
3336:
3335:
3326:(6): 1783–1796,
3247:
3245:
3244:
3239:
3225:
3222:
3117:
3115:
3114:
3109:
3075:
3071:
3070:
3060:
3056:
3016:the identity on
2979:
2977:
2976:
2971:
2969:
2968:
2959:
2958:
2953:
2949:
2945:
2923:
2921:
2920:
2915:
2913:
2912:
2903:
2902:
2889:
2885:
2863:
2861:
2860:
2855:
2850:
2849:
2834:
2833:
2823:
2819:
2801:
2799:
2798:
2793:
2788:
2787:
2728:
2726:
2725:
2720:
2718:
2717:
2708:
2707:
2702:
2698:
2694:
2625:
2623:
2622:
2617:
2600:
2599:
2583:
2581:
2580:
2575:
2526:
2524:
2523:
2518:
2507:
2506:
2481:
2479:
2478:
2473:
2450:
2449:
2362:intersecting in
2334:
2332:
2331:
2326:
2309:
2308:
2277:
2275:
2274:
2269:
2243:+1, so they let
2166:
2164:
2163:
2158:
2153:
2152:
2140:
2139:
2126:
2121:
2095:
2093:
2092:
2087:
2082:
2081:
2069:
2068:
2014:
2012:
2011:
2006:
2004:
2003:
1978:
1976:
1975:
1970:
1968:
1967:
1914:
1912:
1911:
1906:
1895:
1894:
1837:
1835:
1834:
1829:
1827:
1826:
1805:
1804:
1782:
1780:
1779:
1774:
1752:
1750:
1749:
1744:
1742:
1741:
1721:
1719:
1718:
1713:
1699:
1698:
1662:Bratteli diagram
1334:
1332:
1331:
1326:
1321:
1304:
1303:
1285:
1268:
1267:
1249:
1241:
1240:
1215:
1213:
1212:
1207:
1205:
1204:
1195:
1194:
1185:
1184:
1174:
1169:
1148:
1147:
1129:
1128:
1090:
1088:
1087:
1082:
1065:
1064:
1055:
1054:
1044:
1039:
1018:if it satisfies
997:
995:
994:
989:
981:
980:
964:
962:
961:
956:
942:
941:
926:-linear mapping
878:
876:
875:
870:
865:
864:
817:
815:
814:
809:
804:
803:
783:
781:
780:
775:
770:
769:
745:
744:
728:
726:
725:
720:
712:
701:
700:
676:
665:
664:
643:
635:
634:
609:
607:
606:
601:
599:
598:
583:
582:
557:
555:
554:
549:
547:
546:
537:
536:
526:
518:
509:
508:
498:
493:
469:
468:
450:
449:
424:
422:
421:
416:
411:
410:
395:
394:
370:
368:
367:
362:
360:
359:
341:
340:
227:
225:
224:
219:
199:
198:
163:
161:
160:
155:
150:
149:
134:
133:
118:-bimodules from
105:
103:
102:
97:
70:commutative ring
3493:
3492:
3488:
3487:
3486:
3484:
3483:
3482:
3468:
3467:
3427:
3403:
3363:
3339:
3315:
3298:
3180:
3179:
3118:has depth two.
3062:
3037:
3036:
2960:
2951:
2938:
2937:
2904:
2894:
2878:
2877:
2841:
2825:
2812:
2811:
2779:
2771:
2770:
2709:
2700:
2687:
2686:
2644:Hopf algebroids
2640:
2591:
2586:
2585:
2545:
2544:
2498:
2484:
2483:
2482:if and only if
2441:
2436:
2435:
2422:if and only if
2300:
2295:
2294:
2245:
2244:
2144:
2131:
2102:
2101:
2073:
2060:
2052:
2051:
1995:
1981:
1980:
1959:
1945:
1944:
1886:
1872:
1871:
1812:
1790:
1785:
1784:
1759:
1758:
1733:
1728:
1727:
1690:
1682:
1681:
1654:
1644:in H such that
1620:, the depth of
1395:if and only if
1349:
1314:
1295:
1278:
1259:
1242:
1232:
1218:
1217:
1196:
1186:
1176:
1139:
1120:
1109:
1108:
1056:
1046:
1020:
1019:
1008:projective base
972:
967:
966:
933:
928:
927:
856:
848:
847:
795:
790:
789:
758:
736:
731:
730:
705:
692:
669:
656:
636:
626:
612:
611:
590:
574:
560:
559:
538:
528:
500:
460:
441:
427:
426:
402:
386:
381:
380:
351:
332:
327:
326:
309:normal subgroup
190:
185:
184:
141:
125:
120:
119:
108:right depth two
82:
81:
78:
42:Hopf algebroids
12:
11:
5:
3491:
3489:
3481:
3480:
3470:
3469:
3466:
3465:
3438:(2): 367–380,
3425:
3401:
3374:(2): 258–286,
3361:
3337:
3313:
3297:
3294:
3237:
3234:
3231:
3228:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3159:
3158:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3074:
3069:
3065:
3053:
3050:
3047:
3044:
2982:Hopf algebroid
2967:
2963:
2957:
2911:
2907:
2901:
2897:
2893:
2853:
2848:
2844:
2840:
2837:
2832:
2828:
2791:
2786:
2782:
2778:
2716:
2712:
2706:
2639:
2636:
2615:
2612:
2609:
2606:
2603:
2598:
2594:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2516:
2513:
2510:
2505:
2501:
2497:
2494:
2491:
2471:
2468:
2465:
2462:
2459:
2456:
2453:
2448:
2444:
2335:of a subgroup
2324:
2321:
2318:
2315:
2312:
2307:
2303:
2267:
2264:
2261:
2258:
2255:
2252:
2156:
2151:
2147:
2143:
2138:
2134:
2130:
2125:
2120:
2117:
2114:
2110:
2085:
2080:
2076:
2072:
2067:
2063:
2059:
2002:
1998:
1994:
1991:
1988:
1966:
1962:
1958:
1955:
1952:
1904:
1901:
1898:
1893:
1889:
1885:
1882:
1879:
1825:
1822:
1819:
1815:
1811:
1808:
1803:
1800:
1797:
1793:
1772:
1769:
1766:
1740:
1736:
1711:
1708:
1705:
1702:
1697:
1693:
1689:
1653:
1650:
1593:-module or if
1566:represents an
1348:
1345:
1324:
1320:
1317:
1313:
1310:
1307:
1302:
1298:
1294:
1291:
1288:
1284:
1281:
1277:
1274:
1271:
1266:
1262:
1258:
1255:
1252:
1248:
1245:
1239:
1235:
1231:
1228:
1225:
1203:
1199:
1193:
1189:
1183:
1179:
1173:
1168:
1165:
1162:
1158:
1154:
1151:
1146:
1142:
1138:
1135:
1132:
1127:
1123:
1119:
1116:
1080:
1077:
1074:
1071:
1068:
1063:
1059:
1053:
1049:
1043:
1038:
1035:
1032:
1028:
987:
984:
979:
975:
954:
951:
948:
945:
940:
936:
868:
863:
859:
855:
807:
802:
798:
773:
768:
765:
761:
757:
754:
751:
748:
743:
739:
718:
715:
711:
708:
704:
699:
695:
691:
688:
685:
682:
679:
675:
672:
668:
663:
659:
655:
652:
649:
646:
642:
639:
633:
629:
625:
622:
619:
597:
593:
589:
586:
581:
577:
573:
570:
567:
545:
541:
535:
531:
525:
522:
517:
513:
507:
503:
497:
492:
489:
486:
482:
478:
475:
472:
467:
463:
459:
456:
453:
448:
444:
440:
437:
434:
414:
409:
405:
401:
398:
393:
389:
379:epimorphism p:
358:
354:
350:
347:
344:
339:
335:
217:
214:
211:
208:
205:
202:
197:
193:
181:left depth two
153:
148:
144:
140:
137:
132:
128:
95:
92:
89:
77:
74:
66:group algebras
13:
10:
9:
6:
4:
3:
2:
3490:
3479:
3476:
3475:
3473:
3463:
3459:
3455:
3451:
3446:
3441:
3437:
3433:
3432:
3426:
3422:
3417:
3413:
3409:
3408:
3402:
3399:
3395:
3391:
3387:
3382:
3377:
3373:
3369:
3368:
3362:
3358:
3353:
3349:
3345:
3344:
3338:
3334:
3329:
3325:
3321:
3320:
3314:
3312:
3311:0-521-53931-5
3308:
3304:
3300:
3299:
3295:
3293:
3291:
3288:as a natural
3287:
3283:
3279:
3275:
3271:
3267:
3263:
3259:
3255:
3251:
3232:
3229:
3226:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3188:
3185:
3177:
3173:
3169:
3165:
3156:
3152:
3148:
3144:
3140:
3136:
3135:
3134:
3132:
3128:
3124:
3119:
3105:
3102:
3099:
3093:
3084:
3078:
3072:
3067:
3063:
3048:
3045:
3042:
3034:
3031:over subring
3030:
3026:
3021:
3019:
3015:
3011:
3007:
3003:
2999:
2995:
2991:
2987:
2983:
2965:
2961:
2955:
2935:
2931:
2927:
2909:
2905:
2899:
2895:
2891:
2875:
2871:
2867:
2851:
2846:
2842:
2838:
2835:
2830:
2826:
2809:
2805:
2789:
2784:
2780:
2776:
2768:
2764:
2760:
2759:Galois theory
2756:
2752:
2748:
2744:
2740:
2736:
2732:
2714:
2710:
2704:
2684:
2680:
2676:
2671:
2669:
2665:
2661:
2657:
2653:
2649:
2645:
2637:
2635:
2633:
2629:
2610:
2607:
2604:
2596:
2592:
2568:
2565:
2562:
2559:
2556:
2550:
2542:
2539:is normal in
2538:
2534:
2530:
2511:
2503:
2499:
2495:
2492:
2489:
2469:
2466:
2460:
2457:
2454:
2446:
2442:
2433:
2429:
2425:
2421:
2417:
2413:
2409:
2405:
2401:
2397:
2393:
2389:
2385:
2381:
2377:
2373:
2369:
2365:
2361:
2357:
2353:
2349:
2347:
2342:
2338:
2319:
2316:
2313:
2305:
2301:
2291:
2289:
2285:
2281:
2262:
2259:
2256:
2250:
2242:
2238:
2234:
2230:
2226:
2223:extension of
2222:
2218:
2214:
2210:
2206:
2202:
2198:
2194:
2190:
2187:; similarly,
2186:
2182:
2178:
2174:
2170:
2154:
2149:
2145:
2141:
2136:
2132:
2128:
2123:
2118:
2115:
2112:
2108:
2099:
2083:
2078:
2074:
2070:
2065:
2061:
2057:
2049:
2045:
2041:
2036:
2034:
2030:
2026:
2022:
2018:
2000:
1996:
1992:
1989:
1986:
1964:
1960:
1956:
1953:
1950:
1942:
1938:
1934:
1930:
1926:
1922:
1918:
1899:
1891:
1887:
1883:
1880:
1877:
1869:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1823:
1820:
1817:
1813:
1809:
1806:
1801:
1798:
1795:
1791:
1770:
1767:
1764:
1756:
1738:
1734:
1725:
1709:
1706:
1703:
1700:
1695:
1691:
1687:
1679:
1675:
1671:
1667:
1663:
1659:
1651:
1649:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1609:
1604:
1600:
1596:
1592:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1408:
1406:
1402:
1399:is normal in
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1354:
1346:
1344:
1342:
1338:
1318:
1315:
1308:
1300:
1296:
1292:
1289:
1286:
1282:
1279:
1272:
1264:
1260:
1253:
1246:
1243:
1237:
1233:
1229:
1223:
1201:
1197:
1191:
1187:
1181:
1177:
1171:
1166:
1163:
1160:
1156:
1152:
1144:
1140:
1136:
1133:
1130:
1125:
1121:
1114:
1106:
1102:
1098:
1094:
1078:
1075:
1069:
1061:
1057:
1051:
1047:
1041:
1036:
1033:
1030:
1026:
1017:
1013:
1009:
1005:
1001:
985:
982:
977:
973:
965:and elements
952:
946:
943:
938:
934:
925:
921:
917:
913:
909:
905:
901:
897:
896:module theory
893:
888:
886:
882:
866:
861:
857:
853:
845:
841:
838:is normal in
837:
833:
829:
825:
821:
805:
800:
796:
788:in the coset
787:
771:
766:
763:
759:
755:
749:
741:
737:
709:
706:
697:
693:
689:
686:
683:
680:
673:
670:
661:
657:
653:
647:
640:
637:
631:
627:
623:
617:
595:
591:
584:
579:
575:
571:
568:
565:
543:
539:
533:
529:
523:
520:
515:
511:
505:
501:
495:
490:
487:
484:
480:
476:
465:
461:
457:
454:
451:
446:
442:
432:
412:
407:
403:
399:
391:
387:
378:
374:
356:
352:
348:
345:
342:
337:
333:
324:
320:
317:
313:
310:
306:
302:
298:
294:
290:
286:
281:
279:
275:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
231:
215:
212:
209:
206:
203:
200:
195:
191:
182:
178:
174:
170:
167:
151:
146:
142:
138:
130:
126:
117:
113:
109:
93:
90:
87:
75:
73:
71:
67:
63:
59:
55:
51:
47:
46:Galois groups
43:
39:
35:
31:
27:
23:
19:
3435:
3429:
3411:
3405:
3381:math/0107064
3371:
3365:
3347:
3341:
3323:
3317:
3302:
3289:
3285:
3281:
3277:
3273:
3269:
3268:onto all of
3265:
3261:
3257:
3253:
3249:
3175:
3171:
3167:
3163:
3160:
3154:
3150:
3146:
3142:
3138:
3120:
3032:
3028:
3024:
3022:
3017:
3009:
3005:
2997:
2993:
2989:
2929:
2925:
2873:
2869:
2865:
2807:
2803:
2766:
2762:
2754:
2750:
2746:
2745:(those a in
2742:
2738:
2737:, one shows
2734:
2730:
2678:
2674:
2672:
2663:
2659:
2655:
2651:
2647:
2641:
2631:
2627:
2540:
2536:
2532:
2528:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2383:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2351:
2345:
2340:
2336:
2292:
2287:
2283:
2279:
2240:
2236:
2232:
2228:
2224:
2216:
2212:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2097:
2047:
2043:
2039:
2037:
2028:
2024:
2020:
2016:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1867:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1754:
1723:
1677:
1673:
1669:
1665:
1657:
1655:
1645:
1641:
1637:
1629:
1625:
1621:
1613:
1607:
1602:
1598:
1594:
1590:
1579:
1575:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1409:
1404:
1400:
1396:
1392:
1388:
1385:Hopf algebra
1380:
1372:
1368:
1364:
1360:
1356:
1350:
1340:
1336:
1107:by defining
1104:
1100:
1096:
1092:
1015:
1011:
1007:
1003:
999:
923:
919:
915:
911:
907:
903:
899:
889:
884:
880:
843:
839:
835:
831:
827:
823:
819:
785:
376:
372:
322:
318:
311:
304:
297:group theory
292:
288:
284:
282:
277:
273:
269:
265:
261:
257:
253:
249:
245:
241:
237:
233:
229:
180:
176:
172:
168:
115:
111:
107:
106:has (or is)
79:
62:finite group
33:
29:
15:
3478:Ring theory
3350:: 258–281,
2191:has depth 2
1634:annihilator
1612:). In case
1454:is a right
1006:) called a
892:ring theory
610:defined by
26:mathematics
18:ring theory
3414:: 75–121,
3296:References
3131:subfactors
2802:dual over
1502:such that
1355:extension
301:group ring
52:above the
3462:119128079
3445:1210.3178
3292:-module.
3230:∈
3198:∈
3079:λ
3052:→
3043:λ
2896:⊗
2843:⊗
2836:≅
2781:⊗
2757:) with a
2146:⊗
2142:⋯
2133:⊗
2109:⊕
2075:⊗
2071:⋯
2062:⊗
1821:−
1807:≤
1768:≥
1704:≤
1588:generator
1542:+1 times
1363:(such as
1290:⋯
1234:⊗
1188:⊗
1157:∑
1134:⋯
1027:∑
983:∈
950:→
914:. Assume
858:⊗
760:δ
738:γ
694:γ
684:⋯
658:γ
628:⊗
588:→
576:⊗
530:⊗
521:−
481:∑
455:⋯
404:⊗
397:→
346:⋯
213:×
210:…
207:×
143:⊗
136:→
91:⊆
3472:Category
3398:18876684
3008:-coring
2386:in the (
1319:′
1283:′
1247:′
1091:for all
1014:-module
1010:for the
1002:= 1,...,
906:, where
710:′
674:′
641:′
58:subgroup
3174:of the
3149:, then
2654:, then
2406:if the
2046:to be 2
1601:(i.e.,
1570:in the
1418:. Let
1387:, then
166:integer
68:over a
54:subring
3460:
3396:
3309:
3076:
3012:(with
2683:Galois
2681:has a
2171:times
2050:+1 if
1550:as an
1534:⊗⋅⋅⋅⊗
1514:times
1494:as an
1486:as an
1351:For a
729:where
3458:S2CID
3440:arXiv
3394:S2CID
3376:arXiv
2992:over
2348:-sets
2219:is a
2175:) as
1931:, or
1582:is a
1506:⊗⋅⋅⋅⊗
822:into
316:index
268:into
260:over
60:of a
3307:ISBN
2876:and
2402:is 2
2350:and
2282:and
2231:and
1979:and
1668:and
1640:= {
1610:-set
1586:, a
1410:Let
1367:and
826:, a
784:for
299:and
20:and
3450:doi
3436:218
3416:doi
3412:179
3386:doi
3372:163
3352:doi
3348:335
3328:doi
3324:323
3166:in
3057:End
2946:End
2886:End
2820:End
2806:to
2769:in
2753:in
2729:on
2695:End
2630:in
2398:in
2195:in
1919:in
1862:in
1676:in
1656:If
1558:in
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1478:in
1095:in
894:or
425:by
321:in
314:of
252:in
64:as
32:or
16:In
3474::
3456:,
3448:,
3434:,
3410:,
3392:,
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3370:,
3346:,
3322:,
3256:a
3129:,
2928:|
2677:|
2634:.
2290:.
1939:x
1935:=
1864:CG
1860:CH
1646:Qh
1442:=
1359:|
1343:.
887:.
844:pq
242:pq
72:.
3452::
3442::
3418::
3388::
3378::
3354::
3330::
3290:B
3286:A
3282:B
3278:A
3274:B
3270:B
3266:A
3262:B
3258:T
3254:A
3250:T
3236:}
3233:B
3227:b
3219:b
3216:r
3213:=
3210:r
3207:b
3204::
3201:A
3195:r
3192:{
3189:=
3186:R
3176:B
3172:S
3168:A
3164:B
3157:.
3155:B
3151:A
3147:R
3143:B
3139:A
3106:a
3103:b
3100:=
3097:)
3094:a
3091:(
3088:)
3085:b
3082:(
3073:,
3068:B
3064:A
3049:B
3046::
3033:B
3029:A
3025:n
3018:A
3010:S
3006:R
2998:A
2994:B
2990:A
2966:B
2962:A
2956:B
2930:B
2926:A
2910:A
2906:A
2900:B
2892:A
2874:T
2870:A
2866:S
2852:S
2847:R
2839:A
2831:B
2827:A
2808:S
2804:R
2790:A
2785:B
2777:A
2767:T
2763:B
2755:B
2751:b
2747:A
2743:R
2739:S
2735:S
2731:A
2715:B
2711:A
2705:B
2679:B
2675:A
2664:G
2660:B
2656:A
2652:A
2648:G
2632:G
2628:H
2614:)
2611:G
2608:,
2605:H
2602:(
2597:c
2593:d
2572:)
2569:G
2566:R
2563:,
2560:H
2557:R
2554:(
2551:d
2541:G
2537:H
2533:G
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2515:)
2512:H
2509:(
2504:G
2500:C
2496:H
2493:=
2490:G
2470:1
2467:=
2464:)
2461:G
2458:,
2455:H
2452:(
2447:c
2443:d
2432:n
2428:G
2424:H
2420:G
2416:H
2412:n
2408:n
2404:n
2400:G
2396:H
2392:H
2388:n
2384:H
2380:n
2376:H
2372:H
2368:H
2364:G
2360:H
2356:n
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2323:)
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2317:,
2314:H
2311:(
2306:c
2302:d
2288:R
2284:H
2280:G
2266:)
2263:A
2260:,
2257:B
2254:(
2251:d
2241:m
2237:m
2233:B
2229:A
2225:B
2217:A
2213:A
2211:-
2209:B
2205:B
2203:-
2201:A
2197:A
2193:n
2189:B
2185:m
2181:B
2179:-
2177:B
2173:A
2169:n
2167:(
2155:A
2150:B
2137:B
2129:A
2124:m
2119:1
2116:=
2113:i
2098:n
2096:(
2084:A
2079:B
2066:B
2058:A
2048:n
2044:A
2040:B
2029:c
2025:c
2023:,
2021:b
2019:,
2017:a
2001:3
1997:S
1993:C
1990:=
1987:A
1965:2
1961:S
1957:C
1954:=
1951:B
1941:K
1937:H
1933:G
1929:G
1925:H
1921:H
1917:X
1903:)
1900:X
1897:(
1892:G
1888:C
1884:H
1881:=
1878:G
1868:C
1856:G
1852:H
1848:m
1844:m
1840:M
1824:1
1818:m
1814:M
1810:n
1802:1
1799:+
1796:m
1792:M
1771:1
1765:m
1755:M
1739:3
1735:M
1724:n
1710:M
1707:n
1701:M
1696:t
1692:M
1688:M
1678:A
1674:B
1670:A
1666:B
1658:M
1642:h
1638:Q
1630:Q
1626:H
1622:Q
1614:H
1608:G
1603:Q
1599:R
1595:Q
1591:H
1580:Q
1576:R
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1552:H
1548:Q
1544:Q
1540:n
1538:(
1536:Q
1532:Q
1524:R
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1500:n
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1450:°
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1446:/
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1436:H
1432:H
1430:°
1428:R
1424:R
1420:R
1416:H
1412:R
1405:A
1401:A
1397:B
1393:A
1389:B
1381:B
1373:n
1369:B
1365:A
1361:B
1357:A
1341:A
1337:B
1323:)
1316:a
1312:)
1309:a
1306:(
1301:n
1297:f
1293:,
1287:,
1280:a
1276:)
1273:a
1270:(
1265:1
1261:f
1257:(
1254:=
1251:)
1244:a
1238:B
1230:a
1227:(
1224:q
1202:i
1198:a
1192:B
1182:i
1178:x
1172:n
1167:1
1164:=
1161:i
1153:=
1150:)
1145:n
1141:a
1137:,
1131:,
1126:1
1122:a
1118:(
1115:p
1105:A
1101:B
1097:A
1093:a
1079:a
1076:=
1073:)
1070:a
1067:(
1062:i
1058:f
1052:i
1048:x
1042:n
1037:1
1034:=
1031:i
1016:A
1012:B
1004:n
1000:i
998:(
986:A
978:i
974:x
953:B
947:A
944::
939:i
935:f
924:B
920:B
916:A
912:A
908:B
904:B
900:A
885:A
881:B
867:A
862:B
854:A
840:G
836:H
832:B
830:-
828:B
824:B
820:A
806:H
801:j
797:g
786:g
772:g
767:j
764:i
756:=
753:)
750:g
747:(
742:i
717:)
714:)
707:a
703:(
698:n
690:a
687:,
681:,
678:)
671:a
667:(
662:1
654:a
651:(
648:=
645:)
638:a
632:B
624:a
621:(
618:q
596:n
592:A
585:A
580:B
572:A
569::
566:q
544:i
540:g
534:B
524:1
516:i
512:g
506:i
502:a
496:n
491:1
488:=
485:i
477:=
474:)
471:)
466:n
462:a
458:,
452:,
447:1
443:a
439:(
436:(
433:p
413:A
408:B
400:A
392:n
388:A
377:B
375:-
373:A
357:n
353:g
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338:1
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319:n
312:H
305:B
293:k
289:G
285:A
278:A
274:B
270:A
266:B
262:B
258:A
254:A
250:B
246:p
238:q
234:p
230:n
228:(
216:A
204:A
201:=
196:n
192:A
177:A
175:-
173:B
169:n
152:A
147:B
139:A
131:n
127:A
116:B
114:-
112:A
94:A
88:B
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