887:
4064:-dual). Under this correspondence, the commutative finite-dimensional algebras correspond to the cocommutative finite-dimensional coalgebras. So in the finite-dimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, relations diverge in the infinite-dimensional case: while the
2943:
1772:
In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finite-dimensional (unital associative) algebra is a coalgebra, while the dual of a finite-dimensional coalgebra is a (unital associative) algebra. In general, the dual of an algebra may not be a coalgebra.
858:
2631:
371:, having the form above. That they are two different products is emphasized by recalling that the internal tensor product of a vector and a scalar is just simple scalar multiplication. The external product keeps them separated. In this setting, the coproduct is the map
753:
1328:
1755:
2522:
472:
1227:
3230:
358:
3863:
4075:
Every coalgebra is the sum of its finite-dimensional subcoalgebras, something that is not true for algebras. Abstractly, coalgebras are generalizations, or duals, of finite-dimensional unital associative algebras.
2058:
1651:
600:
759:
3039:
3685:
2938:{\displaystyle \sum _{(c)}c_{(1)}\otimes \left(\sum _{(c_{(2)})}(c_{(2)})_{(1)}\otimes (c_{(2)})_{(2)}\right)=\sum _{(c)}\left(\sum _{(c_{(1)})}(c_{(1)})_{(1)}\otimes (c_{(1)})_{(2)}\right)\otimes c_{(2)}.}
3443:
2372:
3731:
3456:. (It's important to understand that the implied summation is significant here: it is not required that all the summands are pairwise equal, only that the sums are equal, a much weaker requirement.)
2243:
3112:
3289:
301:
410:
672:
1233:
1659:
2149:
2105:
562:
537:
4060:
is a finite-dimensional coalgebra, and indeed every finite-dimensional coalgebra arises in this fashion from some finite-dimensional algebra (namely from the coalgebra's
2383:
259:
2620:
2272:
513:, therefore, any homomorphism defined on a subset can be extended to the entire algebra. Examining the lifting in detail, one observes that the coproduct behaves as the
3497:. Contrary to what this naming convention suggests the group-like elements do not always form a group and in general they only form a set. The group-like elements of a
948:
of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative
2591:
2558:
219:
192:
165:
1372:
given above. The shuffle product is appropriate, because it preserves the order of the terms appearing in the product, as is needed by non-commutative algebras.
2169:
495:
1922:
When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element
421:
1116:
3235:
Whenever a variable with lowered and parenthesized index is encountered in an expression of this kind, a summation symbol for that variable is implied.
3123:
367:. A tensor algebra comes with a tensor product (the internal one); it can also be equipped with a second tensor product, the "external" one, or the
313:
613:
The formal definition of the coalgebra, below, abstracts away this particular special case, and its requisite properties, into a general setting.
3739:
4537:
4200:
1964:
1537:
853:{\displaystyle (\mathrm {id} _{C}\otimes \varepsilon )\circ \Delta =\mathrm {id} _{C}=(\varepsilon \otimes \mathrm {id} _{C})\circ \Delta }
567:
2954:
3626:
4598:
4567:
4507:
4414:
4340:
3358:
2291:
4233:
4164:
4147:
304:
138:
4130:
3933:
3690:
607:
4359:
Block, Richard E.; Leroux, Pierre (1985), "Generalized dual coalgebras of algebras, with applications to cofree coalgebras",
4311:
3503:
307:, which extracts the needed quantity from each side of the tensor product. It can be written as an "external" tensor product
4263:
4183:
610:. (The Littlewood–Richardson rule conveys the same idea as the Clebsch–Gordan coefficients, but in a more general setting).
4559:
4499:
1341:
is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called
4243:
4080:
2174:
93:
3050:
4332:
498:
3262:
264:
865:
606:
be linear. As a general rule, the coproduct in representation theory is reducible; the factors are given by the
1093:
748:{\displaystyle (\mathrm {id} _{C}\otimes \Delta )\circ \Delta =(\Delta \otimes \mathrm {id} _{C})\circ \Delta }
602:
is in order to maintain linearity: for this example, (and for representation theory in general), the coproduct
539:
must be kept in sequential order during products of multiple angular momenta (rotations are not commutative).
1323:{\displaystyle \varepsilon (X^{n})={\begin{cases}1&{\mbox{if }}n=0\\0&{\mbox{if }}n>0\end{cases}}}
377:
129:. A primary task, of practical use in physics, is to obtain combinations of systems with different states of
1369:
1101:
3869:
1805:(meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are also dual
1399:
1004:
43:
1750:{\displaystyle \varepsilon ={\begin{cases}1&{\text{if }}x=y,\\0&{\text{if }}x\neq y.\end{cases}}}
1364:. Unlike the polynomial case above, none of these are commutative. Therefore, the coproduct becomes the
122:
89:
2171:; there is only a promise that there are finitely many terms, and that the full sum of all these terms
1510:
2527:
Here it is understood that the sums have the same number of terms, and the same lists of values for
2517:{\displaystyle c=\sum _{(c)}\varepsilon (c_{(1)})c_{(2)}=\sum _{(c)}c_{(1)}\varepsilon (c_{(2)}).\;}
1689:
1264:
4632:
4105:
4027:
4016:
2110:
2066:
880:
622:
66:
58:
4207:
886:
545:
520:
4445:
502:
224:
2596:
2248:
1384:
3044:
Some authors omit the summation symbols as well; in this sumless
Sweedler notation, one writes
4594:
4563:
4533:
4503:
4467:
4410:
4378:
4336:
4307:
1380:
1376:
985:
873:
2563:
2530:
4573:
4543:
4513:
4483:
4457:
4420:
4394:
4368:
4100:
3985:
1353:
949:
130:
110:
4479:
4409:, Pure and Applied Mathematics, vol. 235 (1st ed.), New York, NY: Marcel Dekker,
4390:
197:
170:
4590:
4577:
4547:
4529:
4517:
4487:
4475:
4424:
4398:
4386:
4303:
4267:
4187:
3884:
3877:
1911:
1365:
1086:
640:
514:
51:
47:
144:
69:. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (
2154:
1349:
480:
467:{\displaystyle \Delta :\mathbf {j} \mapsto \mathbf {j} \otimes 1+1\otimes \mathbf {j} }
368:
364:
134:
126:
55:
4462:
1345:, and in fact most of the important coalgebras considered in practice are bialgebras.
4626:
4373:
2282:
1361:
945:
4432:
Gómez-Torrecillas, José (1998), "Coalgebras and comodules over a commutative ring",
363:
The word "external" appears here, in contrast to the "internal" tensor product of a
3733:. In Sweedler's sumless notation, the first of these properties may be written as:
3498:
1357:
1222:{\displaystyle \Delta (X^{n})=\sum _{k=0}^{n}{\dbinom {n}{k}}X^{k}\otimes X^{n-k},}
629:
510:
497:
can be taken to be one of the spin representations of the rotation group, with the
97:
70:
3872:
of two coalgebra morphisms is again a coalgebra morphism, and the coalgebras over
4260:
4180:
3527:
2063:
Note that neither the number of terms in this sum, nor the exact values of each
1333:
Again, because of linearity, this suffices to define Δ and ε uniquely on all of
918:
506:
105:
31:
3225:{\displaystyle c=\varepsilon (c_{(1)})c_{(2)}=c_{(1)}\varepsilon (c_{(2)}).\;}
353:{\displaystyle \mathbf {J} \equiv \mathbf {j} \otimes 1+1\otimes \mathbf {j} }
74:
4471:
4382:
2377:
The fact that ε is a counit can then be expressed with the following formula
4110:
1819:
1809:(meaning that a coalgebra is the dual object of an algebra and conversely).
1342:
4498:, Regional Conference Series in Mathematics, vol. 82, Providence, RI:
4615:
3858:{\displaystyle f(c_{(1)})\otimes f(c_{(2)})=f(c)_{(1)}\otimes f(c)_{(2)}.}
4088:
2053:{\displaystyle \Delta (c)=\sum _{i}c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}}
564:
appear only once in the coproduct, rather than (for example) defining
221:, a particularly important task is to find the total angular momentum
17:
1646:{\displaystyle \Delta =\sum _{y\in }\otimes {\text{ for }}x\leq z\ .}
595:{\displaystyle \mathbf {j} \mapsto \mathbf {j} \otimes \mathbf {j} }
1074:
By linearity, both Δ and ε can then uniquely be extended to all of
921:. Similarly, in the second diagram the naturally isomorphic spaces
4405:
Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Șerban (2001),
4072:-dual of an infinite-dimensional algebra need not be a coalgebra.
62:
3034:{\displaystyle \sum _{(c)}c_{(1)}\otimes c_{(2)}\otimes c_{(3)}.}
2948:
In
Sweedler's notation, both of these expressions are written as
1801:
To distinguish these: in general, algebra and coalgebra are dual
88:
Coalgebras occur naturally in a number of contexts (for example,
77:, gives rise to an algebra, but not in general the other way. In
3680:{\displaystyle (f\otimes f)\circ \Delta _{1}=\Delta _{2}\circ f}
517:, essentially because the two factors above, the left and right
505:
to all of the tensor algebra, by a simple lemma that applies to
4448:(2003), "Cofree coalgebras and multivariable recursiveness",
3438:{\displaystyle c_{(1)}\otimes c_{(2)}=c_{(2)}\otimes c_{(1)}}
2367:{\displaystyle \Delta (c)=\sum _{(c)}c_{(1)}\otimes c_{(2)}.}
65:
of unital associative algebras can be formulated in terms of
4298:
Mikhalev, Aleksandr Vasilʹevich; Pilz, Günter, eds. (2002).
885:
1743:
1316:
4558:, Translations of mathematical monographs, vol. 108,
1082:
becomes a coalgebra with comultiplication Δ and counit ε.
1531:
as basis. The comultiplication and counit are defined as
1387:
holds, e.g. if the coefficients are taken to be a field.
125:, and in particular, in the representation theory of the
121:
One frequently recurring example of coalgebras occurs in
3726:{\displaystyle \varepsilon _{2}\circ f=\varepsilon _{1}}
4617:
A brief introduction to coalgebra representation theory
1298:
1273:
4163:
See also Dăscălescu, Năstăsescu & Raianu (2001).
3742:
3693:
3629:
3361:
3265:
3126:
3053:
2957:
2634:
2599:
2566:
2533:
2386:
2294:
2251:
2177:
2157:
2113:
2069:
1967:
1760:
The intervals of length zero correspond to points of
1662:
1540:
1236:
1164:
1119:
762:
675:
570:
548:
523:
501:
being the common-sense choice. This coproduct can be
483:
424:
380:
316:
267:
227:
200:
173:
147:
3977:
is itself a coalgebra, with the restriction of ε to
944:
The first diagram is the dual of the one expressing
4434:
Revue
Roumaine de Mathématiques Pures et Appliquées
27:
Vector space V equipped with a linear map V → V ⊗ V
3857:
3725:
3679:
3526:. The primitive elements of a Hopf algebra form a
3437:
3283:
3224:
3106:
3033:
2937:
2614:
2585:
2552:
2516:
2366:
2266:
2238:{\displaystyle c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}}
2237:
2163:
2143:
2099:
2052:
1749:
1645:
1322:
1221:
852:
747:
594:
556:
531:
489:
466:
404:
352:
295:
253:
213:
186:
159:
4331:. Cambridge Tracts in Mathematics. Vol. 74.
3107:{\displaystyle \Delta (c)=c_{(1)}\otimes c_{(2)}}
1180:
1167:
1010:, as follows. The elements of this vector space
2625:The coassociativity of Δ can be expressed as
3876:together with this notion of morphism form a
8:
4242:Dăscălescu, Năstăsescu & Raianu (2001).
4232:Dăscălescu, Năstăsescu & Raianu (2001).
4068:-dual of every coalgebra is an algebra, the
3284:{\displaystyle \sigma \circ \Delta =\Delta }
1776:The key point is that in finite dimensions,
296:{\displaystyle |A\rangle \otimes |B\rangle }
290:
276:
4052:is a finite-dimensional unital associative
1022:that map all but finitely many elements of
4030:are valid for coalgebras, so for instance
3943:becomes a coalgebra in a natural fashion.
3221:
2513:
1865:, which when dualized yields a linear map
4461:
4372:
3840:
3812:
3781:
3753:
3741:
3717:
3698:
3692:
3665:
3652:
3628:
3423:
3404:
3385:
3366:
3360:
3264:
3203:
3181:
3162:
3143:
3125:
3092:
3073:
3052:
3016:
2997:
2978:
2962:
2956:
2920:
2896:
2880:
2858:
2842:
2818:
2810:
2789:
2765:
2749:
2727:
2711:
2687:
2679:
2655:
2639:
2633:
2598:
2571:
2565:
2538:
2532:
2495:
2473:
2457:
2438:
2419:
2397:
2385:
2349:
2330:
2314:
2293:
2250:
2223:
2212:
2193:
2182:
2176:
2156:
2129:
2118:
2112:
2085:
2074:
2068:
2038:
2027:
2008:
1997:
1987:
1966:
1723:
1697:
1684:
1661:
1623:
1566:
1539:
1297:
1272:
1259:
1247:
1235:
1204:
1191:
1179:
1166:
1163:
1157:
1146:
1130:
1118:
879:Equivalently, the following two diagrams
835:
827:
808:
800:
775:
767:
761:
730:
722:
688:
680:
674:
587:
579:
571:
569:
549:
547:
524:
522:
482:
459:
439:
431:
423:
379:
345:
325:
317:
315:
282:
268:
266:
245:
232:
226:
205:
199:
178:
172:
146:
4496:Hopf algebras and their actions on rings
1902:, so this defines a comultiplication on
81:, this duality goes in both directions (
4121:
3577:are two coalgebras over the same field
952:. Accordingly, the map Δ is called the
405:{\displaystyle \Delta :J\to J\otimes J}
1847:is a coalgebra. The multiplication of
1383:forms a graded coalgebra whenever the
82:
78:
7:
4450:Journal of Pure and Applied Algebra
4361:Journal of Pure and Applied Algebra
1348:Examples of coalgebras include the
4556:Tensor spaces and exterior algebra
4149:Tensor spaces and exterior algebra
4132:Tensor spaces and exterior algebra
3662:
3649:
3348:. In Sweedler's sumless notation,
3278:
3272:
3054:
2600:
2295:
2252:
1968:
1880:. In the finite-dimensional case,
1541:
1171:
1120:
1085:As a second example, consider the
847:
831:
828:
804:
801:
793:
771:
768:
742:
726:
723:
715:
706:
697:
684:
681:
425:
381:
25:
3352:is co-commutative if and only if
137:. For this purpose, one uses the
109:, with important applications in
4526:An introduction to Hopf algebras
4079:Corresponding to the concept of
1502:, Δ, ε) is a coalgebra known as
1099:. This becomes a coalgebra (the
588:
580:
572:
550:
542:The peculiar form of having the
525:
460:
440:
432:
346:
326:
318:
4300:The Concise Handbook of Algebra
1038:to 1 and all other elements of
305:total angular momentum operator
4407:Hopf Algebras: An introduction
4245:Hopf Algebras: An introduction
4235:Hopf Algebras: An introduction
4166:Hopf Algebras: An introduction
4146:Yokonuma (1992). "Prop. 1.4".
4129:Yokonuma (1992). "Prop. 1.7".
3847:
3841:
3837:
3830:
3819:
3813:
3809:
3802:
3793:
3788:
3782:
3774:
3765:
3760:
3754:
3746:
3642:
3630:
3430:
3424:
3411:
3405:
3392:
3386:
3373:
3367:
3215:
3210:
3204:
3196:
3188:
3182:
3169:
3163:
3155:
3150:
3144:
3136:
3099:
3093:
3080:
3074:
3063:
3057:
3023:
3017:
3004:
2998:
2985:
2979:
2969:
2963:
2927:
2921:
2903:
2897:
2893:
2887:
2881:
2873:
2865:
2859:
2855:
2849:
2843:
2835:
2830:
2825:
2819:
2811:
2796:
2790:
2772:
2766:
2762:
2756:
2750:
2742:
2734:
2728:
2724:
2718:
2712:
2704:
2699:
2694:
2688:
2680:
2662:
2656:
2646:
2640:
2609:
2603:
2578:
2572:
2545:
2539:
2507:
2502:
2496:
2488:
2480:
2474:
2464:
2458:
2445:
2439:
2431:
2426:
2420:
2412:
2404:
2398:
2356:
2350:
2337:
2331:
2321:
2315:
2304:
2298:
2261:
2255:
2230:
2224:
2219:
2213:
2200:
2194:
2189:
2183:
2136:
2130:
2125:
2119:
2092:
2086:
2081:
2075:
2045:
2039:
2034:
2028:
2015:
2009:
2004:
1998:
1977:
1971:
1930:, Δ, ε), there exist elements
1851:can be viewed as a linear map
1678:
1666:
1620:
1608:
1602:
1590:
1585:
1573:
1556:
1544:
1253:
1240:
1136:
1123:
1026:to zero; identify the element
841:
817:
787:
763:
736:
712:
700:
676:
576:
436:
390:
283:
269:
1:
4560:American Mathematical Society
4524:Underwood, Robert G. (2011),
4500:American Mathematical Society
4463:10.1016/S0022-4049(03)00013-6
4201:"Lecture notes for reference"
2593:, as in the previous sum for
2151:, are uniquely determined by
2144:{\displaystyle c_{(2)}^{(i)}}
2100:{\displaystyle c_{(1)}^{(i)}}
1764:and are group-like elements.
621:Formally, a coalgebra over a
94:universal enveloping algebras
4374:10.1016/0022-4049(85)90060-X
3988:of every coalgebra morphism
1034:with the function that maps
557:{\displaystyle \mathbf {j} }
532:{\displaystyle \mathbf {j} }
4583:Chapter III, section 11 in
1892:is naturally isomorphic to
254:{\displaystyle j_{A}+j_{B}}
139:Clebsch–Gordan coefficients
4649:
4585:Bourbaki, Nicolas (1989).
4494:Montgomery, Susan (1993),
4333:Cambridge University Press
3239:Further concepts and facts
2615:{\displaystyle \Delta (c)}
2285:), this is abbreviated to
2267:{\displaystyle \Delta (c)}
608:Littlewood–Richardson rule
509:: the tensor algebra is a
499:fundamental representation
303:. This is provided by the
4179:See also Raianu, Serban.
1014:are those functions from
261:given the combined state
4554:Yokonuma, Takeo (1992),
4261:Coalgebras from Formulas
4181:Coalgebras from Formulas
917:; the two are naturally
42:are structures that are
3314:-linear map defined by
2586:{\displaystyle c_{(2)}}
2553:{\displaystyle c_{(1)}}
1910:is given by evaluating
1504:trigonometric coalgebra
1370:divided power structure
4279:Montgomery (1993) p.61
4041:) is isomorphic to im(
3859:
3727:
3681:
3439:
3285:
3226:
3108:
3035:
2939:
2616:
2587:
2554:
2518:
2368:
2268:
2239:
2165:
2145:
2101:
2054:
1751:
1647:
1516:with set of intervals
1324:
1223:
1162:
905:) is identified with (
893:In the first diagram,
890:
864:(Here ⊗ refers to the
854:
749:
596:
558:
533:
491:
468:
406:
354:
297:
255:
215:
188:
161:
4288:Underwood (2011) p.35
4019:is a subcoalgebra of
3860:
3728:
3682:
3440:
3286:
3227:
3109:
3036:
2940:
2617:
2588:
2555:
2519:
2369:
2269:
2245:have the right value
2240:
2166:
2146:
2102:
2055:
1752:
1648:
1325:
1224:
1142:
889:
855:
750:
597:
559:
534:
492:
469:
407:
355:
298:
256:
216:
214:{\displaystyle j_{B}}
189:
187:{\displaystyle j_{A}}
167:with angular momenta
162:
123:representation theory
90:representation theory
4327:Abe, Eiichi (2004).
4028:isomorphism theorems
3932:. In that case, the
3740:
3691:
3627:
3501:do form a group. A
3359:
3263:
3124:
3051:
2955:
2632:
2597:
2564:
2531:
2384:
2292:
2249:
2175:
2155:
2111:
2067:
1965:
1660:
1538:
1511:locally finite poset
1498:In this situation, (
1234:
1117:
760:
673:
568:
546:
521:
481:
422:
378:
314:
265:
225:
198:
171:
145:
141:. Given two systems
67:commutative diagrams
59:associative algebras
4446:Hazewinkel, Michiel
4106:Measuring coalgebra
2279:Sweedler's notation
2234:
2204:
2140:
2096:
2049:
2019:
1827:-algebra, then its
1823:unital associative
1522:incidence coalgebra
1398:-vector space with
1385:Künneth isomorphism
1078:. The vector space
160:{\displaystyle A,B}
117:Informal discussion
50:sense of reversing
4266:2010-05-29 at the
4186:2010-05-29 at the
4083:for algebras is a
3855:
3723:
3677:
3583:coalgebra morphism
3461:group-like element
3435:
3281:
3222:
3104:
3031:
2973:
2935:
2834:
2800:
2703:
2650:
2612:
2583:
2550:
2514:
2468:
2408:
2364:
2325:
2281:, (so named after
2264:
2235:
2208:
2178:
2161:
2141:
2114:
2097:
2070:
2050:
2023:
1993:
1992:
1926:of the coalgebra (
1912:linear functionals
1839:-linear maps from
1835:consisting of all
1747:
1742:
1643:
1589:
1368:, rather than the
1320:
1315:
1302:
1277:
1219:
1185:
1110:≥ 0 one defines:
984:Take an arbitrary
891:
850:
745:
592:
554:
529:
487:
477:For this example,
464:
402:
350:
293:
251:
211:
184:
157:
48:category-theoretic
4539:978-0-387-72765-3
3504:primitive element
2958:
2806:
2785:
2675:
2635:
2453:
2393:
2310:
2164:{\displaystyle c}
1983:
1918:Sweedler notation
1768:Finite dimensions
1726:
1700:
1639:
1626:
1562:
1381:topological space
1377:singular homology
1301:
1276:
1178:
874:identity function
617:Formal definition
490:{\displaystyle J}
79:finite dimensions
16:(Redirected from
4640:
4604:
4580:
4550:
4520:
4490:
4465:
4441:
4427:
4401:
4376:
4347:
4346:
4324:
4318:
4317:
4306:. p. 307, C.42.
4295:
4289:
4286:
4280:
4277:
4271:
4259:Raianu, Serban.
4257:
4251:
4249:
4239:
4228:
4222:
4221:
4219:
4218:
4212:
4206:. Archived from
4205:
4197:
4191:
4177:
4171:
4170:
4160:
4154:
4153:
4143:
4137:
4136:
4126:
4101:Cofree coalgebra
4085:corepresentation
4008:is a coideal in
4007:
3973:; in that case,
3972:
3931:
3908:
3864:
3862:
3861:
3856:
3851:
3850:
3823:
3822:
3792:
3791:
3764:
3763:
3732:
3730:
3729:
3724:
3722:
3721:
3703:
3702:
3686:
3684:
3683:
3678:
3670:
3669:
3657:
3656:
3622:
3576:
3554:
3525:
3496:
3485:
3467:) is an element
3465:set-like element
3444:
3442:
3441:
3436:
3434:
3433:
3415:
3414:
3396:
3395:
3377:
3376:
3335:
3309:
3290:
3288:
3287:
3282:
3254:
3231:
3229:
3228:
3223:
3214:
3213:
3192:
3191:
3173:
3172:
3154:
3153:
3113:
3111:
3110:
3105:
3103:
3102:
3084:
3083:
3040:
3038:
3037:
3032:
3027:
3026:
3008:
3007:
2989:
2988:
2972:
2944:
2942:
2941:
2936:
2931:
2930:
2912:
2908:
2907:
2906:
2891:
2890:
2869:
2868:
2853:
2852:
2833:
2829:
2828:
2799:
2781:
2777:
2776:
2775:
2760:
2759:
2738:
2737:
2722:
2721:
2702:
2698:
2697:
2666:
2665:
2649:
2621:
2619:
2618:
2613:
2592:
2590:
2589:
2584:
2582:
2581:
2559:
2557:
2556:
2551:
2549:
2548:
2523:
2521:
2520:
2515:
2506:
2505:
2484:
2483:
2467:
2449:
2448:
2430:
2429:
2407:
2373:
2371:
2370:
2365:
2360:
2359:
2341:
2340:
2324:
2273:
2271:
2270:
2265:
2244:
2242:
2241:
2236:
2233:
2222:
2203:
2192:
2170:
2168:
2167:
2162:
2150:
2148:
2147:
2142:
2139:
2128:
2106:
2104:
2103:
2098:
2095:
2084:
2059:
2057:
2056:
2051:
2048:
2037:
2018:
2007:
1991:
1953:
1952:
1941:
1940:
1906:. The counit of
1901:
1891:
1879:
1864:
1798:are isomorphic.
1797:
1787:
1756:
1754:
1753:
1748:
1746:
1745:
1727:
1724:
1701:
1698:
1652:
1650:
1649:
1644:
1637:
1627:
1624:
1588:
1354:exterior algebra
1329:
1327:
1326:
1321:
1319:
1318:
1303:
1299:
1278:
1274:
1252:
1251:
1228:
1226:
1225:
1220:
1215:
1214:
1196:
1195:
1186:
1184:
1183:
1170:
1161:
1156:
1135:
1134:
970:
969:
954:comultiplication
941:are identified.
859:
857:
856:
851:
840:
839:
834:
813:
812:
807:
780:
779:
774:
754:
752:
751:
746:
735:
734:
729:
693:
692:
687:
601:
599:
598:
593:
591:
583:
575:
563:
561:
560:
555:
553:
538:
536:
535:
530:
528:
496:
494:
493:
488:
473:
471:
470:
465:
463:
443:
435:
411:
409:
408:
403:
359:
357:
356:
351:
349:
329:
321:
302:
300:
299:
294:
286:
272:
260:
258:
257:
252:
250:
249:
237:
236:
220:
218:
217:
212:
210:
209:
193:
191:
190:
185:
183:
182:
166:
164:
163:
158:
131:angular momentum
111:computer science
21:
4648:
4647:
4643:
4642:
4641:
4639:
4638:
4637:
4623:
4622:
4611:
4601:
4591:Springer-Verlag
4584:
4570:
4553:
4540:
4530:Springer-Verlag
4523:
4510:
4493:
4444:
4431:
4417:
4404:
4358:
4355:
4353:Further reading
4350:
4343:
4326:
4325:
4321:
4314:
4304:Springer-Verlag
4297:
4296:
4292:
4287:
4283:
4278:
4274:
4268:Wayback Machine
4258:
4254:
4241:
4231:
4229:
4225:
4216:
4214:
4210:
4203:
4199:
4198:
4194:
4188:Wayback Machine
4178:
4174:
4162:
4161:
4157:
4145:
4144:
4140:
4128:
4127:
4123:
4119:
4097:
4056:-algebra, then
4036:
4025:
4014:
4006:
3999:
3989:
3959:
3910:
3899:
3885:linear subspace
3836:
3808:
3777:
3749:
3738:
3737:
3713:
3694:
3689:
3688:
3661:
3648:
3625:
3624:
3621:
3614:
3604:
3598:
3591:
3574:
3567:
3563:
3556:
3552:
3545:
3541:
3534:
3512:
3511:that satisfies
3487:
3472:
3419:
3400:
3381:
3362:
3357:
3356:
3315:
3292:
3261:
3260:
3244:
3241:
3199:
3177:
3158:
3139:
3122:
3121:
3088:
3069:
3049:
3048:
3012:
2993:
2974:
2953:
2952:
2916:
2892:
2876:
2854:
2838:
2814:
2805:
2801:
2761:
2745:
2723:
2707:
2683:
2674:
2670:
2651:
2630:
2629:
2595:
2594:
2567:
2562:
2561:
2534:
2529:
2528:
2491:
2469:
2434:
2415:
2382:
2381:
2345:
2326:
2290:
2289:
2247:
2246:
2173:
2172:
2153:
2152:
2109:
2108:
2065:
2064:
1963:
1962:
1951:
1948:
1947:
1946:
1939:
1936:
1935:
1934:
1920:
1893:
1881:
1866:
1852:
1789:
1777:
1770:
1741:
1740:
1721:
1715:
1714:
1695:
1685:
1658:
1657:
1625: for
1536:
1535:
1410:}, consider Δ:
1366:shuffle product
1314:
1313:
1295:
1289:
1288:
1270:
1260:
1243:
1232:
1231:
1200:
1187:
1165:
1126:
1115:
1114:
1087:polynomial ring
982:
967:
966:
826:
799:
766:
758:
757:
721:
679:
671:
670:
619:
566:
565:
544:
543:
519:
518:
515:shuffle product
479:
478:
420:
419:
376:
375:
312:
311:
263:
262:
241:
228:
223:
222:
201:
196:
195:
174:
169:
168:
143:
142:
119:
103:There are also
28:
23:
22:
15:
12:
11:
5:
4646:
4644:
4636:
4635:
4625:
4624:
4621:
4620:
4614:William Chin:
4610:
4609:External links
4607:
4606:
4605:
4599:
4581:
4568:
4551:
4538:
4521:
4508:
4491:
4442:
4429:
4415:
4402:
4354:
4351:
4349:
4348:
4341:
4335:. p. 59.
4319:
4312:
4290:
4281:
4272:
4252:
4223:
4192:
4172:
4155:
4138:
4120:
4118:
4115:
4114:
4113:
4108:
4103:
4096:
4093:
4081:representation
4034:
4023:
4012:
4004:
3997:
3934:quotient space
3866:
3865:
3854:
3849:
3846:
3843:
3839:
3835:
3832:
3829:
3826:
3821:
3818:
3815:
3811:
3807:
3804:
3801:
3798:
3795:
3790:
3787:
3784:
3780:
3776:
3773:
3770:
3767:
3762:
3759:
3756:
3752:
3748:
3745:
3720:
3716:
3712:
3709:
3706:
3701:
3697:
3676:
3673:
3668:
3664:
3660:
3655:
3651:
3647:
3644:
3641:
3638:
3635:
3632:
3619:
3612:
3596:
3589:
3572:
3565:
3561:
3550:
3543:
3539:
3507:is an element
3446:
3445:
3432:
3429:
3426:
3422:
3418:
3413:
3410:
3407:
3403:
3399:
3394:
3391:
3388:
3384:
3380:
3375:
3372:
3369:
3365:
3280:
3277:
3274:
3271:
3268:
3257:co-commutative
3240:
3237:
3233:
3232:
3220:
3217:
3212:
3209:
3206:
3202:
3198:
3195:
3190:
3187:
3184:
3180:
3176:
3171:
3168:
3165:
3161:
3157:
3152:
3149:
3146:
3142:
3138:
3135:
3132:
3129:
3115:
3114:
3101:
3098:
3095:
3091:
3087:
3082:
3079:
3076:
3072:
3068:
3065:
3062:
3059:
3056:
3042:
3041:
3030:
3025:
3022:
3019:
3015:
3011:
3006:
3003:
3000:
2996:
2992:
2987:
2984:
2981:
2977:
2971:
2968:
2965:
2961:
2946:
2945:
2934:
2929:
2926:
2923:
2919:
2915:
2911:
2905:
2902:
2899:
2895:
2889:
2886:
2883:
2879:
2875:
2872:
2867:
2864:
2861:
2857:
2851:
2848:
2845:
2841:
2837:
2832:
2827:
2824:
2821:
2817:
2813:
2809:
2804:
2798:
2795:
2792:
2788:
2784:
2780:
2774:
2771:
2768:
2764:
2758:
2755:
2752:
2748:
2744:
2741:
2736:
2733:
2730:
2726:
2720:
2717:
2714:
2710:
2706:
2701:
2696:
2693:
2690:
2686:
2682:
2678:
2673:
2669:
2664:
2661:
2658:
2654:
2648:
2645:
2642:
2638:
2611:
2608:
2605:
2602:
2580:
2577:
2574:
2570:
2547:
2544:
2541:
2537:
2525:
2524:
2512:
2509:
2504:
2501:
2498:
2494:
2490:
2487:
2482:
2479:
2476:
2472:
2466:
2463:
2460:
2456:
2452:
2447:
2444:
2441:
2437:
2433:
2428:
2425:
2422:
2418:
2414:
2411:
2406:
2403:
2400:
2396:
2392:
2389:
2375:
2374:
2363:
2358:
2355:
2352:
2348:
2344:
2339:
2336:
2333:
2329:
2323:
2320:
2317:
2313:
2309:
2306:
2303:
2300:
2297:
2263:
2260:
2257:
2254:
2232:
2229:
2226:
2221:
2218:
2215:
2211:
2207:
2202:
2199:
2196:
2191:
2188:
2185:
2181:
2160:
2138:
2135:
2132:
2127:
2124:
2121:
2117:
2094:
2091:
2088:
2083:
2080:
2077:
2073:
2061:
2060:
2047:
2044:
2041:
2036:
2033:
2030:
2026:
2022:
2017:
2014:
2011:
2006:
2003:
2000:
1996:
1990:
1986:
1982:
1979:
1976:
1973:
1970:
1949:
1937:
1919:
1916:
1769:
1766:
1758:
1757:
1744:
1739:
1736:
1733:
1730:
1722:
1720:
1717:
1716:
1713:
1710:
1707:
1704:
1696:
1694:
1691:
1690:
1688:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1654:
1653:
1642:
1636:
1633:
1630:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1565:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1496:
1495:
1488:
1469:
1468:
1446:
1362:Lie bialgebras
1350:tensor algebra
1331:
1330:
1317:
1312:
1309:
1306:
1296:
1294:
1291:
1290:
1287:
1284:
1281:
1271:
1269:
1266:
1265:
1263:
1258:
1255:
1250:
1246:
1242:
1239:
1229:
1218:
1213:
1210:
1207:
1203:
1199:
1194:
1190:
1182:
1177:
1174:
1169:
1160:
1155:
1152:
1149:
1145:
1141:
1138:
1133:
1129:
1125:
1122:
1072:
1071:
1062:) = 1 for all
1042:to 0. Define
995:-vector space
981:
978:
872:and id is the
866:tensor product
862:
861:
849:
846:
843:
838:
833:
830:
825:
822:
819:
816:
811:
806:
803:
798:
795:
792:
789:
786:
783:
778:
773:
770:
765:
755:
744:
741:
738:
733:
728:
725:
720:
717:
714:
711:
708:
705:
702:
699:
696:
691:
686:
683:
678:
639:together with
618:
615:
590:
586:
582:
578:
574:
552:
527:
486:
475:
474:
462:
458:
455:
452:
449:
446:
442:
438:
434:
430:
427:
413:
412:
401:
398:
395:
392:
389:
386:
383:
365:tensor algebra
361:
360:
348:
344:
341:
338:
335:
332:
328:
324:
320:
292:
289:
285:
281:
278:
275:
271:
248:
244:
240:
235:
231:
208:
204:
181:
177:
156:
153:
150:
127:rotation group
118:
115:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4645:
4634:
4631:
4630:
4628:
4619:
4618:
4613:
4612:
4608:
4602:
4600:0-387-19373-1
4596:
4592:
4588:
4582:
4579:
4575:
4571:
4569:0-8218-4564-0
4565:
4561:
4557:
4552:
4549:
4545:
4541:
4535:
4531:
4527:
4522:
4519:
4515:
4511:
4509:0-8218-0738-2
4505:
4501:
4497:
4492:
4489:
4485:
4481:
4477:
4473:
4469:
4464:
4459:
4456:(1): 61–103,
4455:
4451:
4447:
4443:
4439:
4435:
4430:
4426:
4422:
4418:
4416:0-8247-0481-9
4412:
4408:
4403:
4400:
4396:
4392:
4388:
4384:
4380:
4375:
4370:
4366:
4362:
4357:
4356:
4352:
4344:
4342:0-521-60489-3
4338:
4334:
4330:
4329:Hopf Algebras
4323:
4320:
4315:
4309:
4305:
4301:
4294:
4291:
4285:
4282:
4276:
4273:
4269:
4265:
4262:
4256:
4253:
4248:. p. 55.
4247:
4246:
4237:
4236:
4227:
4224:
4213:on 2012-02-24
4209:
4202:
4196:
4193:
4189:
4185:
4182:
4176:
4173:
4168:
4167:
4159:
4156:
4152:. p. 10.
4151:
4150:
4142:
4139:
4135:. p. 12.
4134:
4133:
4125:
4122:
4116:
4112:
4109:
4107:
4104:
4102:
4099:
4098:
4094:
4092:
4090:
4086:
4082:
4077:
4073:
4071:
4067:
4063:
4059:
4055:
4051:
4046:
4044:
4040:
4033:
4029:
4026:. The common
4022:
4018:
4011:
4003:
3996:
3992:
3987:
3982:
3980:
3976:
3971:
3967:
3963:
3957:
3953:
3949:
3944:
3942:
3938:
3935:
3930:
3926:
3922:
3918:
3914:
3906:
3902:
3897:
3893:
3889:
3886:
3881:
3879:
3875:
3871:
3852:
3844:
3833:
3827:
3824:
3816:
3805:
3799:
3796:
3785:
3778:
3771:
3768:
3757:
3750:
3743:
3736:
3735:
3734:
3718:
3714:
3710:
3707:
3704:
3699:
3695:
3674:
3671:
3666:
3658:
3653:
3645:
3639:
3636:
3633:
3618:
3611:
3607:
3602:
3595:
3588:
3584:
3580:
3571:
3560:
3549:
3538:
3531:
3529:
3524:
3520:
3516:
3510:
3506:
3505:
3500:
3494:
3490:
3484:
3480:
3476:
3470:
3466:
3462:
3457:
3455:
3451:
3427:
3420:
3416:
3408:
3401:
3397:
3389:
3382:
3378:
3370:
3363:
3355:
3354:
3353:
3351:
3347:
3343:
3339:
3334:
3330:
3326:
3322:
3318:
3313:
3308:
3304:
3300:
3296:
3275:
3269:
3266:
3258:
3252:
3248:
3238:
3236:
3218:
3207:
3200:
3193:
3185:
3178:
3174:
3166:
3159:
3147:
3140:
3133:
3130:
3127:
3120:
3119:
3118:
3096:
3089:
3085:
3077:
3070:
3066:
3060:
3047:
3046:
3045:
3028:
3020:
3013:
3009:
3001:
2994:
2990:
2982:
2975:
2966:
2959:
2951:
2950:
2949:
2932:
2924:
2917:
2913:
2909:
2900:
2884:
2877:
2870:
2862:
2846:
2839:
2822:
2815:
2807:
2802:
2793:
2786:
2782:
2778:
2769:
2753:
2746:
2739:
2731:
2715:
2708:
2691:
2684:
2676:
2671:
2667:
2659:
2652:
2643:
2636:
2628:
2627:
2626:
2623:
2606:
2575:
2568:
2542:
2535:
2510:
2499:
2492:
2485:
2477:
2470:
2461:
2454:
2450:
2442:
2435:
2423:
2416:
2409:
2401:
2394:
2390:
2387:
2380:
2379:
2378:
2361:
2353:
2346:
2342:
2334:
2327:
2318:
2311:
2307:
2301:
2288:
2287:
2286:
2284:
2283:Moss Sweedler
2280:
2275:
2258:
2227:
2216:
2209:
2205:
2197:
2186:
2179:
2158:
2133:
2122:
2115:
2089:
2078:
2071:
2042:
2031:
2024:
2020:
2012:
2001:
1994:
1988:
1984:
1980:
1974:
1961:
1960:
1959:
1957:
1945:
1933:
1929:
1925:
1917:
1915:
1913:
1909:
1905:
1900:
1896:
1889:
1885:
1877:
1873:
1869:
1863:
1859:
1855:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1821:
1815:
1810:
1808:
1804:
1799:
1796:
1792:
1785:
1781:
1774:
1767:
1765:
1763:
1737:
1734:
1731:
1728:
1718:
1711:
1708:
1705:
1702:
1692:
1686:
1681:
1675:
1672:
1669:
1663:
1656:
1655:
1640:
1634:
1631:
1628:
1617:
1614:
1611:
1605:
1599:
1596:
1593:
1582:
1579:
1576:
1570:
1567:
1563:
1559:
1553:
1550:
1547:
1534:
1533:
1532:
1530:
1526:
1523:
1520:, define the
1519:
1515:
1512:
1507:
1505:
1501:
1493:
1489:
1486:
1482:
1481:
1480:
1478:
1474:
1467:
1463:
1459:
1455:
1451:
1447:
1445:
1441:
1437:
1433:
1429:
1425:
1424:
1423:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1388:
1386:
1382:
1378:
1373:
1371:
1367:
1363:
1359:
1358:Hopf algebras
1355:
1351:
1346:
1344:
1340:
1336:
1310:
1307:
1304:
1292:
1285:
1282:
1279:
1267:
1261:
1256:
1248:
1244:
1237:
1230:
1216:
1211:
1208:
1205:
1201:
1197:
1192:
1188:
1175:
1172:
1158:
1153:
1150:
1147:
1143:
1139:
1131:
1127:
1113:
1112:
1111:
1109:
1106:) if for all
1105:
1103:
1102:divided power
1098:
1095:
1094:indeterminate
1091:
1088:
1083:
1081:
1077:
1069:
1065:
1061:
1057:
1053:
1049:
1045:
1044:
1043:
1041:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1006:
1002:
998:
994:
991:and form the
990:
987:
979:
977:
975:
971:
964:and ε is the
963:
959:
955:
951:
947:
946:associativity
942:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
888:
884:
882:
877:
875:
871:
867:
844:
836:
823:
820:
814:
809:
796:
790:
784:
781:
776:
756:
739:
731:
718:
709:
703:
694:
689:
669:
668:
667:
665:
661:
657:
653:
649:
645:
643:
638:
634:
631:
627:
624:
616:
614:
611:
609:
605:
584:
540:
516:
512:
508:
504:
500:
484:
456:
453:
450:
447:
444:
428:
418:
417:
416:
399:
396:
393:
387:
384:
374:
373:
372:
370:
366:
342:
339:
336:
333:
330:
322:
310:
309:
308:
306:
287:
279:
273:
246:
242:
238:
233:
229:
206:
202:
179:
175:
154:
151:
148:
140:
136:
132:
128:
124:
116:
114:
112:
108:
107:
101:
99:
98:group schemes
95:
91:
86:
84:
80:
76:
72:
68:
64:
60:
57:
53:
49:
45:
41:
37:
33:
19:
4616:
4586:
4555:
4525:
4495:
4453:
4449:
4437:
4433:
4406:
4367:(1): 15–21,
4364:
4360:
4328:
4322:
4299:
4293:
4284:
4275:
4270:, p. 1.
4255:
4250:, Ex. 1.1.5.
4244:
4238:. p. 4.
4234:
4226:
4215:. Retrieved
4208:the original
4195:
4190:, p. 2.
4175:
4169:. p. 3.
4165:
4158:
4148:
4141:
4131:
4124:
4084:
4078:
4074:
4069:
4065:
4061:
4057:
4053:
4049:
4047:
4042:
4038:
4031:
4020:
4009:
4001:
3994:
3990:
3983:
3978:
3974:
3969:
3965:
3961:
3956:subcoalgebra
3955:
3954:is called a
3951:
3947:
3945:
3940:
3936:
3928:
3924:
3920:
3916:
3912:
3904:
3900:
3895:
3894:is called a
3891:
3887:
3882:
3873:
3867:
3616:
3609:
3605:
3603:-linear map
3600:
3593:
3586:
3582:
3578:
3569:
3558:
3547:
3536:
3532:
3522:
3518:
3514:
3508:
3502:
3499:Hopf algebra
3492:
3488:
3482:
3478:
3474:
3468:
3464:
3460:
3458:
3453:
3449:
3447:
3349:
3345:
3341:
3337:
3332:
3328:
3324:
3320:
3316:
3311:
3306:
3302:
3298:
3294:
3256:
3250:
3246:
3243:A coalgebra
3242:
3234:
3116:
3043:
2947:
2624:
2526:
2376:
2278:
2276:
2062:
1955:
1943:
1931:
1927:
1923:
1921:
1907:
1903:
1898:
1894:
1887:
1883:
1875:
1871:
1867:
1861:
1857:
1853:
1848:
1844:
1840:
1836:
1832:
1828:
1824:
1817:
1813:
1811:
1806:
1802:
1800:
1794:
1790:
1783:
1779:
1775:
1771:
1761:
1759:
1528:
1524:
1521:
1517:
1513:
1508:
1503:
1499:
1497:
1491:
1484:
1479:is given by
1476:
1472:
1470:
1465:
1461:
1457:
1453:
1449:
1443:
1439:
1435:
1431:
1427:
1422:is given by
1419:
1415:
1411:
1407:
1403:
1395:
1391:
1389:
1374:
1347:
1338:
1334:
1332:
1107:
1100:
1096:
1089:
1084:
1079:
1075:
1073:
1067:
1063:
1059:
1055:
1051:
1047:
1039:
1035:
1031:
1027:
1023:
1019:
1015:
1011:
1007:
1000:
996:
992:
988:
983:
973:
965:
961:
957:
953:
943:
938:
934:
930:
926:
922:
914:
910:
906:
902:
898:
894:
892:
878:
869:
863:
663:
659:
655:
651:
647:
644:-linear maps
641:
636:
632:
630:vector space
625:
620:
612:
603:
541:
511:free algebra
507:free objects
476:
414:
362:
120:
106:F-coalgebras
104:
102:
87:
71:vector space
39:
35:
29:
3981:as counit.
3946:A subspace
3870:composition
3528:Lie algebra
1958:such that
1820:dimensional
415:that takes
32:mathematics
4633:Coalgebras
4578:0754.15028
4548:1234.16022
4528:, Berlin:
4518:0793.16029
4488:1048.16022
4425:0962.16026
4399:0556.16005
4313:0792370724
4217:2008-10-31
4117:References
4015:, and the
3623:such that
3521:⊗ 1 + 1 ⊗
3471:such that
3255:is called
1343:bialgebras
919:isomorphic
666:such that
36:coalgebras
4472:0022-4049
4440:: 591–603
4383:0022-4049
4230:See also
4111:Dialgebra
3825:⊗
3769:⊗
3715:ε
3705:∘
3696:ε
3672:∘
3663:Δ
3650:Δ
3646:∘
3637:⊗
3581:, then a
3417:⊗
3379:⊗
3279:Δ
3273:Δ
3270:∘
3267:σ
3194:ε
3134:ε
3086:⊗
3055:Δ
3010:⊗
2991:⊗
2960:∑
2914:⊗
2871:⊗
2808:∑
2787:∑
2740:⊗
2677:∑
2668:⊗
2637:∑
2601:Δ
2486:ε
2455:∑
2410:ε
2395:∑
2343:⊗
2312:∑
2296:Δ
2253:Δ
2206:⊗
2021:⊗
1985:∑
1969:Δ
1732:≠
1664:ε
1632:≤
1606:⊗
1571:∈
1564:∑
1542:Δ
1238:ε
1209:−
1198:⊗
1144:∑
1121:Δ
1104:coalgebra
958:coproduct
848:Δ
845:∘
824:⊗
821:ε
794:Δ
791:∘
785:ε
782:⊗
743:Δ
740:∘
719:⊗
716:Δ
707:Δ
704:∘
698:Δ
695:⊗
585:⊗
577:↦
457:⊗
445:⊗
437:↦
426:Δ
397:⊗
391:→
382:Δ
369:coproduct
343:⊗
331:⊗
323:≡
291:⟩
280:⊗
277:⟩
83:see below
4627:Category
4264:Archived
4184:Archived
4095:See also
4089:comodule
3993: :
3878:category
3608: :
3448:for all
3336:for all
3291:, where
1725:if
1699:if
1300:if
1275:if
980:Examples
950:identity
46:(in the
40:cogebras
4587:Algebra
4480:1992043
4391:0782637
3896:coideal
3310:is the
1818:finite-
1807:objects
1803:notions
1471:and ε:
1394:is the
1092:in one
1058:and ε(
881:commute
658:and ε:
75:duality
4597:
4576:
4566:
4546:
4536:
4516:
4506:
4486:
4478:
4470:
4423:
4413:
4397:
4389:
4381:
4339:
4310:
4240:, and
3986:kernel
3903:⊆ ker(
1914:at 1.
1831:-dual
1638:
1509:For a
1352:, the
1337:. Now
968:counit
503:lifted
63:axioms
61:. The
56:unital
52:arrows
18:Counit
4211:(PDF)
4204:(PDF)
4037:/ker(
4017:image
3599:is a
3585:from
3495:) = 1
3249:, Δ,
1816:is a
1527:with
1494:) = 1
1487:) = 0
1400:basis
1379:of a
1005:basis
1003:with
960:) of
868:over
635:over
628:is a
623:field
54:) to
4595:ISBN
4564:ISBN
4534:ISBN
4504:ISBN
4468:ISSN
4411:ISBN
4379:ISSN
4337:ISBN
4308:ISBN
3984:The
3964:) ⊆
3915:) ⊆
3909:and
3868:The
3687:and
3555:and
3517:) =
3486:and
3477:) =
3463:(or
3327:) =
3117:and
2560:and
1942:and
1788:and
1452:) =
1430:) =
1375:The
1360:and
1308:>
1050:) =
956:(or
933:and
913:) ⊗
604:must
194:and
135:spin
133:and
96:and
44:dual
4574:Zbl
4544:Zbl
4514:Zbl
4484:Zbl
4458:doi
4454:183
4421:Zbl
4395:Zbl
4369:doi
4087:or
4048:If
4045:).
3958:if
3950:of
3898:if
3890:in
3592:to
3564:, Δ
3542:, Δ
3533:If
3452:in
3344:in
3293:σ:
3259:if
2277:In
2107:or
1954:in
1950:(2)
1938:(1)
1870:→ (
1843:to
1812:If
1390:If
1066:in
1030:of
1018:to
986:set
972:of
897:⊗ (
876:.)
646:Δ:
100:).
85:).
38:or
30:In
4629::
4593:.
4589:.
4572:,
4562:,
4542:,
4532:,
4512:,
4502:,
4482:,
4476:MR
4474:,
4466:,
4452:,
4438:43
4436:,
4419:,
4393:,
4387:MR
4385:,
4377:,
4365:36
4363:,
4302:.
4091:.
4000:→
3968:⊗
3960:Δ(
3927:⊗
3923:+
3919:⊗
3911:Δ(
3883:A
3880:.
3615:→
3568:,
3546:,
3530:.
3513:Δ(
3481:⊗
3473:Δ(
3459:A
3340:,
3331:⊗
3323:⊗
3305:⊗
3301:→
3297:⊗
2622:.
2274:.
1897:⊗
1886:⊗
1874:⊗
1860:→
1856:⊗
1793:⊗
1782:⊗
1506:.
1490:ε(
1483:ε(
1475:→
1464:⊗
1460:−
1456:⊗
1448:Δ(
1442:⊗
1438:+
1434:⊗
1426:Δ(
1418:⊗
1414:→
1406:,
1356:,
1054:⊗
1046:Δ(
999:=
976:.
937:⊗
929:⊗
925:,
909:⊗
901:⊗
883::
662:→
654:⊗
650:→
113:.
92:,
73:)
34:,
4603:.
4460::
4428:.
4371::
4345:.
4316:.
4220:.
4070:K
4066:K
4062:K
4058:A
4054:K
4050:A
4043:f
4039:f
4035:1
4032:C
4024:2
4021:C
4013:1
4010:C
4005:2
4002:C
3998:1
3995:C
3991:f
3979:D
3975:D
3970:D
3966:D
3962:D
3952:C
3948:D
3941:I
3939:/
3937:C
3929:I
3925:C
3921:C
3917:I
3913:I
3907:)
3905:ε
3901:I
3892:C
3888:I
3874:K
3853:.
3848:)
3845:2
3842:(
3838:)
3834:c
3831:(
3828:f
3820:)
3817:1
3814:(
3810:)
3806:c
3803:(
3800:f
3797:=
3794:)
3789:)
3786:2
3783:(
3779:c
3775:(
3772:f
3766:)
3761:)
3758:1
3755:(
3751:c
3747:(
3744:f
3719:1
3711:=
3708:f
3700:2
3675:f
3667:2
3659:=
3654:1
3643:)
3640:f
3634:f
3631:(
3620:2
3617:C
3613:1
3610:C
3606:f
3601:K
3597:2
3594:C
3590:1
3587:C
3579:K
3575:)
3573:2
3570:ε
3566:2
3562:2
3559:C
3557:(
3553:)
3551:1
3548:ε
3544:1
3540:1
3537:C
3535:(
3523:x
3519:x
3515:x
3509:x
3493:x
3491:(
3489:ε
3483:x
3479:x
3475:x
3469:x
3454:C
3450:c
3431:)
3428:1
3425:(
3421:c
3412:)
3409:2
3406:(
3402:c
3398:=
3393:)
3390:2
3387:(
3383:c
3374:)
3371:1
3368:(
3364:c
3350:C
3346:C
3342:d
3338:c
3333:c
3329:d
3325:d
3321:c
3319:(
3317:σ
3312:K
3307:C
3303:C
3299:C
3295:C
3276:=
3253:)
3251:ε
3247:C
3245:(
3219:.
3216:)
3211:)
3208:2
3205:(
3201:c
3197:(
3189:)
3186:1
3183:(
3179:c
3175:=
3170:)
3167:2
3164:(
3160:c
3156:)
3151:)
3148:1
3145:(
3141:c
3137:(
3131:=
3128:c
3100:)
3097:2
3094:(
3090:c
3081:)
3078:1
3075:(
3071:c
3067:=
3064:)
3061:c
3058:(
3029:.
3024:)
3021:3
3018:(
3014:c
3005:)
3002:2
2999:(
2995:c
2986:)
2983:1
2980:(
2976:c
2970:)
2967:c
2964:(
2933:.
2928:)
2925:2
2922:(
2918:c
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