1035:
769:
686:
785:
753:
22:
1469:
1831:
1301:
1026:
907:
3226:, if an appropriate antipode can be found; thus, all Hopf algebras are examples of bialgebras. Similar structures with different compatibility between the product and comultiplication, or different types of multiplication and comultiplication, include
1963:
2374:
3107:
2168:
1701:
3014:
Taking the product of two random variables, and preparing a copy of the resulting random variable, has the same distribution as preparing copies of each random variable independently of one another, and multiplying them together in
2451:
2871:
1594:
1171:
2561:
3000:
on the remaining variables (the remaining tensor factors). Given the interpretation of (Δ,ε) in terms of probability distributions as above, the bialgebra consistency conditions amount to constraints on (∇,η) as follows:
2626:
1310:
2040:
1707:
1177:
927:
3154:
2233:
2958:
2689:
3193:
826:
1031:
Coassociativity and counit are expressed by the commutativity of the following two diagrams (they are the duals of the diagrams expressing associativity and unit of an algebra):
1875:
2798:
in the case of vectors whose coefficients are all non-negative and sum to 1. An example of suitable comultiplication operators and counits which yield a counital coalgebra are
2289:
2990:
2907:
2768:
3029:
2092:
2736:
1625:
3334:
3313:
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2380:
2804:
1478:
3156:
by linearity; this produces a normalized probability distribution from a distribution on two random variables, and has as a unit the delta-distribution
1095:
3400:
2487:
529:
2567:
3219:, which can be made into a bialgebra by adding the appropriate comultiplication and counit; these are worked out in detail in that article.
1464:{\displaystyle \nabla _{2}((x_{1}\otimes x_{2})\otimes (y_{1}\otimes y_{2}))=\nabla (x_{1}\otimes y_{1})\otimes \nabla (x_{2}\otimes y_{2})}
1057:
These statements are meaningful once we explain the natural structures of algebra and coalgebra in all the vector spaces involved besides
3344:
3011:
Copying a random variable in the distribution given by η is equivalent to having two independent random variables in the distribution η;
3328:
3307:
1826:{\displaystyle \Delta _{2}:=(id\otimes \tau \otimes id)\circ (\Delta \otimes \Delta ):(B\otimes B)\to (B\otimes B)\otimes (B\otimes B)}
1296:{\displaystyle \nabla _{2}:=(\nabla \otimes \nabla )\circ (id\otimes \tau \otimes id):(B\otimes B)\otimes (B\otimes B)\to (B\otimes B)}
3382:
Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. (2010). "Bialgebras and Hopf algebras. Motivation, definitions, and examples".
1021:{\displaystyle (\mathrm {id} _{B}\otimes \epsilon )\circ \Delta =\mathrm {id} _{B}=(\epsilon \otimes \mathrm {id} _{B})\circ \Delta }
1982:
3115:
3438:
3417:
3393:
2187:
2915:
86:
522:
58:
3372:
105:
3005:η is an operator preparing a normalized probability distribution which is independent of all other random variables;
2638:
1472:
65:
3159:
902:{\displaystyle (\mathrm {id} _{B}\otimes \Delta )\circ \Delta =(\Delta \otimes \mathrm {id} _{B})\circ \Delta }
43:
3008:
The product ∇ maps a probability distribution on two variables to a probability distribution on one variable;
515:
3470:
1958:{\displaystyle \Delta \circ \nabla =\nabla _{2}\circ (\Delta \otimes \Delta ):(B\otimes B)\to (B\otimes B)}
72:
3281:
3260:
2369:{\displaystyle \nabla \otimes \nabla \circ \Delta _{2}=\Delta \circ \nabla :(B\otimes B)\to (B\otimes B),}
152:. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the
384:
39:
3102:{\displaystyle \nabla {\bigl (}\mathbf {e} _{g}\otimes \mathbf {e} _{h}{\bigr )}=\mathbf {e} _{gh}\,,}
2966:
2883:
2744:
2163:{\displaystyle \epsilon \circ \nabla =\nabla _{0}\circ (\epsilon \otimes \epsilon ):(B\otimes B)\to K}
54:
3361:
Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Șerban (2001), "4. Bialgebras and Hopf
Algebras",
768:
2996: forgetting the value of a random variable (represented by a single tensor factor) to obtain a
784:
752:
685:
2791:
2719:
475:
32:
165:
2771:
183:
1696:{\displaystyle \epsilon _{2}:=(\epsilon \otimes \epsilon ):(B\otimes B)\to K\otimes K\equiv K}
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8:
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321:
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182:
As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is
175:
Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a
169:
145:
127:
2446:{\displaystyle \nabla _{0}\circ \epsilon _{2}=\epsilon \circ \nabla :(B\otimes B)\to K}
286:
277:
235:
79:
3434:
3413:
3389:
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2866:{\displaystyle \Delta (\mathbf {e} _{g})=\mathbf {e} _{g}\otimes \mathbf {e} _{g}\,,}
1589:{\displaystyle (x_{1}\otimes x_{2})(y_{1}\otimes y_{2})=x_{1}y_{1}\otimes x_{2}y_{2}}
1046:
The four commutative diagrams can be read either as "comultiplication and counit are
306:
153:
1166:{\displaystyle \eta _{2}:=(\eta \otimes \eta ):K\otimes K\equiv K\to (B\otimes B)}
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2556:{\displaystyle \eta _{2}\circ \Delta _{0}=\Delta \circ \eta :K\to (B\otimes B),}
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440:
142:
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291:
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161:
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1051:
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265:
164:, or equivalently, the multiplication and the unit of the algebra both are
134:
3020:
490:
485:
374:
364:
338:
168:. (These statements are equivalent since they are expressed by the same
119:
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3316:
2621:{\displaystyle \eta _{0}\circ \epsilon _{0}=\epsilon \circ \eta :K\to K}
710:
581:
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176:
3235:
820:
664:
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21:
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An example of a bialgebra is the set of functions from a finite
1089:) is a unital associative algebra with unit and multiplication
2713:
230:
157:
3367:, Pure and Applied Mathematics, vol. 235, Marcel Dekker,
2035:{\displaystyle \Delta \circ \eta =\eta _{2}:K\to (B\otimes B)}
198:
is finite-dimensional), then it is automatically a bialgebra.
1050:
of algebras" or, equivalently, "multiplication and unit are
3381:
3406:
Kassel, Christian (2012). "The
Language of Hopf Algebras".
2992:) which represents "tracing out" a random variable —
3385:
Algebras, Rings and
Modules Lie Algebras and Hopf Algebras
3360:
3340:
3324:
3303:
2465:
is a homomorphism of (counital coassociative) coalgebras (
2263:
is a homomorphism of (counital coassociative) coalgebras (
1073:) is a unital associative algebra in an obvious way and (
3149:{\displaystyle \mathbb {R} ^{G}\otimes \mathbb {R} ^{G}}
179:
that is both an algebra and a coalgebra homomorphism.
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3019:
A pair (∇,η) which satisfy these constraints are the
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2228:{\displaystyle \epsilon \circ \eta =\eta _{0}:K\to K}
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is a homomorphism of unital (associative) algebras (
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is a homomorphism of unital (associative) algebras (
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compatibility conditions expressed by the following
3388:. American Mathematical Society. pp. 131–173.
3234:. Additional examples are given in the article on
2953:{\displaystyle \varepsilon (\mathbf {e} _{g})=1\,,}
46:. Unsourced material may be challenged and removed.
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1619:is a coalgebra with counit and comultiplication
2770:consisting of linear combinations of standard
2684:{\displaystyle \epsilon _{0}=id_{K}=\eta _{0}}
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3038:
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2251:Equivalently, diagrams 1 and 2 say that ∇:
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3203:denotes the identity element of the group
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3433:. Springer Science & Business Media.
3426:
3412:. Springer Science & Business Media.
3215:Other examples of bialgebras include the
3188:{\displaystyle \eta =\mathbf {e} _{i}\;,}
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106:Learn how and when to remove this message
3341:Dăscălescu, Năstăsescu & Raianu 2001
3325:Dăscălescu, Năstăsescu & Raianu 2001
3304:Dăscălescu, Năstăsescu & Raianu 2001
679:Multiplication ∇ and comultiplication Δ
3427:Underwood, Robert G. (28 August 2011).
1611:) is a coalgebra in an obvious way and
3453:
3405:
3287:
3271:
663:, Δ, ε) is a (counital coassociative)
562:if it has the following properties:
3222:Bialgebras can often be extended to
2876:which represents making a copy of a
542:
44:adding citations to reliable sources
15:
2963:(again extended linearly to all of
1837:Then, diagrams 1 and 3 say that Δ:
639:-linear maps (comultiplication) Δ:
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3430:An Introduction to Hopf Algebras
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3082:
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3045:
2985:{\displaystyle \mathbb {R} ^{G}}
2927:
2902:{\displaystyle \mathbb {R} ^{G}}
2849:
2834:
2816:
2763:{\displaystyle \mathbb {R} ^{G}}
783:
767:
751:
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628:, ∇, η) is a unital associative
150:counital coassociative coalgebra
20:
2712:(or more generally, any finite
1473:multiplication as juxtaposition
31:needs additional citations for
3364:Hopf Algebras: An introduction
2937:
2922:
2826:
2811:
2738:, which we may represent as a
2612:
2547:
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2532:
2437:
2434:
2422:
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2219:
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2151:
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2133:
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2014:
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762:Comultiplication Δ and unit η
746:Multiplication ∇ and counit ε
1:
3354:
2457:diagrams 3 and 4 say that η:
2062:diagrams 2 and 4 say that ε:
194:(which is always possible if
2731:{\displaystyle \mathbb {R} }
7:
3241:
2695:
1471:or, omitting ∇ and writing
10:
3487:
796:Coassociativity and counit
186:, so if one can define a
3253:
2880:(which we extend to all
2792:probability distribution
2790:, which may represent a
1042:Compatibility conditions
160:are both unital algebra
569:is a vector space over
3401:Download full-text PDF
3189:
3150:
3112:again extended to all
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2998:marginal distribution
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778:Unit η and counit ε
672:commutative diagrams
584:(multiplication) ∇:
427:Group with operators
370:Complemented lattice
205:Algebraic structures
170:commutative diagrams
40:improve this article
3471:Monoidal categories
3199: ∈
2909:by linearity), and
2786: ∈
481:Composition algebra
241:Quasigroup and loop
166:coalgebra morphisms
146:associative algebra
3232:Frobenius algebras
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3440:978-0-387-72766-0
3419:978-1-4612-0783-2
3395:978-0-8218-5262-0
543:Formal definition
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2513:
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2444:
2406:
2405:
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2392:
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2313:
2235:, or simply ε(1
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2231:
2226:
2212:
2211:
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2117:
2116:
2042:, or simply Δ(1
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1054:of coalgebras".
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651:and (counit) ε:
616:) and (unit) η:
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307:Commutative ring
236:Rack and quandle
201:
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154:comultiplication
141:which is both a
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16:
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3248:Quasi-bialgebra
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2878:random variable
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2723:
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2718:
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2703:
2701:Group bialgebra
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994:
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943:
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925:
924:is a counit if
916:-linear map ε:
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834:
833:
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824:
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602:multilinear map
596:(equivalent to
545:
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505:
476:Non-associative
458:
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446:
436:
416:
405:
404:
393:Map of lattices
389:
385:Boolean algebra
380:Heyting algebra
354:
343:
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336:
317:Integral domain
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37:
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12:
11:
5:
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3409:Quantum Groups
3403:
3394:
3379:
3373:
3356:
3353:
3350:
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3333:
3312:
3296:
3280:
3258:
3257:
3255:
3252:
3251:
3250:
3243:
3240:
3228:Lie bialgebras
3217:tensor algebra
3212:
3211:Other examples
3209:
3184:
3178:
3173:
3168:
3165:
3143:
3138:
3133:
3128:
3123:
3110:
3109:
3098:
3092:
3089:
3084:
3079:
3074:
3067:
3062:
3057:
3052:
3047:
3040:
3035:
3017:
3016:
3012:
3009:
3006:
2979:
2974:
2961:
2960:
2949:
2945:
2942:
2939:
2934:
2929:
2924:
2921:
2896:
2891:
2874:
2873:
2862:
2856:
2851:
2846:
2841:
2836:
2831:
2828:
2823:
2818:
2813:
2810:
2777:
2757:
2752:
2726:
2702:
2699:
2697:
2694:
2693:
2692:
2678:
2674:
2670:
2665:
2661:
2657:
2654:
2649:
2645:
2630:
2629:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2591:
2587:
2583:
2578:
2574:
2563:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2511:
2507:
2503:
2498:
2494:
2474:
2470:
2455:
2454:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2404:
2400:
2396:
2391:
2387:
2376:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2312:
2308:
2304:
2301:
2298:
2295:
2276:
2272:
2249:
2248:
2242:
2236:
2224:
2221:
2218:
2215:
2210:
2206:
2202:
2199:
2196:
2193:
2183:
2170:, or simply ε(
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2115:
2111:
2107:
2104:
2101:
2098:
2083:
2079:
2060:
2059:
2049:
2043:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2005:
2001:
1997:
1994:
1991:
1988:
1978:
1965:, or simply Δ(
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1898:
1894:
1890:
1887:
1884:
1881:
1866:
1862:
1835:
1834:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1718:
1714:
1703:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1636:
1632:
1608:
1604:
1583:
1579:
1573:
1569:
1565:
1560:
1556:
1550:
1546:
1542:
1539:
1534:
1530:
1526:
1521:
1517:
1513:
1510:
1505:
1501:
1497:
1492:
1488:
1484:
1460:
1455:
1451:
1447:
1442:
1438:
1434:
1431:
1428:
1425:
1420:
1416:
1412:
1407:
1403:
1399:
1396:
1393:
1390:
1387:
1382:
1378:
1374:
1369:
1365:
1361:
1358:
1355:
1350:
1346:
1342:
1337:
1333:
1329:
1326:
1321:
1317:
1305:
1304:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1188:
1184:
1173:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1106:
1102:
1086:
1082:
1070:
1066:
1043:
1040:
1017:
1014:
1011:
1006:
1001:
998:
993:
990:
987:
984:
979:
974:
971:
966:
963:
960:
957:
954:
951:
946:
941:
938:
933:
898:
895:
892:
887:
882:
879:
874:
871:
868:
865:
862:
859:
856:
853:
850:
845:
840:
837:
832:
797:
794:
793:
792:
791:
790:
789:
788:
776:
775:
774:
773:
772:
760:
759:
758:
757:
756:
744:
743:
742:
691:
690:
689:
676:
675:
668:
633:
574:
544:
541:
538:
537:
535:
534:
527:
520:
512:
509:
508:
504:
503:
498:
493:
488:
483:
478:
473:
467:
466:
465:
459:
453:
452:
449:
448:
445:
444:
441:Linear algebra
435:
434:
429:
424:
418:
417:
411:
410:
407:
406:
403:
402:
399:Lattice theory
395:
388:
387:
382:
377:
372:
367:
362:
356:
355:
349:
348:
345:
344:
335:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
283:
282:
276:
275:
272:
271:
262:
261:
256:
251:
245:
244:
243:
238:
233:
224:
218:
212:
211:
208:
207:
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
3483:
3472:
3469:
3467:
3464:
3462:
3459:
3458:
3456:
3447:
3442:
3436:
3432:
3431:
3425:
3421:
3415:
3411:
3410:
3404:
3402:
3397:
3391:
3387:
3386:
3380:
3376:
3374:0-8247-0481-9
3370:
3366:
3365:
3359:
3358:
3346:
3342:
3337:
3330:
3326:
3321:
3319:
3317:
3309:
3305:
3300:
3293:
3289:
3284:
3277:
3273:
3268:
3266:
3264:
3259:
3249:
3246:
3245:
3239:
3237:
3233:
3229:
3225:
3224:Hopf algebras
3220:
3218:
3208:
3206:
3202:
3198:
3182:
3176:
3166:
3163:
3141:
3131:
3126:
3096:
3090:
3087:
3077:
3065:
3055:
3050:
3026:
3025:
3024:
3022:
3013:
3010:
3007:
3004:
3003:
3002:
2999:
2995:
2977:
2947:
2943:
2940:
2932:
2919:
2912:
2911:
2910:
2894:
2879:
2860:
2854:
2844:
2839:
2829:
2821:
2801:
2800:
2799:
2797:
2793:
2789:
2785:
2780:
2776:
2773:
2772:basis vectors
2755:
2741:
2715:
2711:
2708:
2676:
2672:
2668:
2663:
2659:
2655:
2652:
2647:
2643:
2635:
2634:
2633:
2615:
2609:
2606:
2603:
2600:
2597:
2594:
2589:
2585:
2581:
2576:
2572:
2564:
2550:
2544:
2541:
2538:
2529:
2526:
2523:
2520:
2514:
2509:
2501:
2496:
2492:
2484:
2483:
2482:
2480:
2468:
2464:
2460:
2440:
2431:
2428:
2425:
2419:
2413:
2410:
2407:
2402:
2398:
2394:
2389:
2377:
2363:
2357:
2354:
2351:
2339:
2336:
2333:
2327:
2321:
2315:
2310:
2302:
2296:
2286:
2285:
2284:
2282:
2270:
2266:
2262:
2258:
2254:
2245:
2239:
2222:
2216:
2213:
2208:
2204:
2200:
2197:
2194:
2191:
2184:
2181:
2177:
2173:
2157:
2148:
2145:
2142:
2136:
2130:
2127:
2124:
2118:
2113:
2105:
2099:
2096:
2089:
2088:
2087:
2077:
2074:, ∇, η) and (
2073:
2069:
2065:
2056:
2052:
2046:
2026:
2023:
2020:
2011:
2008:
2003:
1999:
1995:
1992:
1989:
1979:
1976:
1972:
1968:
1949:
1946:
1943:
1931:
1928:
1925:
1919:
1910:
1901:
1896:
1888:
1882:
1872:
1871:
1870:
1860:
1856:
1853:, ∇, η) and (
1852:
1848:
1844:
1840:
1817:
1814:
1811:
1805:
1799:
1796:
1793:
1781:
1778:
1775:
1769:
1760:
1751:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1721:
1716:
1704:
1690:
1687:
1684:
1681:
1678:
1669:
1666:
1663:
1657:
1651:
1648:
1645:
1639:
1634:
1630:
1622:
1621:
1620:
1618:
1614:
1602:
1597:
1581:
1577:
1571:
1567:
1563:
1558:
1554:
1548:
1544:
1540:
1532:
1528:
1524:
1519:
1515:
1503:
1499:
1495:
1490:
1486:
1474:
1453:
1449:
1445:
1440:
1436:
1426:
1418:
1414:
1410:
1405:
1401:
1391:
1380:
1376:
1372:
1367:
1363:
1356:
1348:
1344:
1340:
1335:
1331:
1319:
1287:
1284:
1281:
1269:
1266:
1263:
1257:
1251:
1248:
1245:
1239:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1209:
1200:
1191:
1186:
1174:
1157:
1154:
1151:
1142:
1139:
1136:
1133:
1130:
1127:
1121:
1118:
1115:
1109:
1104:
1100:
1092:
1091:
1090:
1080:
1076:
1064:
1060:
1055:
1053:
1052:homomorphisms
1049:
1048:homomorphisms
1036:
1032:
1029:
1012:
1004:
991:
988:
982:
977:
964:
958:
952:
949:
944:
923:
919:
915:
910:
893:
885:
872:
863:
857:
848:
843:
822:
821:coassociative
818:
814:
810:
806:
804:
786:
782:
781:
780:
779:
777:
770:
766:
765:
764:
763:
761:
754:
750:
749:
748:
747:
745:
740:
736:
732:
728:
724:
720:
716:
713:defined by τ(
712:
708:
704:
700:
696:
692:
687:
683:
682:
681:
680:
678:
677:
673:
669:
666:
662:
659:, such that (
658:
654:
650:
646:
642:
638:
634:
631:
627:
624:, such that (
623:
619:
615:
611:
607:
603:
599:
595:
591:
587:
583:
579:
575:
572:
568:
565:
564:
563:
561:
557:
553:
552:, ∇, η, Δ, ε)
551:
533:
528:
526:
521:
519:
514:
513:
511:
510:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
468:
464:
461:
460:
456:
451:
450:
443:
442:
438:
437:
433:
430:
428:
425:
423:
420:
419:
414:
409:
408:
401:
400:
396:
394:
391:
390:
386:
383:
381:
378:
376:
373:
371:
368:
366:
363:
361:
358:
357:
352:
347:
346:
341:
340:
333:
330:
328:
327:Division ring
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
279:
274:
273:
268:
267:
260:
257:
255:
252:
250:
249:Abelian group
247:
246:
242:
239:
237:
234:
232:
228:
225:
223:
220:
219:
215:
210:
209:
206:
203:
202:
199:
197:
193:
189:
185:
180:
178:
173:
171:
167:
163:
162:homomorphisms
159:
155:
151:
147:
144:
140:
136:
132:
129:
125:
121:
110:
107:
99:
96:December 2009
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
3429:
3408:
3384:
3363:
3336:
3299:
3283:
3221:
3214:
3204:
3200:
3196:
3111:
3018:
2993:
2962:
2875:
2795:
2787:
2783:
2778:
2774:
2740:vector space
2709:
2704:
2631:
2478:
2466:
2462:
2458:
2456:
2280:
2268:
2264:
2260:
2256:
2252:
2250:
2243:
2237:
2179:
2175:
2171:
2075:
2071:
2067:
2063:
2061:
2054:
2050:
2044:
1974:
1970:
1966:
1858:
1854:
1850:
1846:
1842:
1838:
1836:
1616:
1612:
1600:
1599:similarly, (
1598:
1306:
1078:
1074:
1062:
1058:
1056:
1045:
1030:
921:
917:
913:
911:
816:
812:
808:
802:
799:
738:
734:
730:
726:
722:
718:
714:
706:
702:
698:
694:
660:
656:
652:
648:
644:
640:
636:
625:
621:
617:
613:
609:
605:
597:
593:
589:
585:
577:
570:
566:
559:
555:
549:
547:
546:
501:Hopf algebra
495:
439:
432:Vector space
397:
337:
266:Group theory
264:
229: /
195:
191:
181:
174:
138:
135:vector space
130:
123:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
3446:Online Book
3288:Kassel 2012
3272:Kassel 2012
3021:convolution
805:-linear map
582:linear maps
486:Lie algebra
471:Associative
375:Total order
365:Semilattice
339:Ring theory
120:mathematics
55:"Bialgebra"
3466:Coalgebras
3461:Bialgebras
3455:Categories
3355:References
3343:, p.
3327:, p.
3306:, p.
3290:, p.
3274:, p.
3236:coalgebras
711:linear map
635:there are
576:there are
177:linear map
66:newspapers
3164:η
3132:⊗
3056:⊗
3034:∇
3023:operator
2920:ε
2845:⊗
2809:Δ
2782:for each
2673:η
2644:ϵ
2613:→
2604:η
2601:∘
2598:ϵ
2586:ϵ
2582:∘
2573:η
2542:⊗
2533:→
2524:η
2521:∘
2518:Δ
2506:Δ
2502:∘
2493:η
2481:, Δ, ε):
2438:→
2429:⊗
2417:∇
2414:∘
2411:ϵ
2399:ϵ
2395:∘
2386:∇
2355:⊗
2346:→
2337:⊗
2325:∇
2322:∘
2319:Δ
2307:Δ
2303:∘
2300:∇
2297:⊗
2294:∇
2283:, Δ, ε):
2220:→
2205:η
2198:η
2195:∘
2192:ϵ
2155:→
2146:⊗
2131:ϵ
2128:⊗
2125:ϵ
2119:∘
2110:∇
2103:∇
2100:∘
2097:ϵ
2024:⊗
2015:→
2000:η
1993:η
1990:∘
1987:Δ
1947:⊗
1938:→
1929:⊗
1914:Δ
1911:⊗
1908:Δ
1902:∘
1893:∇
1886:∇
1883:∘
1880:Δ
1815:⊗
1806:⊗
1797:⊗
1788:→
1779:⊗
1764:Δ
1761:⊗
1758:Δ
1752:∘
1740:⊗
1737:τ
1734:⊗
1713:Δ
1688:≡
1682:⊗
1676:→
1667:⊗
1652:ϵ
1649:⊗
1646:ϵ
1631:ϵ
1564:⊗
1525:⊗
1496:⊗
1446:⊗
1430:∇
1427:⊗
1411:⊗
1395:∇
1373:⊗
1357:⊗
1341:⊗
1316:∇
1285:⊗
1276:→
1267:⊗
1258:⊗
1249:⊗
1228:⊗
1225:τ
1222:⊗
1210:∘
1204:∇
1201:⊗
1198:∇
1183:∇
1155:⊗
1146:→
1140:≡
1134:⊗
1122:η
1119:⊗
1116:η
1101:η
1016:Δ
1013:∘
992:⊗
989:ϵ
962:Δ
959:∘
953:ϵ
950:⊗
897:Δ
894:∘
873:⊗
870:Δ
861:Δ
858:∘
852:Δ
849:⊗
693:where τ:
665:coalgebra
556:bialgebra
496:Bialgebra
302:Near-ring
259:Lie group
227:Semigroup
184:self-dual
124:bialgebra
3242:See also
2696:Examples
1307:so that
729:for all
332:Lie ring
297:Semiring
156:and the
2632:where
2477:) and (
2279:) and (
709:is the
630:algebra
463:Algebra
455:Algebra
360:Lattice
351:Lattice
126:over a
80:scholar
3437:
3416:
3392:
3371:
3195:where
3015:pairs.
2714:monoid
2174:) = ε(
1969:) = Δ(
491:Graded
422:Module
413:Module
312:Domain
231:Monoid
158:counit
148:and a
143:unital
82:
75:
68:
61:
53:
3254:Notes
2994:i.e.,
2794:over
2716:) to
2707:group
2241:) = 1
2048:) = 1
558:over
554:is a
457:-like
415:-like
353:-like
322:Field
280:-like
254:Magma
222:Group
216:-like
214:Group
137:over
133:is a
128:field
87:JSTOR
73:books
3435:ISBN
3414:ISBN
3390:ISBN
3369:ISBN
3230:and
2178:) ε(
1973:) Δ(
912:The
800:The
733:and
721:) =
287:Ring
278:Ring
188:dual
122:, a
59:news
3345:151
3329:148
3308:147
2473:, ε
2469:, Δ
2275:, ε
2271:, Δ
2086:):
2082:, η
2078:, ∇
1865:, η
1861:, ∇
1607:, ε
1603:, Δ
1085:, η
1081:, ∇
1069:, η
1065:, ∇
1061:: (
823:if
819:is
807:Δ:
737:in
604:∇:
292:Rng
190:of
172:.)
118:In
42:by
3457::
3315:^
3292:45
3276:46
3262:^
3238:.
3207:.
2461:→
2267:⊗
2259:→
2255:⊗
2172:xy
2066:→
2053:⊗
1977:),
1967:xy
1869:)
1857:⊗
1845:⊗
1841:→
1722::=
1640::=
1615:⊗
1596:;
1475:,
1192::=
1110::=
1077:⊗
1028:.
920:→
909:.
815:⊗
811:→
725:⊗
717:⊗
705:⊗
701:→
697:⊗
655:→
647:⊗
643:→
620:→
612:→
608:×
592:→
588:⊗
3443:.
3422:.
3398:.
3378:.
3347:.
3331:.
3310:.
3294:.
3278:.
3205:G
3201:G
3197:i
3183:,
3177:i
3172:e
3167:=
3142:G
3137:R
3127:G
3122:R
3097:,
3091:h
3088:g
3083:e
3078:=
3073:)
3066:h
3061:e
3051:g
3046:e
3039:(
2978:G
2973:R
2948:,
2944:1
2941:=
2938:)
2933:g
2928:e
2923:(
2895:G
2890:R
2861:,
2855:g
2850:e
2840:g
2835:e
2830:=
2827:)
2822:g
2817:e
2812:(
2796:G
2788:G
2784:g
2779:g
2775:e
2756:G
2751:R
2725:R
2710:G
2691:.
2677:0
2669:=
2664:K
2660:d
2656:i
2653:=
2648:0
2628:,
2616:K
2610:K
2607::
2595:=
2590:0
2577:0
2551:,
2548:)
2545:B
2539:B
2536:(
2530:K
2527::
2515:=
2510:0
2497:2
2479:B
2475:0
2471:0
2467:K
2463:B
2459:K
2453:;
2441:K
2435:)
2432:B
2426:B
2423:(
2420::
2408:=
2403:2
2390:0
2364:,
2361:)
2358:B
2352:B
2349:(
2343:)
2340:B
2334:B
2331:(
2328::
2316:=
2311:2
2281:B
2277:2
2273:2
2269:B
2265:B
2261:B
2257:B
2253:B
2247:.
2244:K
2238:B
2223:K
2217:K
2214::
2209:0
2201:=
2182:)
2180:y
2176:x
2158:K
2152:)
2149:B
2143:B
2140:(
2137::
2134:)
2122:(
2114:0
2106:=
2084:0
2080:0
2076:K
2072:B
2068:K
2064:B
2058:;
2055:B
2051:B
2045:B
2030:)
2027:B
2021:B
2018:(
2012:K
2009::
2004:2
1996:=
1975:y
1971:x
1953:)
1950:B
1944:B
1941:(
1935:)
1932:B
1926:B
1923:(
1920::
1917:)
1905:(
1897:2
1889:=
1867:2
1863:2
1859:B
1855:B
1851:B
1847:B
1843:B
1839:B
1833:.
1821:)
1818:B
1812:B
1809:(
1803:)
1800:B
1794:B
1791:(
1785:)
1782:B
1776:B
1773:(
1770::
1767:)
1755:(
1749:)
1746:d
1743:i
1731:d
1728:i
1725:(
1717:2
1691:K
1685:K
1679:K
1673:)
1670:B
1664:B
1661:(
1658::
1655:)
1643:(
1635:2
1617:B
1613:B
1609:0
1605:0
1601:K
1582:2
1578:y
1572:2
1568:x
1559:1
1555:y
1549:1
1545:x
1541:=
1538:)
1533:2
1529:y
1520:1
1516:y
1512:(
1509:)
1504:2
1500:x
1491:1
1487:x
1483:(
1459:)
1454:2
1450:y
1441:2
1437:x
1433:(
1424:)
1419:1
1415:y
1406:1
1402:x
1398:(
1392:=
1389:)
1386:)
1381:2
1377:y
1368:1
1364:y
1360:(
1354:)
1349:2
1345:x
1336:1
1332:x
1328:(
1325:(
1320:2
1303:,
1291:)
1288:B
1282:B
1279:(
1273:)
1270:B
1264:B
1261:(
1255:)
1252:B
1246:B
1243:(
1240::
1237:)
1234:d
1231:i
1219:d
1216:i
1213:(
1207:)
1195:(
1187:2
1161:)
1158:B
1152:B
1149:(
1143:K
1137:K
1131:K
1128::
1125:)
1113:(
1105:2
1087:2
1083:2
1079:B
1075:B
1071:0
1067:0
1063:K
1059:B
1010:)
1005:B
1000:d
997:i
986:(
983:=
978:B
973:d
970:i
965:=
956:)
945:B
940:d
937:i
932:(
922:K
918:B
914:K
891:)
886:B
881:d
878:i
867:(
864:=
855:)
844:B
839:d
836:i
831:(
817:B
813:B
809:B
803:K
741:,
739:B
735:y
731:x
727:x
723:y
719:y
715:x
707:B
703:B
699:B
695:B
674::
667:;
661:B
657:K
653:B
649:B
645:B
641:B
637:K
632:;
626:B
622:B
618:K
614:B
610:B
606:B
600:-
598:K
594:B
590:B
586:B
580:-
578:K
573:;
571:K
567:B
560:K
550:B
548:(
531:e
524:t
517:v
196:B
192:B
139:K
131:K
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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