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Zinbiel algebra

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411: 130: 235: 201: 446: 46: 641: 406:{\displaystyle (x_{0}\otimes \cdots \otimes x_{p})\circ (x_{p+1}\otimes \cdots \otimes x_{p+q})=x_{0}\sum _{(p,q)}(x_{1},\ldots ,x_{p+q}),} 636: 613: 584:
Zinbiel, Guillaume W. (2012), "Encyclopedia of types of algebras 2010", in Guo, Li; Bai, Chengming; Loday, Jean-Louis (eds.),
504: 156: 631: 601: 590:, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 9, pp. 217–298, 29: 591: 513: 609: 566: 543: 523: 499: 136: 33: 535: 419: 576: 531: 495: 448: 144: 547: 605: 226: 214: 625: 585: 37: 527: 482:
Dzhumadil'daev, A.S.; Tulenbaev, K.M. (2005). "Nilpotency of Zinbiel algebras".
218: 207: 17: 570: 143:). The name was proposed by Jean-Michel Lemaire as being "opposite" to 125:{\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).} 596: 518: 548:"Cup-product for Leibniz cohomology and dual Leibniz algebras" 422: 238: 159: 49: 440: 405: 195: 124: 575:. Lecture Notes in Mathematics. Vol. 1763. 150:In any Zinbiel algebra, the symmetrised product 8: 196:{\displaystyle a\star b=a\circ b+b\circ a} 40:product satisfying the defining identity: 595: 517: 421: 385: 366: 341: 331: 309: 284: 265: 246: 237: 158: 48: 466: 464: 460: 502:(1994). "Koszul duality for operads". 470: 140: 7: 135:Zinbiel algebras were introduced by 217:concept to a Leibniz algebra. The 14: 572:Dialgebras and related operads 435: 423: 397: 359: 354: 342: 321: 277: 271: 239: 116: 104: 92: 80: 62: 50: 1: 587:Operads and universal algebra 528:10.1215/s0012-7094-94-07608-4 642:Algebra of random variables 658: 416:where the sum is over all 505:Duke Mathematical Journal 213:A Zinbiel algebra is the 637:Non-associative algebras 442: 407: 197: 126: 443: 441:{\displaystyle (p,q)} 408: 221:Zinbiel algebra over 198: 127: 484:J. Dyn. Control Syst 420: 236: 157: 137:Jean-Louis Loday 47: 26:dual Leibniz algebra 606:2011arXiv1101.0267Z 438: 403: 358: 193: 122: 567:Loday, Jean-Louis 544:Loday, Jean-Louis 500:Kapranov, Mikhail 337: 649: 618: 599: 580: 579:. pp. 7–66. 562: 552: 539: 521: 496:Ginzburg, Victor 491: 474: 468: 447: 445: 444: 439: 412: 410: 409: 404: 396: 395: 371: 370: 357: 336: 335: 320: 319: 295: 294: 270: 269: 251: 250: 202: 200: 199: 194: 131: 129: 128: 123: 34:commutative ring 657: 656: 652: 651: 650: 648: 647: 646: 622: 621: 616: 583: 577:Springer Verlag 565: 550: 542: 494: 481: 478: 477: 469: 462: 457: 418: 417: 381: 362: 327: 305: 280: 261: 242: 234: 233: 155: 154: 145:Leibniz algebra 45: 44: 22:Zinbiel algebra 12: 11: 5: 655: 653: 645: 644: 639: 634: 624: 623: 620: 619: 614: 581: 563: 540: 492: 476: 475: 459: 458: 456: 453: 437: 434: 431: 428: 425: 414: 413: 402: 399: 394: 391: 388: 384: 380: 377: 374: 369: 365: 361: 356: 353: 350: 347: 344: 340: 334: 330: 326: 323: 318: 315: 312: 308: 304: 301: 298: 293: 290: 287: 283: 279: 276: 273: 268: 264: 260: 257: 254: 249: 245: 241: 227:tensor algebra 204: 203: 192: 189: 186: 183: 180: 177: 174: 171: 168: 165: 162: 133: 132: 121: 118: 115: 112: 109: 106: 103: 100: 97: 94: 91: 88: 85: 82: 79: 76: 73: 70: 67: 64: 61: 58: 55: 52: 13: 10: 9: 6: 4: 3: 2: 654: 643: 640: 638: 635: 633: 630: 629: 627: 617: 615:9789814365116 611: 607: 603: 598: 593: 589: 588: 582: 578: 574: 573: 568: 564: 561:(2): 189–196. 560: 556: 549: 545: 541: 537: 533: 529: 525: 520: 515: 511: 507: 506: 501: 497: 493: 490:(2): 195–213. 489: 485: 480: 479: 472: 467: 465: 461: 454: 452: 450: 432: 429: 426: 400: 392: 389: 386: 382: 378: 375: 372: 367: 363: 351: 348: 345: 338: 332: 328: 324: 316: 313: 310: 306: 302: 299: 296: 291: 288: 285: 281: 274: 266: 262: 258: 255: 252: 247: 243: 232: 231: 230: 229:with product 228: 224: 220: 216: 211: 209: 190: 187: 184: 181: 178: 175: 172: 169: 166: 163: 160: 153: 152: 151: 148: 146: 142: 138: 119: 113: 110: 107: 101: 98: 95: 89: 86: 83: 77: 74: 71: 68: 65: 59: 56: 53: 43: 42: 41: 39: 35: 31: 27: 23: 19: 632:Lie algebras 586: 571: 558: 554: 509: 503: 487: 483: 473:, p. 45 415: 222: 212: 205: 149: 134: 25: 21: 15: 555:Math. Scand 512:: 203–273. 215:Koszul dual 208:associative 18:mathematics 626:Categories 471:Loday 2001 455:References 597:1101.0267 519:0709.1228 376:… 339:∑ 303:⊗ 300:⋯ 297:⊗ 275:∘ 259:⊗ 256:⋯ 253:⊗ 188:∘ 176:∘ 164:⋆ 111:∘ 102:∘ 87:∘ 78:∘ 66:∘ 57:∘ 569:(2001). 546:(1995). 449:shuffles 38:bilinear 602:Bibcode 536:1301191 225:is the 139: ( 36:with a 32:over a 612:  534:  30:module 592:arXiv 551:(PDF) 514:arXiv 28:is a 610:ISBN 219:free 141:1995 20:, a 524:doi 206:is 24:or 16:In 628:: 608:, 600:, 559:77 557:. 553:. 532:MR 530:. 522:. 510:76 508:. 498:; 488:11 486:. 463:^ 451:. 210:. 147:. 604:: 594:: 538:. 526:: 516:: 436:) 433:q 430:, 427:p 424:( 401:, 398:) 393:q 390:+ 387:p 383:x 379:, 373:, 368:1 364:x 360:( 355:) 352:q 349:, 346:p 343:( 333:0 329:x 325:= 322:) 317:q 314:+ 311:p 307:x 292:1 289:+ 286:p 282:x 278:( 272:) 267:p 263:x 248:0 244:x 240:( 223:V 191:a 185:b 182:+ 179:b 173:a 170:= 167:b 161:a 120:. 117:) 114:b 108:c 105:( 99:a 96:+ 93:) 90:c 84:b 81:( 75:a 72:= 69:c 63:) 60:b 54:a 51:(

Index

mathematics
module
commutative ring
bilinear
Jean-Louis Loday
1995
Leibniz algebra
associative
Koszul dual
free
tensor algebra
shuffles


Loday 2001
Ginzburg, Victor
Kapranov, Mikhail
Duke Mathematical Journal
arXiv
0709.1228
doi
10.1215/s0012-7094-94-07608-4
MR
1301191
Loday, Jean-Louis
"Cup-product for Leibniz cohomology and dual Leibniz algebras"
Loday, Jean-Louis
Dialgebras and related operads
Springer Verlag
Operads and universal algebra

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