Knowledge (XXG)

Cartan subgroup

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622: 260: 420: 355: 388: 164: 663: 193: 315: 291: 213: 122: 95: 75: 55: 656: 596: 515: 549: 297:(in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser and thus Cartan subgroups of 692: 493: 687: 649: 580: 218: 77:
is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If
682: 575: 525: 533: 35: 393: 328: 364: 127: 570: 262:. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of 629: 169: 592: 566: 545: 511: 489: 20: 633: 537: 606: 559: 602: 555: 441: 436: 294: 27: 431: 300: 276: 198: 107: 80: 60: 40: 591:, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston, 676: 263: 621: 503: 390:
already before any base extension), and it can be shown to be maximal. Since
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Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field
215:) is isomorphic to the product of a finite number of copies of the 357:
are reductive. The diagonal subgroup is clearly a torus (indeed a
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is algebraically closed, they are all conjugate to each other.
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is reductive, the diagonal subgroup is a Cartan subgroup.
637: 396: 367: 331: 303: 279: 221: 201: 172: 130: 110: 83: 63: 43: 57:
over a (not necessarily algebraically closed) field
414: 382: 349: 309: 285: 255:{\displaystyle \mathbf {G} _{m}=\mathbf {GL} _{1}} 254: 207: 187: 158: 116: 89: 69: 49: 100:Notice that in the context of algebraic groups a 657: 8: 664: 650: 361:torus, since it is product of n copies of 19:For a Cartan subgroup of a Lie group, see 406: 398: 395: 374: 369: 366: 341: 333: 330: 302: 278: 246: 238: 228: 223: 220: 200: 174: 173: 171: 140: 139: 135: 129: 109: 82: 62: 42: 21:Cartan subalgebra § Cartan subgroup 453: 472: 460: 7: 618: 616: 636:. You can help Knowledge (XXG) by 16:Maximal connected Abelian subgroup 14: 415:{\displaystyle \mathbf {GL} _{n}} 350:{\displaystyle \mathbf {GL} _{n}} 620: 402: 399: 383:{\displaystyle \mathbf {G} _{m}} 370: 337: 334: 317:are precisely the maximal tori. 242: 239: 224: 159:{\displaystyle T_{({\bar {k}})}} 179: 151: 145: 136: 1: 124:such that the base extension 484:Borel, Armand (1991-12-31). 195:is the algebraic closure of 587:Springer, Tonny A. (1998), 576:Encyclopedia of Mathematics 709: 615: 534:Cambridge University Press 325:The general linear groups 188:{\displaystyle {\bar {k}}} 18: 693:Algebraic geometry stubs 688:Linear algebraic groups 589:Linear algebraic groups 486:Linear algebraic groups 632:–related article is a 416: 384: 351: 311: 287: 256: 209: 189: 160: 118: 104:is an algebraic group 91: 71: 51: 36:linear algebraic group 542:10.1017/9781316711736 417: 385: 352: 312: 288: 257: 210: 190: 161: 119: 92: 72: 52: 463:, Proposition 17.44. 394: 365: 329: 301: 277: 219: 199: 170: 128: 108: 81: 61: 41: 683:Algebraic geometry 630:algebraic geometry 475:, Corollary 17.84. 412: 380: 347: 307: 283: 252: 205: 185: 156: 114: 87: 67: 47: 645: 644: 598:978-0-8176-4021-7 571:"Cartan subgroup" 517:978-0-387-95385-4 310:{\displaystyle G} 286:{\displaystyle G} 266:in the theory of 208:{\displaystyle k} 182: 148: 117:{\displaystyle T} 90:{\displaystyle k} 70:{\displaystyle k} 50:{\displaystyle G} 26:In the theory of 700: 666: 659: 652: 624: 617: 609: 583: 562: 521: 499: 476: 470: 464: 458: 421: 419: 418: 413: 411: 410: 405: 389: 387: 386: 381: 379: 378: 373: 356: 354: 353: 348: 346: 345: 340: 316: 314: 313: 308: 292: 290: 289: 284: 261: 259: 258: 253: 251: 250: 245: 233: 232: 227: 214: 212: 211: 206: 194: 192: 191: 186: 184: 183: 175: 165: 163: 162: 157: 155: 154: 150: 149: 141: 123: 121: 120: 115: 96: 94: 93: 88: 76: 74: 73: 68: 56: 54: 53: 48: 28:algebraic groups 708: 707: 703: 702: 701: 699: 698: 697: 673: 672: 671: 670: 613: 599: 586: 565: 552: 524: 518: 502: 496: 483: 480: 479: 471: 467: 459: 455: 450: 442:Algebraic torus 437:Algebraic group 428: 397: 392: 391: 368: 363: 362: 332: 327: 326: 323: 299: 298: 275: 274: 237: 222: 217: 216: 197: 196: 168: 167: 131: 126: 125: 106: 105: 79: 78: 59: 58: 39: 38: 34:of a connected 32:Cartan subgroup 24: 17: 12: 11: 5: 706: 704: 696: 695: 690: 685: 675: 674: 669: 668: 661: 654: 646: 643: 642: 625: 611: 610: 597: 584: 563: 551:978-1107167483 550: 522: 516: 500: 494: 478: 477: 465: 452: 451: 449: 446: 445: 444: 439: 434: 432:Borel subgroup 427: 424: 409: 404: 401: 377: 372: 344: 339: 336: 322: 319: 306: 282: 249: 244: 241: 236: 231: 226: 204: 181: 178: 153: 147: 144: 138: 134: 113: 86: 66: 46: 15: 13: 10: 9: 6: 4: 3: 2: 705: 694: 691: 689: 686: 684: 681: 680: 678: 667: 662: 660: 655: 653: 648: 647: 641: 639: 635: 631: 626: 623: 619: 614: 608: 604: 600: 594: 590: 585: 582: 578: 577: 572: 568: 564: 561: 557: 553: 547: 543: 539: 535: 531: 527: 523: 519: 513: 509: 505: 501: 497: 495:3-540-97370-2 491: 487: 482: 481: 474: 469: 466: 462: 457: 454: 447: 443: 440: 438: 435: 433: 430: 429: 425: 423: 407: 375: 360: 342: 320: 318: 304: 296: 280: 271: 269: 265: 247: 234: 229: 202: 176: 142: 132: 111: 103: 98: 84: 64: 44: 37: 33: 29: 22: 638:expanding it 627: 612: 588: 574: 567:Popov, V. L. 529: 526:Milne, J. S. 510:. Springer. 507: 485: 473:Milne (2017) 468: 461:Milne (2017) 456: 358: 324: 272: 264:maximal tori 101: 99: 31: 25: 504:Lang, Serge 677:Categories 448:References 268:Lie groups 581:EMS Press 569:(2001) , 295:reductive 180:¯ 146:¯ 528:(2017), 506:(2002). 426:See also 607:1642713 560:3729270 508:Algebra 321:Example 166:(where 605:  595:  558:  548:  514:  492:  628:This 359:split 102:torus 634:stub 593:ISBN 546:ISBN 512:ISBN 490:ISBN 30:, a 538:doi 293:is 273:If 679:: 603:MR 601:, 579:, 573:, 556:MR 554:, 544:, 536:, 532:, 488:. 270:. 665:e 658:t 651:v 640:. 540:: 520:. 498:. 408:n 403:L 400:G 376:m 371:G 343:n 338:L 335:G 305:G 281:G 248:1 243:L 240:G 235:= 230:m 225:G 203:k 177:k 152:) 143:k 137:( 133:T 112:T 85:k 65:k 45:G 23:.

Index

Cartan subalgebra § Cartan subgroup
algebraic groups
linear algebraic group
maximal tori
Lie groups
reductive
Borel subgroup
Algebraic group
Algebraic torus
Milne (2017)
Milne (2017)
ISBN
3-540-97370-2
Lang, Serge
ISBN
978-0-387-95385-4
Milne, J. S.
Cambridge University Press
doi
10.1017/9781316711736
ISBN
978-1107167483
MR
3729270
Popov, V. L.
"Cartan subgroup"
Encyclopedia of Mathematics
EMS Press
ISBN
978-0-8176-4021-7

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