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History,人物,霍赫希尔德,上同调,代数群,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc="" name="description" content="本文介绍了数学家格哈德·霍赫希尔德的生平及其主要贡献，包括他创立的霍赫希尔德上同调理论、HKR定理、Hochschild–Serre谱序列，以及对代数群与Hopf代数的研究，这些工作深刻影响了同调代数、非交换几何和表示论等领域 - MHuiG - Magicland"><meta property="og:type" content="website"><meta property="og:title" content="Magicland"><meta property="og:url" content="https://blog.mhuig.top/notes/person/23"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="本文介绍了数学家格哈德·霍赫希尔德的生平及其主要贡献，包括他创立的霍赫希尔德上同调理论、HKR定理、Hochschild–Serre谱序列，以及对代数群与Hopf代数的研究，这些工作深刻影响了同调代数、非交换几何和表示论等领域"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2026-05-16T06:07:00.000Z"><meta property="article:modified_time" content="2026-05-16T06:10:00.000Z"><meta property="article:author" content="MHuiG"><meta property="article:tag" content="History"><meta 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itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="Hochschild数学家霍赫希尔德：生平、贡献与影响"><meta itemprop="description" content="本文介绍了数学家格哈德·霍赫希尔德的生平及其主要贡献，包括他创立的霍赫希尔德上同调理论、HKR定理、Hochschild–Serre谱序列，以及对代数群与Hopf代数的研究，这些工作深刻影响了同调代数、非交换几何和表示论等领域"></span><span hidden=""><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><span hidden="" itemprop="name headline"></span></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p> <span class="p logo center large">Hochschild</span></p><br><h1 hidden="">Hochschild</h1><div class="story post-story"><h2 id="生平"><a href="#生平" class="headerlink" title="生平"></a>生平</h2><p>格哈德・保罗・霍赫希尔德（Gerhard Paul Hochschild，1915-2010）出生在德国柏林一个中产阶级家庭。</p><p>1924 年，他的母亲 Lilli 被诊断为肺结核，他九岁时随母亲前往达沃斯（Davos）附近的疗养院（Sanatorium near Davos），陪伴她接受长期治疗。</p><p>在疗养期间，他母亲精神状况恶化，1926 年被送入精神病院后丧生。这一经历对 Hochschild 的童年影响深远，他曾在回忆中提到读 Thomas Mann 的《魔山》时，常回想自己在疗养院的经历。</p><p>童年时期，他在柏林开始培养对摄影和登山的兴趣，这些爱好贯穿一生。1933 年，他父亲为安全起见，将 Hochschild 和兄长 Ulrich 送往南非开普敦（Cape Town），成为众多逃离纳粹的德国人之一。</p><p>在南非，由于纳粹限制资金流出，他们被迫自力更生。Hochschild 曾在摄影店工作，同时加入一个知识分子与艺术家圈子，这段经历培养了他终生的自由思想与社会责任感。</p><p>1934 年，Hochschild 入读 Cape Town 大学理学士课程，1936 年获理学学士学位，1937 年获理学硕士学位。</p><p>他所修课程覆盖应用数学、物理、化学、纯数学、微分几何、调和分析、张量方法等，导师是利特伍德（John Edensor Littlewood，1885-1977）的学生斯坦利・斯奎斯（Stanley Skewes，1899-1988）。</p><p>Skewes 在推荐信中写道：“他是一个好学生，也是一位非常有前途的数学家…… 我相信他极其适合从事这一职业”，这为他申请 Princeton 大学博士提供了有力支持。</p><p>1938 年，Hochschild 前往 Princeton 大学攻读博士，师从谢瓦莱（Claude Chevalley，1909-1984）。</p><p>博士期间，他选修课程广泛，包括黎曼几何、连续群、复变函数理论、拓扑群、分析在几何中的应用等。</p><p>他在 Chevalley 的微分方程课程中展现出极高的专注力，课程结束时往往只有 Hochschild、John von Neumann 和 Hermann Weyl 留下来继续讨论。他的博士论文题为《半单代数与广义导子》（Semisimple algebras and generalized derivations），委员会高度评价其独立研究能力，称其研究 “包含了许多新结果（contains many new results）”。</p><p>二战期间，Hochschild 被征召入美国陆军，先驻扎于锡尔堡（Ft. Sill），后调至阿伯丁试验场（Aberdeen Proving Grounds）工作。在军队的数学科，他与费德勒（Herbert Federer，1920-2010）等著名数学家共事（维布伦（Oswald Veblen，1880-1960）也曾作为顾问偶尔造访），在此期间他开始发表关于 Hochschild 上同调（Hochschild cohomology）的开创性论文。</p><p>他的长期合作者莫斯托（George Daniel Mostow，1923-2017）曾这样评价他：“如果不提及他的个人魅力，就无法全面地刻画 Hochschild …… 他以其精湛的‘爆粗口’艺术给战友们留下了深刻印象。”</p><p>1945 年战后，他回到 Princeton 大学担任兼职讲师。1946–1948 年出任 Harvard 大学的 Benjamin Peirce 讲师，随后加入伊利诺伊大学厄巴纳 - 香槟分校（University of Illinois at Urbana‑Champaign），并于 1952 年晋升为正教授。</p><p>1958 年他调任 Berkeley，长期在此执教并指导了 26 名博士生，其中包括莱杰（George Leger）（1951）和纳赫鲁斯（Nazih Nahlus）（1986）。</p><p>Hochschild 个人生活同样丰富。在厄巴纳（Urbana）时，他遇到未来的妻子露丝（Ruth Heinsheimer），两人 1950 年结婚，育有两个孩子：Ann（1955）与 Peter（1957）。</p><p>他与学生关系密切，被学生评价为耐心、支持、极具独立精神的导师。在 Berkeley 任教期间，他曾因对当时少数族裔政策抱有强烈的原则性异议，辞去了相关委员会的职务。</p><p>Hochschild 于 1979 年当选为美国国家科学院院士及美国艺术与科学院院士，并于 1980 年荣获美国数学会 Steele 奖（AMS Steele Prize），以表彰他在 1945–1952 年间关于同调代数（Homological Algebra）及其应用的开创性论文。</p><p>1982 年，Hochschild 退休，但继续兼职教书直到 1985 年。2010 年 7 月 8 日，Hochschild 在家中去世，享年 95 岁。</p></div><div class="story post-story"><h2 id="贡献"><a href="#贡献" class="headerlink" title="贡献"></a>贡献</h2><h3 id="一，-Hochschild-上同调理论"><a href="#一，-Hochschild-上同调理论" class="headerlink" title="一， Hochschild 上同调理论"></a>一， Hochschild 上同调理论</h3><p>对于一个域<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.179ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 521 705"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="k"><g data-mml-node="mi" data-latex="k"><path data-c="1D458" d="M409 353C409 327 423 314 450 314 485 314 508 345 508 379 508 418 476 445 437 445 392 445 344 415 291 356 250 311 217 282 190 269L291 679C289 688 287 694 274 694 242 694 166 685 154 684 139 682 132 675 132 660 132 650 141 645 159 645 178 645 204 646 204 632L59 43C56 32 55 25 55 21 55 0 66-11 87-11 104-11 117-3 124 12 129 21 147 92 179 226 231 221 286 196 286 146 286 131 279 101 279 91 279 34 316-11 373-11 431-11 470 41 490 145 490 154 485 159 475 159 466 159 460 152 457 138 435 59 408 19 375 19 357 19 348 33 348 61 348 77 360 131 360 147 360 204 314 239 221 253 244 269 270 292 298 322 326 352 346 371 359 382 386 404 412 415 435 415 445 415 453 413 459 409 432 404 409 379 409 353Z"></path></g></g></g></svg></mjx-container> 上的结合代数（associative algebra）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container> 及其<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container> - 双模（bi-module）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1044 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="M"><g data-mml-node="mi" data-latex="M"><path data-c="1D440" d="M594 16C594 5 600 0 613 0L736 3C759 4 840 0 859 0 874 0 882 8 882 24 882 34 872 39 851 39 810 39 790 43 790 52 790 55 792 62 795 75L927 602C932 623 942 636 955 641 961 643 979 644 1008 644 1033 644 1044 645 1044 668 1044 678 1034 683 1013 683L882 683C853 683 853 681 840 662L484 108 408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660 204 649 215 644 236 644 256 644 297 647 297 631 297 629 296 623 293 613L167 110C158 72 135 49 100 42 75 39 42 43 42 16 42 5 48 0 60 0 78 0 141 3 159 3 178 3 242 0 261 0 276 0 283 8 283 24 283 33 276 38 261 39 219 39 198 52 198 78 198 82 199 89 202 100L332 621 415 26C418 9 424 0 434 0 442 0 450 7 459 20L848 627 712 82C706 60 696 47 681 42 675 40 656 39 625 39 604 39 594 37 594 16Z"></path></g></g></g></svg></mjx-container> ，他将 Hochschild 上同调群（Hochschild cohomology groups）定义为：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:37.615ex"><svg style="vertical-align:-.601ex;min-width:37.615ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.333ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -765.5)"><g data-mml-node="math" data-latex="
HH^n(A, M):= \operatorname{Ext}_{A^e}^n(A, M)
"><g data-mml-node="mtable" data-latex="
HH^n(A, M):= \operatorname{Ext}_{A^e}^n(A, M)
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39 302 39 262 39 242 43 242 52 242 52 243 55 245 68L312 336 605 336 538 68C535 53 523 43 503 40 498 39 481 39 452 39 433 39 424 31 424 15 424 5 430 0 442 0L568 3 631 2C642 2 683 0 696 0 712 0 720 8 720 24 720 34 709 39 688 39 647 39 627 43 627 52 627 52 628 55 630 68L764 602C769 623 778 636 791 641 797 643 814 644 843 644 869 644 881 644 881 668Z"></path></g><g data-mml-node="msup" data-latex="H^n" transform="translate(881,0)"><g data-mml-node="mi" data-latex="H"><path data-c="1D43B" d="M881 668C881 678 875 683 863 683L736 680 609 683C593 683 586 674 586 659 586 652 589 647 594 646 604 645 612 644 617 644 648 643 665 641 670 640 675 639 678 636 678 631 677 628 676 622 674 613L615 375 321 375 379 602C384 623 393 636 406 641 413 643 430 644 458 644 485 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 683 200 674 200 659 200 652 203 647 209 646 219 645 227 644 232 644 263 643 281 641 286 640 291 639 293 636 293 631 293 629 292 623 289 613L156 82C151 60 141 47 126 42 119 40 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transform="translate(2083.8,0)"></g></g></g></svg><mjx-break size="3"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="6.758ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2987.2 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A^e:= A\otimes_k A^{op}"><g data-mml-node="msub" data-latex="\otimes_k"><g data-mml-node="mo" data-latex="\otimes"><path data-c="2297" d="M48 249C48 60 201-93 390-93 579-93 732 60 732 249 732 438 579 591 390 591 201 591 48 438 48 249M615 58 423 250 614 441C658 389 685 322 685 249 685 176 659 110 615 58M198 475C250 519 316 545 389 545 462 545 528 519 580 475L389 284M581 24C529-20 462-47 389-47 316-47 249-20 197 24L389 216M164 441 355 250 163 58C119 110 93 176 93 249 93 322 120 389 164 441Z"></path></g><g data-mml-node="mi" transform="translate(815,-150) scale(0.707)" data-latex="k"><path data-c="1D458" d="M409 353C409 327 423 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541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="TeXAtom" transform="translate(783,363) scale(0.707)" data-latex="{o p}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="o"><path data-c="1D45C" d="M308 442C239 442 176 412 122 353 68 294 41 229 41 159 41 63 106-11 202-11 272-11 334 19 388 78 442 137 469 201 469 272 469 369 405 442 308 442M389 310C389 287 384 255 374 213 353 130 319 73 272 42 248 26 225 18 203 18 149 18 121 65 121 122 121 180 155 288 178 324 216 383 259 413 307 413 361 413 389 367 389 310Z"></path></g><g data-mml-node="mi" data-latex="p" transform="translate(485,0)"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 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xmlns="http://www.w3.org/2000/svg" width="11.253ex" height="2.448ex" role="img" focusable="false" viewBox="0 -833.9 4973.7 1081.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="HH^0(A,A)"><g data-mml-node="mi" data-latex="H"><path data-c="1D43B" d="M881 668C881 678 875 683 863 683L736 680 609 683C593 683 586 674 586 659 586 652 589 647 594 646 604 645 612 644 617 644 648 643 665 641 670 640 675 639 678 636 678 631 677 628 676 622 674 613L615 375 321 375 379 602C384 623 393 636 406 641 413 643 430 644 458 644 485 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 683 200 674 200 659 200 652 203 647 209 646 219 645 227 644 232 644 263 643 281 641 286 640 291 639 293 636 293 631 293 629 292 623 289 613L156 82C151 60 141 47 126 42 119 40 101 39 70 39 48 39 39 37 39 15 39 5 45 0 57 0L183 3 246 2C257 2 299 0 310 0 326 0 334 8 334 24 334 34 323 39 302 39 262 39 242 43 242 52 242 52 243 55 245 68L312 336 605 336 538 68C535 53 523 43 503 40 498 39 481 39 452 39 433 39 424 31 424 15 424 5 430 0 442 0L568 3 631 2C642 2 683 0 696 0 712 0 720 8 720 24 720 34 709 39 688 39 647 39 627 43 627 52 627 52 628 55 630 68L764 602C769 623 778 636 791 641 797 643 814 644 843 644 869 644 881 644 881 668Z"></path></g><g data-mml-node="msup" data-latex="H^0" transform="translate(881,0)"><g data-mml-node="mi" data-latex="H"><path data-c="1D43B" d="M881 668C881 678 875 683 863 683L736 680 609 683C593 683 586 674 586 659 586 652 589 647 594 646 604 645 612 644 617 644 648 643 665 641 670 640 675 639 678 636 678 631 677 628 676 622 674 613L615 375 321 375 379 602C384 623 393 636 406 641 413 643 430 644 458 644 485 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 683 200 674 200 659 200 652 203 647 209 646 219 645 227 644 232 644 263 643 281 641 286 640 291 639 293 636 293 631 293 629 292 623 289 613L156 82C151 60 141 47 126 42 119 40 101 39 70 39 48 39 39 37 39 15 39 5 45 0 57 0L183 3 246 2C257 2 299 0 310 0 326 0 334 8 334 24 334 34 323 39 302 39 262 39 242 43 242 52 242 52 243 55 245 68L312 336 605 336 538 68C535 53 523 43 503 40 498 39 481 39 452 39 433 39 424 31 424 15 424 5 430 0 442 0L568 3 631 2C642 2 683 0 696 0 712 0 720 8 720 24 720 34 709 39 688 39 647 39 627 43 627 52 627 52 628 55 630 68L764 602C769 623 778 636 791 641 797 643 814 644 843 644 869 644 881 644 881 668Z"></path></g><g data-mml-node="mn" transform="translate(966.5,363) scale(0.707)" data-latex="0"><path data-c="30" d="M249-22C390-22 460 92 460 320 460 473 428 575 365 625 330 652 291 666 250 666 109 666 39 551 39 320 39 136 88-22 249-22M361 524C368 489 371 425 371 332 371 240 367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z"></path></g></g><g data-mml-node="mo" data-latex="(" transform="translate(2251.1,0)"><path data-c="28" d="M318-248C327-248 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541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(4584.7,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 刻画了代数的外导子（outer derivations），在物理上对应着无穷小对称性；Murray Gerstenhaber 后续的工作证明，<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="11.253ex" height="2.448ex" role="img" focusable="false" viewBox="0 -833.9 4973.7 1081.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="HH^2(A,A)"><g data-mml-node="mi" data-latex="H"><path data-c="1D43B" d="M881 668C881 678 875 683 863 683L736 680 609 683C593 683 586 674 586 659 586 652 589 647 594 646 604 645 612 644 617 644 648 643 665 641 670 640 675 639 678 636 678 631 677 628 676 622 674 613L615 375 321 375 379 602C384 623 393 636 406 641 413 643 430 644 458 644 485 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 683 200 674 200 659 200 652 203 647 209 646 219 645 227 644 232 644 263 643 281 641 286 640 291 639 293 636 293 631 293 629 292 623 289 613L156 82C151 60 141 47 126 42 119 40 101 39 70 39 48 39 39 37 39 15 39 5 45 0 57 0L183 3 246 2C257 2 299 0 310 0 326 0 334 8 334 24 334 34 323 39 302 39 262 39 242 43 242 52 242 52 243 55 245 68L312 336 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2C257 2 299 0 310 0 326 0 334 8 334 24 334 34 323 39 302 39 262 39 242 43 242 52 242 52 243 55 245 68L312 336 605 336 538 68C535 53 523 43 503 40 498 39 481 39 452 39 433 39 424 31 424 15 424 5 430 0 442 0L568 3 631 2C642 2 683 0 696 0 712 0 720 8 720 24 720 34 709 39 688 39 647 39 627 43 627 52 627 52 628 55 630 68L764 602C769 623 778 636 791 641 797 643 814 644 843 644 869 644 881 644 881 668Z"></path></g><g data-mml-node="mn" transform="translate(966.5,363) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612 69 576 50 534 50 483 50 449 75 424 106 424 136 424 161 450 161 480 161 513 137 536 105 536 102 536 100 536 98 535 117 584 161 627 224 627 306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z"></path></g></g><g data-mml-node="mo" data-latex="(" 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0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="mo" data-latex="," transform="translate(3390.1,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50 86 20 109-5 139-5 153-5 165-1 174 8L175 0C175-63 154-117 112-160 105-168 101-174 101-178 101-188 105-193 114-193 123-193 135-181 152-158 186-110 203-57 203 0 203 53 185 106 139 106Z"></path></g><g data-mml-node="mi" data-latex="A" transform="translate(3834.7,0)"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(4584.7,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 精确分类了代数的形式形变（formal deformations），决定了将古典交换代数 “量子化” 为非交换代数的所有可能路径。</p><p>这套庞大理论框架的原始论文仅有 10 页，却凭借定义的简洁性与极强的普适性，迅速成为代数学的通用语言。</p><h3 id="二、HKR-定理（The-Hochschild-Kostant-Rosenberg-theorem）"><a href="#二、HKR-定理（The-Hochschild-Kostant-Rosenberg-theorem）" class="headerlink" title="二、HKR 定理（The Hochschild-Kostant-Rosenberg theorem）"></a>二、HKR 定理（The Hochschild-Kostant-Rosenberg theorem）</h3><p>1962 年，Gerhard Hochschild 与科斯坦特（Bertram Kostant，1928-2017）及罗森堡（Alex F. T. W. Rosenberg，1926-2007）共同证明了 HKR 定理，明确界定了在光滑情形下，纯粹的代数同调结构与几何对象之间的同构关系。</p><p>定理指出，设<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container> 是一个复数域<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.043ex" xmlns="http://www.w3.org/2000/svg" width="1.633ex" height="1.636ex" role="img" focusable="false" viewBox="0 -704 722 723"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{C}"><g data-mml-node="TeXAtom" data-latex="\mathbb{C}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="C"><path data-c="2102" d="M653 87C674 108 685 123 685 131 685 147 677 155 660 155 653 155 646 150 637 140 576 71 499 36 406 36 334 36 284 68 256 131 231 188 219 258 219 341 219 416 230 483 252 540 282 621 329 661 392 661 437 661 481 643 526 607 571 571 600 531 614 487 619 473 626 466 637 466 650 466 657 478 657 502L657 667C657 692 650 704 637 704 626 704 618 696 614 681 610 666 601 658 586 658 566 658 536 666 494 681 452 696 417 704 389 704 298 704 219 676 151 620 75 558 37 466 37 343 37 220 75 127 150 66 219 9 300-19 391-19 492-19 580 16 653 87M615 620 615 577C607 586 595 599 579 616L587 616C598 616 608 617 615 620M256 635C247 625 239 613 231 600 200 549 176 429 176 341 176 203 206 105 266 46 141 94 79 193 79 342 79 490 138 588 256 635Z"></path></g></g></g></g></svg></mjx-container> 上的光滑、交换代数，则其 Hochschild 同调（Hochschild homology）与该代数的外微分形式（differential forms）空间之间存在自然的同构：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:25.874ex"><svg style="vertical-align:-.828ex;min-width:25.874ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.787ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -865.9)"><g data-mml-node="math" data-latex="
HH_n(A) \cong \Omega_{A/\mathbb{C} }^n
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HH_n(A) \cong \Omega_{A/\mathbb{C} }^n
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preserveAspectRatio="xMaxYMid" viewBox="1278 -865.9 1 1231.9"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:2" transform="translate(0,863.9)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(2)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="32" d="M237 666C186 666 143 648 106 612 69 576 50 534 50 483 50 449 75 424 106 424 136 424 161 450 161 480 161 513 137 536 105 536 102 536 100 536 98 535 117 584 161 627 224 627 306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>这个优美的等式结合了纯代数与几何微积分。</p><p>当代数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g></g></g></svg></mjx-container> 变得非交换时，传统的微分几何工具完全失效，但左侧的<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.561ex" xmlns="http://www.w3.org/2000/svg" width="8.478ex" height="2.253ex" role="img" focusable="false" viewBox="0 -748 3747.3 996"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="HH_n(A)"><g data-mml-node="mi" data-latex="H"><path data-c="1D43B" d="M881 668C881 678 875 683 863 683L736 680 609 683C593 683 586 674 586 659 586 652 589 647 594 646 604 645 612 644 617 644 648 643 665 641 670 640 675 639 678 636 678 631 677 628 676 622 674 613L615 375 321 375 379 602C384 623 393 636 406 641 413 643 430 644 458 644 485 644 496 644 496 668 496 678 490 683 478 683L351 680 223 683C207 683 200 674 200 659 200 652 203 647 209 646 219 645 227 644 232 644 263 643 281 641 286 640 291 639 293 636 293 631 293 629 292 623 289 613L156 82C151 60 141 47 126 42 119 40 101 39 70 39 48 39 39 37 39 15 39 5 45 0 57 0L183 3 246 2C257 2 299 0 310 0 326 0 334 8 334 24 334 34 323 39 302 39 262 39 242 43 242 52 242 52 243 55 245 68L312 336 605 336 538 68C535 53 523 43 503 40 498 39 481 39 452 39 433 39 424 31 424 15 424 5 430 0 442 0L568 3 631 2C642 2 683 0 696 0 712 0 720 8 720 24 720 34 709 39 688 39 647 39 627 43 627 52 627 52 628 55 630 68L764 602C769 623 778 636 791 641 797 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720 24 720 34 709 39 688 39 647 39 627 43 627 52 627 52 628 55 630 68L764 602C769 623 778 636 791 641 797 643 814 644 843 644 869 644 881 644 881 668Z"></path></g><g data-mml-node="mi" transform="translate(864,-150) scale(0.707)" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18 427 18 420 28 420 47 420 61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g></g><g data-mml-node="mo" data-latex="(" transform="translate(2219.3,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="A" transform="translate(2608.3,0)"><path data-c="1D434" d="M137 3C155 3 217 0 236 0 251 0 258 8 258 23 258 32 252 38 239 39 210 40 196 50 196 69 196 78 205 98 224 128 251 173 270 206 283 228L526 228C526 221 527 207 530 185 537 112 541 72 541 67 541 48 519 39 474 39 455 39 446 31 446 15 446 5 452 0 464 0 488 0 564 3 588 3 608 3 679 0 699 0 714 0 721 8 721 24 721 34 712 39 694 39 666 39 649 41 644 44 639 47 635 56 634 71L574 689C571 708 572 716 551 716 539 716 529 710 522 697L178 120C148 70 108 43 59 39 43 38 35 30 35 15 35 5 41 0 52 0 68 0 121 3 137 3M492 577 522 267 307 267Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(3358.3,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> 依然有明确的代数定义。</p><p>基于此，孔涅（Alain Connes，1947-）等数学家直接将 Hochschild 同调视为非交换空间（noncommutative spaces）中的微分形式，从而正式推开了非交换几何（noncommutative geometry）的大门。</p><h3 id="三、Hochschild–Serre-谱序列（The-Hochschild–Serre-spectral-sequence）"><a href="#三、Hochschild–Serre-谱序列（The-Hochschild–Serre-spectral-sequence）" class="headerlink" title="三、Hochschild–Serre 谱序列（The Hochschild–Serre spectral sequence）"></a>三、Hochschild–Serre 谱序列（The Hochschild–Serre spectral sequence）</h3><p>20 世纪 50 年代初，Gerhard Hochschild 与塞尔（Jean-Pierre Serre，1926-）合作，提出了一种计算复杂群上同调的工具，即 Hochschild–Serre 谱序列。对于任意群<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="G"><g data-mml-node="mi" data-latex="G"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 741 705 735 701 727 692L662 619C623 676 568 705 497 705 442 705 388 692 333 667 222 615 141 533 89 421 63 366 50 310 50 253 50 93 164-22 324-22Z"></path></g></g></g></svg></mjx-container> 、其正规子群（normal subgroup）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.993ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 881 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="N"><g data-mml-node="mi" data-latex="N"><path data-c="1D441" d="M864 683C845 683 783 680 764 680 745 680 682 683 663 683 648 683 641 675 641 659 641 650 648 645 663 644 705 644 726 631 726 605 726 599 725 592 724 585L616 157 401 662C393 681 391 683 366 683L233 683C209 683 200 681 200 659 200 649 211 644 232 644 274 644 295 643 296 640L163 110C154 71 131 49 95 42 69 39 39 43 39 15 39 5 45 0 56 0 74 0 136 3 155 3 174 3 238 0 257 0 272 0 279 8 279 24 279 33 271 38 255 39 214 40 194 53 194 78 194 83 195 90 197 100L326 611 576 21C582 7 590 0 599 0 608 0 614 8 618 24L757 574C770 625 798 643 860 644 874 645 881 653 881 669 878 678 877 683 864 683Z"></path></g></g></g></svg></mjx-container> ，以及一个<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="G"><g data-mml-node="mi" data-latex="G"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 741 705 735 701 727 692L662 619C623 676 568 705 497 705 442 705 388 692 333 667 222 615 141 533 89 421 63 366 50 310 50 253 50 93 164-22 324-22Z"></path></g></g></g></svg></mjx-container> - 模（<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="G"><g data-mml-node="mi" data-latex="G"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 741 705 735 701 727 692L662 619C623 676 568 705 497 705 442 705 388 692 333 667 222 615 141 533 89 421 63 366 50 310 50 253 50 93 164-22 324-22Z"></path></g></g></g></svg></mjx-container>-module）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="2.362ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 1044 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="M"><g data-mml-node="mi" data-latex="M"><path data-c="1D440" d="M594 16C594 5 600 0 613 0L736 3C759 4 840 0 859 0 874 0 882 8 882 24 882 34 872 39 851 39 810 39 790 43 790 52 790 55 792 62 795 75L927 602C932 623 942 636 955 641 961 643 979 644 1008 644 1033 644 1044 645 1044 668 1044 678 1034 683 1013 683L882 683C853 683 853 681 840 662L484 108 408 657C404 682 402 683 373 683L237 683C215 683 204 682 204 660 204 649 215 644 236 644 256 644 297 647 297 631 297 629 296 623 293 613L167 110C158 72 135 49 100 42 75 39 42 43 42 16 42 5 48 0 60 0 78 0 141 3 159 3 178 3 242 0 261 0 276 0 283 8 283 24 283 33 276 38 261 39 219 39 198 52 198 78 198 82 199 89 202 100L332 621 415 26C418 9 424 0 434 0 442 0 450 7 459 20L848 627 712 82C706 60 696 47 681 42 675 40 656 39 625 39 604 39 594 37 594 16Z"></path></g></g></g></svg></mjx-container> ，他们构造了一个谱序列：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width:53.474ex"><svg style="vertical-align:-.704ex;min-width:53.474ex" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.54ex" role="img" focusable="false"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -811.3)"><g data-mml-node="math" data-latex="
E_2^{p,q} := H^p(G/N, H^q(N, M)) \Rightarrow H^{p+q}(G, M)
"><g data-mml-node="mtable" data-latex="
E_2 ^{p,q} := H^p(G/N, H^q(N, M)) \Rightarrow H^{p+q}(G, M)
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683Z"></path></g></g></g></svg></mjx-container> 和商群<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="4.903ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2167 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="G/N"><g data-mml-node="mi" data-latex="G"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 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d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg></mjx-container> 的统一</h3><p>Hochschild 是首批深刻认识到研究群的几何结构可以完全等价于研究群上函数环（ring of functions）的代数结构的数学家之一。</p><p>在他的经典著作《代数群与李代数基础理论》（Basic theory of algebraic groups and Lie algebras）中，他系统地将这一思想应用到了具有乘法结构的代数群（algebraic groups）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="G"><g data-mml-node="mi" data-latex="G"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 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460 152 457 138 435 59 408 19 375 19 357 19 348 33 348 61 348 77 360 131 360 147 360 204 314 239 221 253 244 269 270 292 298 322 326 352 346 371 359 382 386 404 412 415 435 415 445 415 453 413 459 409 432 404 409 379 409 353Z"></path></g></g></g></svg></mjx-container> ；群的求逆映射诱导出了对极映射（antipode）<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.459ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 645 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="S: k[G] \to k[G] "><g data-mml-node="mi" data-latex="S"><path data-c="1D446" d="M133 157C133 181 136 201 141 217 141 227 136 232 125 232 120 232 116 230 114 228 109 223 52 8 52-8 52-17 57-22 67-22 72-22 79-17 88-6L133 47C168 1 224-22 300-22 366-22 424 5 476 58 528 111 554 170 554 236 554 283 538 323 505 355 490 368 470 379 445 388 422 393 400 399 378 405L312 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transform="translate(2862.8,0)"><path data-c="5D" d="M45-250 164-250 164 750 45 750C30 750 22 742 22 726 22 710 30 702 45 702L119 702 119-202 45-202C30-202 22-210 22-226 22-242 30-250 45-250Z"></path></g></g></g></svg></mjx-container> 。</p><p>掌握了这个带有余乘法结构的交换环<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="4.215ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1863 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="k[G] "><g data-mml-node="mi" data-latex="k"><path data-c="1D458" d="M409 353C409 327 423 314 450 314 485 314 508 345 508 379 508 418 476 445 437 445 392 445 344 415 291 356 250 311 217 282 190 269L291 679C289 688 287 694 274 694 242 694 166 685 154 684 139 682 132 675 132 660 132 650 141 645 159 645 178 645 204 646 204 632L59 43C56 32 55 25 55 21 55 0 66-11 87-11 104-11 117-3 124 12 129 21 147 92 179 226 231 221 286 196 286 146 286 131 279 101 279 91 279 34 316-11 373-11 431-11 470 41 490 145 490 154 485 159 475 159 466 159 460 152 457 138 435 59 408 19 375 19 357 19 348 33 348 61 348 77 360 131 360 147 360 204 314 239 221 253 244 269 270 292 298 322 326 352 346 371 359 382 386 404 412 415 435 415 445 415 453 413 459 409 432 404 409 379 409 353Z"></path></g><g data-mml-node="mo" data-latex="[" transform="translate(521,0)"><path data-c="5B" d="M233-202 159-202 159 702 233 702C248 702 256 710 256 726 256 742 248 750 233 750L114 750 114-250 233-250C248-250 256-242 256-226 256-214 245-202 233-202Z"></path></g><g data-mml-node="mi" data-latex="G" transform="translate(799,0)"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 741 705 735 701 727 692L662 619C623 676 568 705 497 705 442 705 388 692 333 667 222 615 141 533 89 421 63 366 50 310 50 253 50 93 164-22 324-22Z"></path></g><g data-mml-node="mo" data-latex="]" transform="translate(1585,0)"><path data-c="5D" d="M45-250 164-250 164 750 45 750C30 750 22 742 22 726 22 710 30 702 45 702L119 702 119-202 45-202C30-202 22-210 22-226 22-242 30-250 45-250Z"></path></g></g></g></svg></mjx-container> ，就能反向重构出原来的代数群<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727"><g 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。</p><p>为了研究群<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="G"><g data-mml-node="mi" data-latex="G"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 741 705 735 701 727 692L662 619C623 676 568 705 497 705 442 705 388 692 333 667 222 615 141 533 89 421 63 366 50 310 50 253 50 93 164-22 324-22Z"></path></g></g></g></svg></mjx-container> 的表示，Hochschild 系统化了代表函数（representative functions）的理论。如果一个定义在群<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.05ex" xmlns="http://www.w3.org/2000/svg" width="1.778ex" height="1.645ex" role="img" focusable="false" viewBox="0 -705 786 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="G"><g data-mml-node="mi" data-latex="G"><path data-c="1D43A" d="M324-22C412-22 481 5 532 59 538 46 564 1 578 1 583 1 586 4 588 8 590 12 597 32 606 68L624 144C630 167 634 184 637 195 648 239 648 238 702 239 715 239 721 247 721 263 721 273 716 278 705 278 686 278 620 274 601 275L462 278C446 278 438 270 438 254 438 245 444 241 456 240 513 237 543 235 546 233 549 231 550 227 550 222 550 215 543 185 530 133 510 61 436 17 343 17 221 17 148 98 148 220 148 239 150 263 153 290 164 372 218 487 261 540 311 603 403 666 505 666 609 666 660 587 660 479 660 470 657 438 657 429 657 420 663 415 676 415 681 415 685 416 688 417 693 424 696 431 698 438L760 691C760 700 755 705 745 705 741 705 735 701 727 692L662 619C623 676 568 705 497 705 442 705 388 692 333 667 222 615 141 533 89 421 63 366 50 310 50 253 50 93 164-22 324-22Z"></path></g></g></g></svg></mjx-container> 上的函数<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 552 910"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg></mjx-container> ，在群元素的平移作用下产生的所有新函数能够张成一个有限维的向量空间，那么<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 552 910"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg></mjx-container> 就被称为代表函数。</p><p>他严格证明对于线性代数群（linear algebraic groups），其所有的多项式函数都是代表函数。这意味着可以用有限维线性代数的工具，精确控制和计算复杂的连续群结构。</p><p>在复数域上，Lie 代数通过指数映射（exponential map）与 Lie 群紧密相连。然而，在特征<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.439ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg></mjx-container> （characteristic<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.439ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg></mjx-container> ）的域上，由于分母中<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.767ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 781 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p! = 0"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g><g data-mml-node="mo" data-latex="!" transform="translate(503,0)"><path data-c="21" d="M139 716C110 716 83 691 86 661L122 211C123 194 129 185 139 185 149 185 155 193 156 210L192 661C195 691 168 716 139 716M192 56C192 87 169 113 139 113 109 113 86 87 86 56 86 26 109 0 139 0 169 0 192 26 192 56Z"></path></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="3.52ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1555.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p! = 0"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367 80 367C64 367 56 359 56 344 56 329 64 321 80 321L698 321C714 321 722 329 722 344 722 356 711 367 698 367M698 179 80 179C64 179 56 171 56 156 56 141 64 133 80 133L698 133C714 133 722 141 722 156 722 169 711 179 698 179Z"></path></g><g data-mml-node="mn" data-latex="0" transform="translate(1055.8,0)"><path data-c="30" d="M249-22C390-22 460 92 460 320 460 473 428 575 365 625 330 652 291 666 250 666 109 666 39 551 39 320 39 136 88-22 249-22M361 524C368 489 371 425 371 332 371 240 367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z"></path></g></g></g></svg></mjx-container> ，经典的指数映射失效。</p><p>Hochschild 基于雅各布森（Nathan Jacobson，1910-1999）提出的限制 Lie 代数（restricted Lie algebras，也称<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.439ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg></mjx-container> -Lie 代数）理论，引入带有<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.439ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg></mjx-container> - 幂映射（<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.439ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg></mjx-container> -power map）的限制 Lie 代数，用以弥补特征<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.439ex" xmlns="http://www.w3.org/2000/svg" width="1.138ex" height="1.439ex" role="img" focusable="false" viewBox="0 -442 503 636"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="p"><g data-mml-node="mi" data-latex="p"><path data-c="1D45D" d="M355 442C311 442 269 419 228 373 217 419 186 442 137 442 96 442 66 409 45 344 35 311 30 292 30 286 30 277 35 272 46 272 51 272 54 273 57 275 62 284 65 291 66 298 84 374 107 412 134 412 152 412 161 398 161 371 161 356 159 339 154 321L44-118C35-151 34-155-5-155-23-155-32-163-32-178-32-189-26-194-15-194 0-194 52-191 67-191 86-191 146-194 165-194 180-194 187-186 187-170 187-160 178-155 159-155 141-155 114-156 114-144 114-131 155 26 159 43 179 6 209-13 249-13 314-13 371 20 421 87 467 148 490 213 490 280 490 367 438 442 355 442M352 412C391 412 411 382 411 323 411 296 405 260 394 215 371 128 342 71 307 42 286 25 267 16 248 16 219 16 199 29 188 56 179 77 174 92 174 100L225 309C230 331 248 354 276 377 304 400 329 412 352 412Z"></path></g></g></g></svg></mjx-container> 域上丢失的高阶信息。他还深入研究并发展了该理论的上同调与表示论。</p><p>Hochschild 通过代表函数（representative functions）将群与其表示论紧密绑定，这启发了后续数学家仅通过表示范畴来反向定义群，为后来的淡中对偶（Tannaka Duality）与淡中范畴（Tannakian Categories）提供了重要基础。同时也在后续催生了量子群（quantum groups）理论。</p></div></div><div class="footer"></div><div class="article-meta" id="bottom"><div class="new-meta-box"></div></div><div class="prev-next"><a class="prev" 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class="name">Notes</span></a></header><div class="content"></div></section><section class="widget list group desktop mobile pjax"><header><i class="fa-duotone fa-p fa-fw" aria-hidden="true"></i> <span class="name">History of mathematics</span></header><div class="content"><ul class="list entry navigation"><li><a class="flat-box" title="/notes/person/" href="/notes/person/" active-action="action-notesperson"><div class="name"> Welcome</div></a></li><li><a class="flat-box" title="/notes/person/1" href="/notes/person/1" active-action="action-notesperson1"><div class="name"> Hilbert</div></a></li><li><a class="flat-box" title="/notes/person/2" href="/notes/person/2" active-action="action-notesperson2"><div class="name"> Thompson</div></a></li><li><a class="flat-box" title="/notes/person/3" href="/notes/person/3" active-action="action-notesperson3"><div class="name"> Conway</div></a></li><li><a class="flat-box" title="/notes/person/4" href="/notes/person/4" active-action="action-notesperson4"><div class="name"> Borcherds</div></a></li><li><a class="flat-box" title="/notes/person/5" href="/notes/person/5" active-action="action-notesperson5"><div class="name"> Wall</div></a></li><li><a class="flat-box" title="/notes/person/6" href="/notes/person/6" active-action="action-notesperson6"><div class="name"> Waldhausen</div></a></li><li><a class="flat-box" title="/notes/person/7" href="/notes/person/7" active-action="action-notesperson7"><div class="name"> Dehn</div></a></li><li><a class="flat-box" title="/notes/person/8" href="/notes/person/8" active-action="action-notesperson8"><div class="name"> Whitehead</div></a></li><li><a class="flat-box" title="/notes/person/9" href="/notes/person/9" active-action="action-notesperson9"><div class="name"> Iwaniec</div></a></li><li><a class="flat-box" title="/notes/person/10" href="/notes/person/10" active-action="action-notesperson10"><div class="name"> Sarnak</div></a></li><li><a class="flat-box" title="/notes/person/11" href="/notes/person/11" active-action="action-notesperson11"><div class="name"> Ramanujan</div></a></li><li><a class="flat-box" title="/notes/person/12" href="/notes/person/12" active-action="action-notesperson12"><div class="name"> Hardy</div></a></li><li><a class="flat-box" title="/notes/person/13" href="/notes/person/13" active-action="action-notesperson13"><div class="name"> Littlewood</div></a></li><li><a class="flat-box" title="/notes/person/14" href="/notes/person/14" active-action="action-notesperson14"><div class="name"> von Neumann</div></a></li><li><a class="flat-box" title="/notes/person/15" href="/notes/person/15" active-action="action-notesperson15"><div class="name"> Irving Segal</div></a></li><li><a class="flat-box" title="/notes/person/16" href="/notes/person/16" active-action="action-notesperson16"><div class="name"> Birkhoff</div></a></li><li><a class="flat-box" title="/notes/person/17" href="/notes/person/17" active-action="action-notesperson17"><div class="name"> Wiener</div></a></li><li><a class="flat-box" title="/notes/person/18" href="/notes/person/18" active-action="action-notesperson18"><div class="name"> Shannon</div></a></li><li><a class="flat-box" title="/notes/person/19" href="/notes/person/19" active-action="action-notesperson19"><div class="name"> Hedlund</div></a></li><li><a class="flat-box" title="/notes/person/20" href="/notes/person/20" active-action="action-notesperson20"><div class="name"> Gibbs</div></a></li><li><a class="flat-box" title="/notes/person/21" href="/notes/person/21" active-action="action-notesperson21"><div class="name"> Veblen</div></a></li><li><a class="flat-box" title="/notes/person/22" href="/notes/person/22" active-action="action-notesperson22"><div class="name"> Spanier</div></a></li><li><a class="flat-box" title="/notes/person/23" href="/notes/person/23" active-action="action-notesperson23"><div class="name"> Hochschild</div></a></li><li><a class="flat-box" title="/notes/person/24" href="/notes/person/24" active-action="action-notesperson24"><div class="name"> Erdős</div></a></li><li><a class="flat-box" title="/notes/person/25" href="/notes/person/25" active-action="action-notesperson25"><div class="name"> Pólya</div></a></li><li><a class="flat-box" title="/notes/person/26" href="/notes/person/26" active-action="action-notesperson26"><div class="name"> Atiyah</div></a></li><li><a class="flat-box" title="/notes/person/27" href="/notes/person/27" active-action="action-notesperson27"><div class="name"> Singer</div></a></li><li><a class="flat-box" title="/notes/person/28" href="/notes/person/28" active-action="action-notesperson28"><div class="name"> Grothendieck</div></a></li><li><a class="flat-box" title="/notes/person/29" href="/notes/person/29" active-action="action-notesperson29"><div class="name"> Loo-Keng Hua</div></a></li><li><a class="flat-box" title="/notes/person/30" href="/notes/person/30" active-action="action-notesperson30"><div class="name"> Poisson</div></a></li><li><a class="flat-box" title="/notes/person/31" href="/notes/person/31" active-action="action-notesperson31"><div class="name"> Dyson</div></a></li></ul></div></section><div class="widget-sticky pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div class="pjax"></div><div 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