James Edward Humphreys

Professor

University of Massachusetts, USA

教授

马萨诸塞大学，美国

I recall vividly my thoughts when I went to the departmental mailroom on February 15, 1979 and found a small airmail letter from Shanghai containing a message typed in Chinese along with an English translation, signed by Cao Xi Hua. It was just two weeks after the resumption of diplomatic relations between the US and the PRC, negotiated at the Jing Jiang Hotel in Shanghai by Nixon and Zhou.[ Zongzhu Lin's remark on some historic facts: Nixon and Zhou signed shanghai Communique on February 27, 1972, in Jin Jiang Hotel. U.S. China established diplomatic relation on January 1, 1979 (the few weeks Jim were referring). ] The letter invited me to visit ECNU and give some lectures there, though of course I had no idea then who Professor Cao was. I soon discovered that he was a student of Richard Brauer at U. Michigan, who had returned to China soon after the revolution. When I went to Shanghai in the spring of 1980, I met Cao and learned how influential he had been in that university, as well as how friendly he was and how helpful in my stay at a modern annex of the Jing Jiang Hotel (the university having no guesthouse at the time).

在我的脑海中往事历历在目——1979年2月15日，我去系邮递室，发现一封来自上海的航空邮件，内含由曹锡华签名一封中文信件并附英文翻译。此时距离经过尼克松和周恩来在上海的锦江饭店的谈判而使中美恢复外交关系刚刚两个星期。[ 林宗柱关于一些史实的附注：1972年2月27日，尼克松和周恩来在锦江饭店签署了《上海公报》；1979年1月1日，美国与中国建立了外交关系（就是Jim所说的几个星期）。] 这封信邀请我到华东师范大学讲学。当然我当时并不知道曹教授是谁， 但我很快弄清了他是Richard Brauer在密西根大学的学生，革命胜利后不久他便回国了。1980年春我到了上海，遇见了曹，才知道他在那所大学有多大的影响力。他又是那么友好，帮我安排住在锦江宾馆的现代化的附楼里（当时学校还没有宾馆）。

His list of students is quite impressive and should perhaps include Jian-yi Shi, who recently retired from ECNU and earlier got his Ph.D. at Warwick (UK) with Roger Carter. Cao was certainly a continuous force at ECNU, pushing students toward a modern view of algebraic groups and modular representations.

曹教授的学生名单相当令人印象深刻，也许还应该包括新近从华东师范大学退休的时俭益，他早年在英国的Warwick大学跟随Roger Carter获得博士学位。 曹在华东师范大学无疑是一个持续的推动力，促使学生走向现代意义的代数群和模表示。

Henning Haahr-Andersen

TGMRC, Yichang, China.

高级研究员

三峡数学研究中心，宜昌

Back in 1987 I was invited to East China Normal University for 5 weeks. This was the first time I met Professor Xihua Cao in person. I had already learned from previous visitors (Jim Humphreys 1980, Jens Carsten Jantzen 1984) about the emerging research group in algebra build up by Cao, and I was truly impressed by the well-prepared and very energetic audience at my lectures there.

时间回溯到1987年，我应邀到华东师范大学讲学5周。这是我第一次见到曹锡华教授本人。访问之前，我已从先前的访问者（Jim Humphreys 1980 和 Jens Carsten Jantzen 1984）那里了解了曹在代数领域所建立的一个新兴研究组。在华东师大讲课时，听众们准备充分、精力充沛，给我留下了极为深刻的印象。

It became clear to me that Professor Cao was the inspiring initiator of bringing in western research specialists in his field. He was at the time in the process of handing over to the next generation (his PhD students and postdocs) the task of developing further Chinese research groups both at ECNU and in due time at many other universities and research centers in China.

毋庸置疑，在引进其研究领域的西方研究专家上，曹教授是一位富有激情的开路者。当时他正致力于把在华东师范大学以及中国其他高校与研究中心中继续发展中国研究群体的接力棒向下一代（他的博士生和博士后）传递。

My experience during my first visit to China in 1987 and at many later visits to exciting conferences at spectacular places in China together with my contact with several of his former students (and their students) have played a big role in my own career in research. I'm very grateful to Professor Cao for backing my first visit to ECNU and for his kindness and hospitality during my stay there. At that time I considered him a rather old man (he was 30 years my senior), but I had to revise this view when I saw his abilities at the Ping-Pong table. He was able to challenge his younger colleagues and PhD-students — many of them less than half his age. In some ways the vitality he demonstrated at the Ping-Pong games illustrated for me his great visions on how to build up Chinese mathematics: work hard, keep finding new approaches, and never give up.

我1987年首访中国及以后数度到来华参加精彩的学术会议，并与曹的学生 (以及学生的学生) 有一系列接触，这些经历大大有益于我自身的研究生涯。我非常感谢曹教授支持我第一次访问华东师大，并在我的逗留期间的所表达的热情好客。当时我把他视为老者 (他长我30岁)，但他在乒乓球桌上的能力改变了我的看法。他能够挑战他的年轻同事和博士生——其中许多人的年龄还不到他的一半。在某种程度上，他在乒乓球比赛中表现出的活力向我展示了他对如何发展中国数学的伟大愿景：刻苦努力，探寻新路，永不言弃。

Now looking back at the last 40 years of amazing developments in many different parts of China in the various subfields in algebra, especially algebraic groups and quantum groups, it is easy to forget the difficulties facing the research communities in China 40 years ago. However, we should never forget those who struggled hard to overcome these difficulties. Professor Cao's energy, determination, visions and endurance should serve as a role model for all of us who over these years have benefited so much from his pioneering work.

40年来，中国的许多地方在代数学的不同领域，尤其在代数群和量子群方面取得了令人惊叹的发展，这容易让人不再记得40年前中国研究界所面临的巨大困难。然而，我们绝不应该忘记为克服那些困难而努力奋斗的人。曹教授的活力、决心、远见和耐力应该成为我们这些多年来从他的开创性工作中获益良多的人的榜样。

Leonard L. Scott, Jr.

Professor

University of Virginia, USA

教授

弗吉尼亚大学，美国

First, let me thank the organizers for the opportunity to write a few words for the conference and Professor Cao, which I am honored to do. Let me also thank them for providing me with a brief CV of Professor Cao and a list of his students. I had forgotten that Professor Cao had been a student of Brauer, or that he had such a number of distinguished students. It appears that Jian-Pan Wang was the first, sometime around 1981. I refereed at that time a paper by Wang, published in the Journal of Algebra in 1982, that resulted from his doctoral dissertation and from conversations he and Professor Cao had with Jim Humphreys. Professor Humphreys had been a visitor to ECNU for a couple of months in 1980. Wang's paper thanks both Cao and Humphreys “for many things.”

首先，非常荣幸能为此次会议和曹教授写一段文字。感谢组织者给我提供这个机会以及曹教授的简历和学生名单，让我想起曹教授是Brauer的学生，又有那么多优秀的学生。王建磐似乎是他的第一个学生，约1981年毕业。当时我是王发表在1982年《代数杂志》上的一篇论文的审稿人，此文源于他的博士论文，是他和曹教授与Jim Humphreys的交流成果。 Humphreys教授在1980年曾到华东师范大学访问过两个月。王的论文感谢了曹和Humphreys“给了他很多帮助”。

The paper by Wang was remarkably strong. It went much beyond early work of Seshadri and his collaborators on tensor products of Weyl modules, where it had been a struggle just to prove the existence of suspected Weyl module inclusions. Wang's work provided the first results of any kind showing tensor products of Weyl modules had filtrations with all sections equal to Weyl modules. It showed this always occurred in type A, and showed that it always held for other types except, possibly, for finitely many prime characteristics for any given Lie rank. I had no idea that it was a first paper, based on a dissertation of a new mathematician, or that he was the first of a new group of strong mathematicians to come from the same university under the leadership of Professor Cao.

王的论文非常优秀， 远远超出了Seshadri及其合作者之前所做的关于Weyl模张量积的工作。Seshadri等人做了很大的努力，仅仅证明了张量积中含有所期望的Weyl模。王的工作证明了对A型，Weyl模的张量积总有一个以Weyl模为截面的滤过；此结论对其他类型同样成立，但可能对给定的李秩，要排除有限个素特征。这是针对任何型提供的这方面的第一个结果。我当时不知道这是一位新晋数学家的第一篇论文，基于其学位论文，也不知道他是在曹教授的领导下出自同一所大学的一批优秀数学家中的第一位。

Wang's published paper records on its title page that it was communicated by the managing editor, Walter Feit. Of course, the referee was not mentioned, but Feit had sent the paper to me to referee. Probably, I was chosen by Feit because several papers in the paper's bibliography had been written by me with collaborators (CPS). Also, I had been Feit's PhD student at Yale in 1964-68. (1964 was the first year at Yale for both myself and Feit.) Apparently, the Wang paper had been sent to Feit by Humphreys, who had, in turn been sent the paper by Wang, asking for advice on submitting it.

王所发表的论文的标题页上写着，由主编Walter Feit递交。当然，没有提到评阅人，但是Feit把论文发给了我。我被Feit选中或许是因为论文参考书目中的几篇论文是我和合作者 (CPS) 共同完成的，也许还因为1964至1968年我是Feit在耶鲁大学的博士生（1964年，对我和Feit而言都是在耶鲁的第一个年头）。显然，王把论文寄给了Humphreys ，征求他的投稿意见，而Humphreys把稿子寄给了Feit。

I should mention that Humphreys and Feit knew each other, having had conversations at Yale in 1964-66 while Humphreys completed his dissertation on Lie p-algebras at Yale (1962-66) under Professor George Seligman. Ironically, Humphreys had become interested also in Brauer theory, a subject on which Feit was a world expert. Like Professor Cao, Feit had been a PhD student of Brauer at the University of Michigan, though Professor Robert Thrall became Feit's official advisor after Brauer left Michigan for Harvard.

我要说明一下，Humphreys和Feit彼此相识，1964年至1966年，他们在耶鲁大学有许多交流，当时 (1962年至1966年) Humphreys在George Seligman教授指导下，在耶鲁完成了关于李p-代数的学位论文。颇有戏剧性的是，汉弗莱斯还对Brauer的理论产生了兴趣，而Feit正是这一领域的世界级专家。和曹教授一样，Feit也曾是Brauer在密歇根大学的博士生，只不过当Brauer离开密歇根去哈佛后， Robert Thrall教授成为了费特的正式导师。

I first met Jian-Pan Wang (or Wang Jian-pan, as he signed his name on his paper) in person at a conference (Arcata, California) in 1986, and again in 1987 when Brian Parshall and I visited ECNU. Later, beginning in 1989 Jian-Pan visited the University of Virginia (UVA) for about a year and a half. This was the beginning of a long and productive collaboration between Jian-Pan and Brian. I also was a coauthor on one of their papers in the year 2000.

我第一次见到王建磐是在1986年的一次会议上 (加州的Arcata)，而1987年我和Brian Parshall一起访问华东师范大学时又与他再次相逢。1989年，建磐到弗吉尼亚大学访问了大约一年半的时间，这是他与Brian进行长期富有成果的合作的开始。我也是他们2000年的一篇论文的合作者。

The ties between ECNU and my University, the University of Virginia, are strong. In 1988 another of Cao's PhD students, Jie Du, began a two-year stay at UVA as my postdoctoral student. Although Professor Du is in some sense a kind of mathematical descendent of mine, we could each be viewed as having Brauer as the same mathematical grandfather (Du through Cao, and myself through Feit)! In any case, the two of us have had a long and productive collaboration, as well as Jie with Brian. In the late 90's the three of us (DPS!) began a joint collaboration that continues today. I also coauthored a paper published 2010 with Nanhua Xi, who had visited UVA earlier for five months, and is a well-known Cao student success story. I want to mention that Cao student Hebing Rui also visited the UVA for about six weeks, and I visited him briefly when he was at Shenzhen. There have been other visitors to UVA from ECNU (not Cao's students) such as Bin Shu and Li Luo, who has written papers with my colleague Weiqiang Wang. Finally, Brian Parshall and I have visited ECNU many times since 1987, and have seen the legacy of Professor Cao in the continuing strength of its mathematics programs.

华东师范大学和我所在的弗吉尼亚大学有非常紧密的联系。1988年，曹的另一名博士生杜杰作为我的博士后开始在弗吉尼亚大学进行为期两年的研究。尽管从某种意义上说，杜教授是我在数学上的后继者，但我们两人都可以把Brauer认作共同的师爷 (杜通过曹，我通过Feit)！无论如何，我与杜杰已经建立了长期而富有成效的合作，杜杰与Brian也是如此。90年代后期，我们三人 (合称为DPS!) 开展了一项合作研究，持续至今。我还与席南华合写过一篇论文，2010年发表。席在弗吉尼亚访问过五个月，曹锡华的这个学生的成功故事为众人所知。还要提及的是，曹的学生芮和兵也曾在弗吉尼亚大学待了大约六周，我也曾到深圳顺访过他。除曹的学生外，华东师范大学还有其他来访者，比如舒斌、罗栗，他们和我的同事王伟强一起合写过论文。最后，从1987年起，我和Brian Parshall多次访问华东师范大学，亲眼目睹了曹教授持续推动数学发展的传奇。

Claus Michael Ringel

Professor

Bielefeld University, Germany

教授

比勒菲尔德大学，德国

1. My first visit to China was in 1987. This was the time when I met Professor Cao XiHua. My travel to China was arranged by Professor Liu ShaoXue from BNU. I gave a series of lectures at Beijing, and the plan was that I should stay also for a week at ECNU. But I arrived at Shanghai later than expected. On the way from Beijing to Shanghai, I wanted to climb mount TaiShan, and I got stuck for 3 days at QuFu waiting to get a train ticket for the continuation (but in this way, I was at least able to see the former homes of KongZi and MengZi).

At Beijing, I had met Professor Tuan Hsio-Fu and now, at Shanghai, there was Professor Cao XiHua. It was very remarkable to see in this way that the spirit of Richard Brauer was really alive in China, and that there was a tremendous interest in the structure and representation theory of groups and algebras, despite the long isolation of the country.

1. 我第一次访问中国是在1987年。这也是我见到曹锡华教授的时间。我在中国的旅行是由北京师范大学的刘绍学教授安排的。我在北京做了一系列的讲座，并且计划在华东师范大学也待上一个星期。但我到达上海比预想的要晚几天，因为从北京到上海的路上，我想登泰山，但在曲阜为拿到火车票继续行程而等了3天（不过这样，我至少看到孔子和孟子的故居了）。

在北京，我见过段学复教授；在上海，又见到了曹锡华教授。如此一来，我看到了Richard Brauer的精神在中国依然富有活力，人们对群与代数的结构和表示理论有着极大的兴趣，尽管这个国家曾被长期孤立。这是非常令人瞩目的。

2. At that time, I had started to work on Hall polynomials trying to understand the root combinatorics of representations of Dynkin quivers. Knowing that the interest at Shanghai included questions on Lie theory, I decided to lecture here on these Hall polynomials — it was my first presentation of this topic. As I remember, there were many fruitful discussions with Professor Cao and his collaborateurs. During my stay at Shanghai, I also got some more insight into Chinese life, I went to a Shanghai Opera and saw the emerging city life of Shanghai (at that time, Beijing was still "the largest village in the world"). My second visit to Shanghai was in 1998, and the changes were very drastic: There was now the Oriental Pearl Tower, there were already many high-rises and there was the perfectly renovated Bund.

2. 当时，我开始研究Hall多项式，试图理解Dynkin quiver的组合性质。知道上海同事关注李理论的问题后，我决定在这里讲讲Hall多项式――这是我第一次做这个话题的讲演。我还记得，我与曹教授及其同事进行了许多富有成果的讨论。在上海期间，我还对中国人的生活有了更多的了解，我去了上海大剧院，看到了上海新的城市生活（当时，北京仍然是“世界上最大的村庄”）。我第二次来上海是在1998年，变化更是惊人：有东方明珠塔，有更多高楼大厦，还有改造一新的外滩。

3. What I found very impressive was that both Professor Liu and Professor Cao felt their responsibility to take care of the development of algebra in all of China, with Professor Liu surveying mainly northern China, and Professor Cao southern China. They were completely aware of young talents in all parts of China, arranging their study at Beijing or Shanghai (later also in foreign countries), and asking them to participate for a while in the mathematical life of BNU and ECNU, before they were sent them back to their home universities in order to spread their knowledge. In this way, a unified mathematical culture was emerging across the country. In addition, both (Liu and Cao) were very eager to build up international contacts and to join the world-wide endeavor of creating the mathematical basis for understanding the universe.

3. 我印象非常深刻的是，刘教授和曹教授都深感有责任推动代数在整个中国的发展，刘教授主要关注北方，而曹教授则在南方。他们完全了解中国各地的青年人才，安排他们短期到北京或上海（有的后来还到了国外），让他们在北京师范大学或者华东师范大学体验数学的生活，然后回到他们的母校传播知识。就这样，一个统一的数学文化在全国各地形成。此外，他们二人（刘和曹）都非常渴望建立国际联系，融入世界范围的努力，共同为理解世界而创造数学基础。

4. There was also a thematic division: the interests of Professor Liu were focussed more on abstract ring and module theory, whereas those of Professor Cao were on group and Lie theory. The Hall algebra approach provides a relationship in-between, using the representations of quivers to get information on the structure of semi-simple Lie algebras. It should be stressed that nearly from the beginning, Chinese mathematicians got strongly involved in this topic and made remarkable contributions. As an important land mark, let me mention just the book by Deng, Du, Parshall and Wang.

I remember with great pleasure several conferences which were organized by ECNU, with exciting lectures (and often at very famous locations, say at Kunming or Lhasa). I should also mention numerous visits to Shanghai, partially by joint invitations by SJTU and ECNU. On the other hand, many mathematicians from China, in particular, from the ECNU-group, gave lectures at Bielefeld and made us aware of the progress achieved by them.

4. 这里有一个主题的区分：刘教授的兴趣更多地聚焦在抽象的环与模的理论上，而曹教授的兴趣则在群与李理论上。Hall代数方法提供了两者之间的联系，用quiver表示来获取半单李代数结构的信息。应该强调的是，中国数学家几乎从一开始就深度介入这一课题，并做出了卓越的贡献。在这里我仅仅提一下邓（邦明）、杜（杰）、Parshall和王（建磐）合写的书，它就是一个重要的标志。

我非常兴奋地回忆起由华东师范大学牵头组织的系列会议，这些会议有精彩的报告（并且在有名的地方如昆明或拉萨等地召开）。我还必须提到多次对上海的访问，其中有些是由上海交通大学和华东师范大学联合邀请的。另一方面，许多中国数学家，特别是华东师范大学群体的数学家在Bielefeld作了演讲，使我们也了解了他们取得的进展。

5. The conference Forty Years of Algebraic Groups, Algebraic Geometry, and Representation Theory in China will provide an overview of the history and the present state of the subject, and surely will outline future directions. I am certain that this conference will be a great success. Let me send all my best wishes for the conference, and the celebration of the 100th anniversary of Professor Cao's birth, praising him for having started an exciting development.

5. “代数群、代数几何和表示理论四十年”会议将概述该学科的历史和现状，并必将勾勒出未来的方向。我确信这次会议会取得圆满成功。让我向这次会议和曹教授诞辰100周年致以最良好的祝愿，并真诚赞颂曹教授所开创的令人振奋的事业。

Vlastimil Dlab

Professor

Carleton University, Canada

教授

卡尔顿大学，加拿大

我记得和曹锡华教授的交往是源自刘绍学教授的介绍，大致是1988年1月份我接受刘绍学教授的邀请先访问北京师范大学，由他安排我访问华东师范大学和曹锡华教授，我从北京飞到上海，在华东师大大概访问了一周时间，期间做了报告。我对上海的访问印象深刻。即使离我访问华东师大已经过去了30年，关于曹锡华教授有两点我现在还能清楚的记得， 一是他对数学和年轻一代数学家教育的热爱， 二是他非常照顾我的安全。当时在北京，从北师大借来自行车，骑着几乎跑遍北京，从颐和园到天安门广场，使我对北京有更多的了解。在上海访问，曹锡华教授非常关心我的安全，提出的建议是在上海不能骑自行车，我外出时曹教授总会安排人陪伴我，虽然我在华东师大只访问了短暂一周， 我对这个地方有美好回忆，我还能记得豫园湖中的茶馆。我记得我和曹锡华教授等在华东师大校门口一起合影留念，可惜一时无法找到。2007年我因去参加在拉萨举办的ICRT4，再次途经上海，感受到中国几十年的变化。因为学术专题的不同，日后我和曹锡华教授碰面机会不多，我知道Scott他们与华东师大有更多交流。感谢纪念曹锡华教授100周年诞辰的会议能再次让我回忆起和曹锡华教授的交往。

（陈学庆根据Dlab教授的邮件和电话交谈整理翻译）