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# In Memory of Earl Jay Taft (1931–2021)

Communicated by *Notices* Associate Editor Steven Sam

My first correspondence with Professor Earl J. Taft was when I was a post-doctoral visitor at the Harish-Chandra Research Institute, India, in 1999. I do not remember what exactly we corresponded about, but I do remember the thrill of receiving an answer from him for a question I had within a day. That generosity of a much renowned mathematician towards a post-doc he had never met is what I remember him the most for.

Years later, in 2006, I met him at the City University of New York (CUNY). Professor Taft regularly participated in several seminars at CUNY. In collaborative works with A. Lauve and S. Rodriguez-Romo, he constructed left Hopf algebras that are not Hopf algebras modeled after Our joint work .2 was a search for a left-Hopf algebra which is not a Hopf algebra containing While we were unsuccessful in identifying such an algebra, we were able to look at one non-example closely. Together with Professor Jonathan D.H. Smith, we were able to study the connections between one-sided Hopf algebras and one-sided quantum quasigroups. .

Working with Professor Taft never seemed like work. He talked about various things: about his childhood in New York City, about his travels all around the world, about his amazing wife, Hessy Levinsons Taft, and her family. He could read and speak several languages fluently. He could discuss opera and literature as comfortably as he could discuss mathematics.

At the risk of repeating myself, I am grateful for his kindness and generosity. What a privilege it has been to have worked with him, to have had interacted with him and his wife. My deepest condolences to his family. *Om Shanti.*

I first met Earl in the fall of 1971, when I visited Rutgers. I also met his wife Hessy and family, who were very welcoming. Of course, Earl suggested that I should look at Hopf algebras. At the time I was not interested; if an algebra had a multiplication, why did it need a comultiplication? Over the years, I would run into Earl at meetings, and again he would suggest I should look at Hopf algebras.

In 1981, I started working with Miriam Cohen on the duality between group actions on rings and rings graded by groups; suddenly the fact that we had two Hopf algebras and were looking at their duals made everything clear. Then I was hooked.

This began my 40 year interest in Hopf algebras. Later, I realized that one needed comultiplication in order to take the tensor product of modules, and having a suitable comultiplcation on an algebra means that the category of , is a tensor category. Beautiful! -modules,

So Earl was right: I should have looked at Hopf algebras earlier.

In the 70s general Hopf algebras, which are neither commutative nor cocommutative, were believed to be pathological as they did not directly arise from algebraic geometry or Lie theory. As a result, some Hopf algebra papers were published as algebraic geometry in disguise.

Drinfeld’s fundamental ICM talk in 1986 on quantum groups changed this narrative; he gave solutions to the quantum Yang Baxter equation by using the representations of certain noncommutative, noncocommutative Hopf algebras. Suddenly general Hopf algebras were of wide interest (and some of the naysayers said that in retrospect they had been too harsh). But Drinfeld made one small error; he said his quantum groups were the first examples of non-commutative non-cocommutative Hopf algebras.

But in 1971 Earl constructed infinite families of such objects. He was interested in constructing examples in which the order of the antipode could be arbitrarily large (In the commutative and cocommutative cases, it is always true that Earl’s examples have turned out to be fundamental in the classification of Hopf algebras. They are now known as Taft algebras. In a joint paper with Yevgenia Kashina and Siu-Hung Ng ).3, we proved, by computing the 2nd indicators, that the Taft algebras are completely distinguished by their representation tensor categories.

As more people worked in the area, Earl came to all of our meetings and was very encouraging.

In 1988 there was an algebra meeting at the Banach Institute in Warsaw. Earl, Hessy, and I were put up at a hotel with a nice restaurant, but the menus were only in Polish. No Polish-English dictionaries were available in Warsaw. Hessy found a Polish-Spansh dictionary, which solved our problem, as she spoke Spanish fluently. So at meals, she would translate the Polish into Spanish, and then tell us what the items were. We had a lot of fun there.

I owe him a great debt, both mathematically and personally.

Some conversations with Earl from when I became his PhD student have stuck in my mind to this day. On several occasions, he expressed to me that he would continue to work as long as he could do the job mentally and physically. Earl sent me a message in November 2016 announcing his retirement in 2017. He passed away on August 9, 2021.

Earl joined Rutgers University in 1959 after his three-year instructorship at Columbia University. I was admitted to Rutgers in 1992 and became Earl’s PhD student in 1994. Before asking Earl to be my advisor, I took a graduate abstract algebra course and a course on quantum groups that he taught. He was a serious lecturer, and I liked his style perhaps because of my cultural background. Meanwhile, Earl was a pioneer of Hopf algebras. He discovered a groundbreaking family of Hopf algebras, now called the Taft algebras. I decided to ask Earl to be my PhD supervisor after understanding some of the interplays among Hopf algebras and many other areas of mathematics and physics.

During the time when I was his graduate student, he was working on Lie bialgebras and the Hopf algebra structure of linearly recursive sequences. Since I had some background in mathematics from my master’s degree from Hong Kong, Earl showed me what he had been thinking in some of his research articles, and I began to work on his proposed problems. We met weekly and I reported what I discovered, even when he was on sabbatical at the Institute for Advanced Study.

In 12, Earl proved that for any Lie algebra if , satisfies the classical Yang Baxter Equation (CYBE), then the map , defines a Lie bialgebra structure on The pair . is called a *coboundary* Lie bialgebra. If is the Witt or Virasoro algebra, then the solutions of the CYBE of the form were classified in 12. With Earl’s guidance, I proved in our joint paper 7 that if is the one-sided Witt algebra (the Lie algebra of derivations of then every Lie bialgebra structure on ), is a coboundary Lie bialgebra given by a solution of the CYBE of the form I was thrilled by the result and grateful for Earl’s encouragement to complete the paper. .

In another paper 8 of Earl’s, the set of linearly recursive sequences over a field was proven to be a Hopf algebra isomorphic to the dual Hopf algebra of , In particular, they are closed under the convolution product; if . and are recursive sequences, then so is -linearly He asked whether they are also closed under the . product, which is similar to the convolution product but using the -convolution coefficients -binomial where is a given root of unity. We ended up answering the question affirmatively with two different approaches in 4: the first one is from the view point of dual Hopf algebras in the category of representations of a finite abelian group and the second one is a direct combinatorial method. This joint work with Earl also brought me to the representation categories of Hopf algebras, which are fundamental examples of finite tensor categories.

In the course of working on these joint articles, Earl demonstrated how I should carry on my career as a mathematician. We tested and discussed some ideas, wrote up notes on our discussions, and organized manuscripts for publication. He carefully edited all the drafts of these articles and meticulously verified the mathematics written on the drafts. In my PhD thesis, I extended the result of 6 on Lie bialgebras to those base fields of finite characteristic. I am deeply grateful for his guidance on my graduate research projects.

After finishing these collaborations, I returned to my focus on Hopf algebras. Earl recommended that I communicate with Susan Montgomery, who first coined the term “Taft algebra.” I had a few email communications with Montgomery soon after I graduated from Rutgers, and I attended some seminars at MSRI in 1999, where I was excited by an open question presented by Montgomery (cf. 4): If is a prime number, must a nonsemisimple Hopf algebra over of dimension be a Taft algebra? An affirmative answer to the question is of fundamental importance to the theory of Hopf algebras.

In Earl’s 1971 paper 11, he constructed the Hopf algebra later called the Taft algebra, for any primitive , root of unity th The underlying . of -algebra is generated by subjected to the relations , and and its coalgebra structure , is given by , , and In 1998, Andruskiewitsch and Schneider .1 proved Susan’s open question for *pointed* Hopf algebras of dimension Under Earl’s encouragement in 2002, long after my graduation, I finally settled this open question in .5 over by proving that Hopf algebras over -dimensional are pointed. I could not have done this without the continuous support of Earl. The result is not only a mathematical achievement, but also a reminder of Earl’s unique place in my heart.

Beside mathematics, Earl had many other interests, particularly in languages. He was fluent in Spanish and French. I still remember that he studied Chinese before his trip to Hong Kong and mainland China in 1997. It was apparent to me that he was entertained by learning new languages. He also immensely enjoyed the city life in New York because of the proximity to good food and Broadway theatre. He took great pleasure in traveling. Before 2010, I met Earl quite often in conferences. He appreciated wine with meals. I usually joined him for a bottle because he was always more relaxed and in good humor after a couple of glasses of wine. Accompanying Earl to a conference was always a delight for me.

In addition to doing mathematics research, Earl promoted mathematics in different communities. He was the founding editor of *Communications in Algebra* in 1974. In his condolence message, Professor Kar-ping Shum, former president of the Southeast Asian Mathematical Society, described Earl’s contributions to the Southeast Asian mathematics community. Earl initiated the first *International Congress in Algebra and Combinatorics* (ICAC) in Hong Kong in 1997 10, and proposed organizing the ICAC every ten years. He was a keynote speaker at the ICAC in 1997 and 2007, and helped edit the proceedings of the ICAC in 1997, 2007, and 2017, which were all published by World Scientific Publishing. Earl also helped develop the *Southeast Asian Bulletin of Mathematics*, which now publishes six issues per year. Professor Shum emphasized that ICAC and its proceedings have had a great impact on the mathematics community.

After I graduated from Rutgers, Earl provided his down-to-earth advice on every major decision of my career. In retrospect, Earl was my thesis supervisor, my life-long career advisor, and a great friend. I am wholeheartedly thankful to him for his unfettered support all these years and I am proud to have been his student.

I am forever grateful for the role Earl played in shaping my career. I have very fond memories of visits to Rutgers University and of my association with Earl there, as well as at numerous mathematical meetings. Earl was a gracious host, funny, thoughtful, and always considerate.

He invited me to give a talk in the Algebra Seminar at Rutgers in 1974 and invited me to spend the year there as a visitor in 1975–1976, which turned out to be mathematically very stimulating and productive. Among other things, Earl led the seminar discussion of Warren Nichols’s PhD thesis, the origin of the Nichols algebras. Together with Robert Wilson, we wrote a paper on the antipode of a Hopf algebra 9.

That year also provided a great opportunity for me to look for another job with very little stress. It was a fruitful transition. From Rutgers, I went to the University of Illinois Chicago in 1976 where I remained on the faculty for the rest of my academic career. I was a visitor again at Rutgers in 1979–1980.

Earl was the first to construct finite-dimensional Hopf algebras with antipode of a given even order 11, later called the Taft algebras. In this 1971 paper, he describes the important variation of the commutative law, namely where , is a root of unity. This variation is fundamental in his construction and the theory of quantum groups, which are important mathematical structures discovered in the 1980s.

Taft algebras are basic examples of Hopf algebras that have been studied over the years in many contexts. Quantum groups are replete with them.

## References

- [1]
- Nicolás Andruskiewitsch and Hans-Jürgen Schneider,
*Hopf algebras of order and braided Hopf algebras of order*, J. Algebra**199**(1998), no. 2, 430–454, DOI 10.1006/jabr.1997.7175. MR1489920,## Show rawAMSref

`\bib{AS}{article}{ author={Andruskiewitsch, Nicol\'{a}s}, author={Schneider, Hans-J\"{u}rgen}, title={Hopf algebras of order $p^2$ and braided Hopf algebras of order $p$}, journal={J. Algebra}, volume={199}, date={1998}, number={2}, pages={430--454}, issn={0021-8693}, review={\MR {1489920}}, doi={10.1006/jabr.1997.7175}, }`

- [2]
- Uma N. Iyer and Earl J. Taft,
*The dual of a certain left quantum group*, J. Algebra Appl.**15**(2016), no. 4, 1650072, 6, DOI 10.1142/S0219498816500729. MR3462203,## Show rawAMSref

`\bib{IT}{article}{ author={Iyer, Uma N.}, author={Taft, Earl J.}, title={The dual of a certain left quantum group}, journal={J. Algebra Appl.}, volume={15}, date={2016}, number={4}, pages={1650072, 6}, issn={0219-4988}, review={\MR {3462203}}, doi={10.1142/S0219498816500729}, }`

- [3]
- Yevgenia Kashina, Susan Montgomery, and Siu-Hung Ng,
*On the trace of the antipode and higher indicators*, Israel J. Math.**188**(2012), 57–89, DOI 10.1007/s11856-011-0092-7. MR2897723,## Show rawAMSref

`\bib{Ng}{article}{ author={Kashina, Yevgenia}, author={Montgomery, Susan}, author={Ng, Siu-Hung}, title={On the trace of the antipode and higher indicators}, journal={Israel J. Math.}, volume={188}, date={2012}, pages={57--89}, issn={0021-2172}, review={\MR {2897723}}, doi={10.1007/s11856-011-0092-7}, }`

- [4]
- Susan Montgomery,
*Classifying finite-dimensional semisimple Hopf algebras*, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997), Contemp. Math., vol. 229, Amer. Math. Soc., Providence, RI, 1998, pp. 265–279, DOI 10.1090/conm/229/03323. MR1676225,## Show rawAMSref

`\bib{NT1}{article}{ author={Montgomery, Susan}, title={Classifying finite-dimensional semisimple Hopf algebras}, conference={ title={Trends in the representation theory of finite-dimensional algebras}, address={Seattle, WA}, date={1997}, }, book={ series={Contemp. Math.}, volume={229}, publisher={Amer. Math. Soc., Providence, RI}, }, date={1998}, pages={265--279}, review={\MR {1676225}}, doi={10.1090/conm/229/03323}, }`

- [5]
- Siu-Hung Ng,
*Non-semisimple Hopf algebras of dimension*, J. Algebra**255**(2002), no. 1, 182–197. MR1935042,## Show rawAMSref

`\bib{KMN}{article}{ author={Ng, Siu-Hung}, title={Non-semisimple {H}opf algebras of dimension {$p^2$}}, date={2002}, issn={0021-8693}, journal={J. Algebra}, volume={255}, number={1}, pages={182\ndash 197}, url={https://doi-org.libezp.lib.lsu.edu/10.1016/S0021-8693(02)00139-4}, review={\MR {1935042}}, }`

- [6]
- Siu-Hung Ng and Earl J. Taft,
*Quantum convolution of linearly recursive sequences*, J. Algebra**198**(1997), no. 1, 101–119. MR1482977,## Show rawAMSref

`\bib{M}{article}{ author={Ng, Siu-Hung}, author={Taft, Earl~J.}, title={Quantum convolution of linearly recursive sequences}, date={1997}, issn={0021-8693}, journal={J. Algebra}, volume={198}, number={1}, pages={101\ndash 119}, url={https://doi-org.libezp.lib.lsu.edu/10.1006/jabr.1997.7143}, review={\MR {1482977}}, }`

- [7]
- Siu-Hung Ng and Earl J. Taft,
*Classification of the Lie bialgebra structures on the Witt and Virasoro algebras*, J. Pure Appl. Algebra**151**(2000), no. 1, 67–88. MR1770644,## Show rawAMSref

`\bib{NT2}{article}{ author={Ng, Siu-Hung}, author={Taft, Earl~J.}, title={Classification of the {L}ie bialgebra structures on the {W}itt and {V}irasoro algebras}, date={2000}, issn={0022-4049}, journal={J. Pure Appl. Algebra}, volume={151}, number={1}, pages={67\ndash 88}, url={https://doi-org.libezp.lib.lsu.edu/10.1016/S0022-4049(99)00045-6}, review={\MR {1770644}}, }`

- [8]
- Brian Peterson and Earl J. Taft,
*The Hopf algebra of linearly recursive sequences*, Aequationes Math.**20**(1980), no. 1, 1–17. MR569945,## Show rawAMSref

`\bib{T3}{article}{ author={Peterson, Brian}, author={Taft, Earl~J.}, title={The {H}opf algebra of linearly recursive sequences}, date={1980}, issn={0001-9054}, journal={Aequationes Math.}, volume={20}, number={1}, pages={1\ndash 17}, url={https://doi-org.libezp.lib.lsu.edu/10.1007/BF02190488}, review={\MR {569945}}, }`

- [9]
- David E. Radford, Earl J. Taft, and Robert Lee Wilson,
*Forms of certain Hopf algebras*, Manuscripta Math.**17**(1975), no. 4, 333–338. MR396651,## Show rawAMSref

`\bib{RTW}{article}{ author={Radford, David~E.}, author={Taft, Earl~J.}, author={Wilson, Robert~Lee}, title={Forms of certain {H}opf algebras}, date={1975}, issn={0025-2611}, journal={Manuscripta Math.}, volume={17}, number={4}, pages={333\ndash 338}, url={https://doi-org.libezp.lib.lsu.edu/10.1007/BF01170730}, review={\MR {396651}}, }`

- [10]
- Kar-Ping Shum, Earl J. Taft, and Zhe-Xian Wan (eds.),
*Algebras and combinatorics*, Springer-Verlag Singapore, Singapore, 1999. Papers from the International Congress (ICAC’97) held in Hong Kong, August 1997. MR1733162,## Show rawAMSref

`\bib{ICAC}{book}{ editor={Shum, Kar-Ping}, editor={Taft, Earl~J.}, editor={Wan, Zhe-Xian}, title={Algebras and combinatorics}, publisher={Springer-Verlag Singapore, Singapore}, date={1999}, isbn={981-4021-31-8}, note={Papers from the International Congress (ICAC'97) held in Hong Kong, August 1997}, review={\MR {1733162}}, }`

- [11]
- Earl J. Taft,
*The order of the antipode of finite-dimensional Hopf algebra*, Proc. Nat. Acad. Sci. U.S.A.**68**(1971), 2631–2633. MR286868,## Show rawAMSref

`\bib{T1}{article}{ author={Taft, Earl~J.}, title={The order of the antipode of finite-dimensional {H}opf algebra}, date={1971}, issn={0027-8424}, journal={Proc. Nat. Acad. Sci. U.S.A.}, volume={68}, pages={2631\ndash 2633}, url={https://doi-org.libezp.lib.lsu.edu/10.1073/pnas.68.11.2631}, review={\MR {286868}}, }`

- [12]
- Earl J. Taft,
*Witt and Virasoro algebras as Lie bialgebras*, J. Pure Appl. Algebra**87**(1993), no. 3, 301–312. MR1228159,## Show rawAMSref

`\bib{T2}{article}{ author={Taft, Earl~J.}, title={Witt and {V}irasoro algebras as {L}ie bialgebras}, date={1993}, issn={0022-4049}, journal={J. Pure Appl. Algebra}, volume={87}, number={3}, pages={301\ndash 312}, url={https://doi-org.libezp.lib.lsu.edu/10.1016/0022-4049(93)90116-B}, review={\MR {1228159}}, }`

## Credits

Figures 1 and 2 are courtesy of Siu-Hung Ng.

Photo of Uma N. Iyer is courtesy of Uma N. Iyer.

Photo of Susan Montgomery is courtesy of Susan Montgomery.

Photo of Siu-Hung Ng is courtesy of Ling Long.

Photo of David Radford is courtesy of David Radford.