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Saint Petersburg School of the Theory of Linear Groups. I. Prehistory

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Abstract

The present survey describes the contribution of St. Petersburg mathematicians to the development of the theories of linear, classical, and algebraic groups. The first part is dedicated to the prehistory of the studies in the theory of linear groups in St. Petersburg, specifically, to the pedigree of the algebra schools created by Tartakovsky and Faddeev, and to an outline of the origin of the works by Borewicz and Suslin of the mid-1970s, which initiated systematic study in the field of classical groups and algebraic K-theory in St. Petersburg.

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Notes

  1. Here is the list of the special courses from my 1974 Diploma: “Class field theory,” “Adeles and algebraic groups,” “Commutative algebra,” “The theory of Γ-extensions.” Here, for a change, further typical special courses proposed at the time at the Chair of Algebra and Number Theory, which I attended: “Galois theory of rings,” “Integral representations,” “Modular forms,” “Quadratic fields,” “Arithmetic theory of quadratic forms,” … . Obviously, this was a professional training of a (n algebraic) number theorist, rather than that of an algebraist in any conventional sense of the word.

  2. Thus, for instance, the 1976 result by Suslin and Tulenbaev [12] on injective stability of K2 was not generalised in its full form to any other group!

  3. The Russian translations of all papers by Korkine and Zolotarev can be found in the complete works of Zolotarev [15], published in 1931–1932. These volumes are edited by B.A. Venkov, Ya.V. Uspensky and N.G. Chebotarev and commented by B.N. Delaunay, V.A. Tartakovsky, and N.I. Ahiezer. The second volume also reproduces the charming correspondence between Korkine and Zolotarev, where one can trace the genesis of these results. As a bonus, one finds there many amusing observations of general character: historical, cultural, and everyday life. Many comments on the living conditions of scientists are of interest even today: “You rightly say that there are many swindlers in Berlin, they defraud you even in the restaurants.”

  4. A depiction of the mutual arrangement of the 120 of these spheres, lying in the positive half-space, can be found, for instance, in [20] or in my papers [8, 21].

  5. “All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA).”

  6. Or 219, should we identify enantiomorphic pairs, in other words, allowing to change the orientation.

  7. Up to now the sequence https://oeis.org/A006227 only advanced to n = 5, the following value being 222  097.

  8. Without preserving the orientation, there should be 4783 crystallographic groups in \({{\mathbb{R}}^{4}}\), see [45]. As of today, the sequence https://oeis.org/A004029 advanced to n = 6, the following values being 222  018 and 28  927  915.

  9. John Conway delivered a talk thereabout at the “Groups and Geometries” conference in Bressanone 2004. I still have a pizza packaging on which he delineated the 17 wallpaper groups in orbifold notation and then all the way to his talk trained me to recognise them. Unfortunately even so I failed to remember all 219 three-dimensional Fedorov groups in this notation.

  10. The publication was solicited by A.N. Krylov, who himself the same 1936 edited Chebyshev’s course on the probability theory: “Lectures given in 1879–1880, notes by A.M. Lyapunov.”

  11. I used to relate to students that this happened during his trip to the Mathematics and Mechanics Department in Peterhof. However, in reality this happened during his trip to a dacha in Tsarskoye Selo.

  12. By the way, quite advanced for his time. It suffices to say that his gymnasium algebra course began with the definition of fields and complex numbers, and his university course—with Galois theory!

  13. I cite an assesment by Sergei Sergeevich Demidov: “Kiev University had very moderate mathematical achievements in the XIX century, but thanks to the endeavours of a remarkable representative of the St. Petersburg school D.A. Grave, who moved there in 1901, it sharply raised its mathematical level” [54]. Faddeev in [53] characterizes Grave’s mathematical school in Kiev of that period as “brilliant.” It was, by any standard!

  14. However, for that matter, Alexander Gennadievich Kurosh (1908–1971) was a student of Pavel Sergeevich Alexandrov, and Anatoly Ivanovich Maltsev (1909–1967) was a student of Andrei Nikolaevich Kolmogorov, so that the rest of Soviet Algebra ascends not to the Chebyshev St. Petersburg school, but to the Moscow school of function theory.

  15. Airplane or rocket flight, ship movement, ocean currents, weather forecast, climate prediction, etc.

  16. Percy Diaconis and Sandy Zabell write: “Despite his wide range of mathematical interests, Uspensky was first and foremost a number theorist. Of the five leading number theorists in Leningrad in the 1920s who did not leave Russia—Delone, Ivanov, Kuzmin, Venkov, and Vinogradov ([54], p. 89)—three were students of Uspensky” [58].

  17. https://www.pdmi.ras.ru/_olymp/lmo_history_1.pdf

  18. In other words, until the death of Linnik, of which I write more in [62].

  19. Later, when I started to converse a lot with colleagues from various countries, I discovered that the name of Borewicz was known to virtually everyone, not just algebraists and number theorists. First and foremost, as the author of this book.

  20. Here I deviate a little from the historical truth. Unlike D.K., whom everyone would call simply “D.K.” (“D.K. would not have done that”), in those years behind his back everyone would refer to Zenon Ivanovich merely as “Zenon” and not “Z. I.” However, this form seems inappropriate for public use, and the full spelling “Zenon Ivanovich”—too long.

  21. Of course, for an algebraist M(n, K[t]) = M(n, K)[t]. In fact, a major role in the optimal control theory belongs to the “quasi-polynomial” matrices, i.e., matrices with entries in the ring of Laurent polynomials M(n, K[t, t–1]).

  22. Various authors merchandise as the Gauß decomposition three or four completely different things. In the lingo of the computational linear algebra, it is usually something like G = \(\overline {LU} \) or G = LUP. But here it means G = ULU.

  23. A couple of years later, in 1977 it seems, D.K., who had just turned 70 at the time, arranged a seminar on Chevalley groups, where we read also the very fresh then [81, 82]. It made a huge impact on me, with what enthusiasm he, together with us, doctoral students back then, scrutinized all technical details of the construction of integral bases in universal enveloping algebras and representations. Exactly at that time, the quantum leap occurred, from the state where practically nobody in St. Petersburg understood the classification of simple Lie algebras, to the state where it suddenly became a common knowledge shared by everyone.

  24. Sasha Merkuriev [90] remembers how it happened. The point is that before that Andrei tried to prove the nonexistence of finite projective planes of order 10, but he could not do it right away—and, apropos, for quite a long time nobody could, even using a computer [90]. Then Andrei decided to prove the non-existence of non-trivial projective modules of finite rank. As far as I remember, Andrei, who was a student Mark Ivanovich Bashmakov then, learned about Serre’s problem from the lectures by Yuri Ivanovich Manin [91].

  25. Slava Kopeiko reminded me that the Serre’s problem is stated in this form in Exercise 12 on page 395 of Serge Leng’s book [94] so that Andrei could have learned about Serre’s problem also from there.

  26. Here we mean general structural theorems of a qualitative character. There are many obvious classification results and estimates that are impossible to generalise anywhere from fields, this issue is discussed in detail in our survey [95]. Thus, for example, description of conjugacy classes in GL(n, \(\mathbb{Z}\))—or even in GL(n, \(\mathbb{Z}\)/p2\(\mathbb{Z}\))—are wild problems, i.e., they are equivalent to the pairs of matrices problem. In other words, there is not and there cannot be any integer analogue of Jordan form. It has recently become clear that many finiteness theorems do not generalise beyond the class of zero-dimensional rings, or some one-dimensional rings, but this, of course, was no longer that surprising.

  27. Well, of course, again, apart from explicit isomorphisms and stabilisation theorems.

  28. Both of whom closely cooperated with Andrei, [96].

  29. Today, I understand how one could come up with such a thing, and I even more or less understand how such a calculation could be carried through. But I still can’t imagine that anyone other than Andrei could do it on mere 3–4 pages!

  30. Which has a virtually zero intersection with [98]!

  31. However, he never demonstrated his superiority. As an anecdote, I will mention that [50] re-proves Bruhat decomposition. With his understanding of mathematical reality and technical prowess, Andrei simply had no need to know such trifles. Slava Kopeiko recalls how exactly this happened. In February 1976 Andrei gave to Slava a draft of [50] and suggested to generalise the results thereof to symplectic groups. Very quickly, Slava managed to generalise all the main results, except for the injective stability of the symplectic K1-functor over polynomial rings. The problem was that in his proof of injective stability, Andrei invoked an auxiliary result, proved with the use of K2-functor and the Steinberg symbol from Milnor’s book [79], whereas the symplectic analogue of the Steinberg symbol was not yet constructed at that time. In a week after Slava told Andrei about this obstacle, Andrei brought him to the university dormitory on the Detskaya street the next draft of the same paper, with a new section “Auxiliary results,” which, among other things, contained Bruhat decomposition. This new approach is used both in the final version of [99] and in Kopeiko’s article [104].

  32. Vanya Panin recalled me that in the 1978–1979 academic year, Andrei was teaching five special courses: “Algebraic geometry,” “Homological algebra,” “Grothendieck topologies,” “Quillen’s constructions of higher K-theory,” “Computation of K‑theory of finite fields,” and in the 1979–1980 academic year he was only teaching two special courses: “The Riemann–Roch–Hirzebruch theorem in Grothendieck form” and “Etale cohomology,” but also directed three special seminars: “Abelian varieties,” “Algebraic surfaces,” and “Serre’s duality.” This is both as far as his impact, his enthusiasm and as a pendent to the first footnote, to illustrate to what extent the content of algebraic education at the St. Petersburg State University changed between the early and the late 1970s.

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Funding

The research reflected in the subsequent parts of the present survey were supported by a number of grants and research projects, of which one should especially mention (1) the RSF project 14-11-00335 “Decomposition of unipotents in reductive groups”, (2) the RSF project 17-11-01261 “Split reductive groups over rings and their relatives”, and the current ones (3) the “Basis” Foundation project 20-7-1-27-1 “Higher symbols in algebraic K-theory”, (4) the RSF project 22-21-00257 “Algebraic groups over rings and Steinberg groups”.

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    N. A. Vavilov

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Dedicated to the 100 anniversary of my teacher Zenon Ivanovich Borewicz

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Vavilov, N.A. Saint Petersburg School of the Theory of Linear Groups. I. Prehistory. Vestnik St.Petersb. Univ.Math. 56, 273–288 (2023). https://doi.org/10.1134/S106345412303010X

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