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Andrei Roiter

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Andrei Vladimirovich Roiter (Russian: Андрей Владимирович Ройтер; Ukrainian: Андрій Володимирович Ройтер, November 30, 1937, Dnipro – July 26, 2006, Riga, Latvia) was a Ukrainian mathematician, specializing in algebra.[1]

A. V. Roiter's father was the Ukrainian physical chemist V. A. Roiter, a leading expert on catalysis.[2] In 1955 Andrei V. Roiter matriculated at Taras Shevchenko National University of Kyiv, where he met a fellow mathematics major Lyudmyla Nazarova. In 1958 he and Nazarova transferred to Saint Petersburg State University (then named Leningrad State University). They married and began a lifelong collaboration on representation theory. He received in 1960 his Diploma (M.S.) and in 1963 his Candidate of Sciences degree (PhD).[3] His PhD thesis was supervised by Dmitry Konstantinovich Faddeev,[4] who also supervised Ludmila Nazarova's PhD.[5] A. V. Roiter was hired in 1961 as a researcher at the Institute of Mathematics of the Academy of Sciences of Ukraine, where he worked until his death in 2006 and since 1991 was Head of the Department of Algebra. He received his Doctor of Sciences degree (habilitation) in 1969.[3] In 1978 he was an invited speaker at the International Congress of Mathematicians in Helsinki.[6]

In his first published paper, Roiter in 1960[7] proved an important result that eventually led several other mathematicians to establish that a finite group has finitely many non-isomorphic indecomposable integral representations if and only if, for each prime p, its Sylow p-subgroup is cyclic of order at most p2.[8][3]

In a 1966 paper[9] he proved an important theorem in the theory of the integral representation of rings.[3] In a famous 1968 paper[10] he proved the first Brauer-Thrall conjecture.[11][3]

Roiter proved the first Brauer-Thrall conjecture for finite-dimensional algebras; his paper[10] never mentioned Artin algebras, but his techniques work for Artin algebras as well. There is an important line of research inspired by the paper[10] and started by Maurice Auslander and Sverre Olaf Smalø in a 1980 paper.[12] Auslander and Smalø's paper and its follow-ups by several researchers introduced, among other things, covariantly and contravariantly finite subcategories of the category of finitely generated modules over an Artin algebra, which led to the theory of almost split sequences in subcategories.[13]

According to Auslander and Smalø:

... it is perhaps surprising that the original impetus for our work did not come from the theory of hereditary artin algebras or those stably equivalent to hereditary artin algebras. Rather, the research came from an effort to explain a much older result of Gabriel and Roiter ... concerning artin algebras of finite representation type in terms of the technics and ideas developed by Auslander and Reiten in connection with almost split sequences and irreducible morphisms ...[12]

Roiter did important research on p-adic representations,[3] especially his 1967 paper with Yuriy Drozd and Vladimir V. Kirichenko on hereditary and Bass orders[14][15][16] and the Drozd-Roiter criterion for a commutative order to have finitely many non-isomorphic indecomposable representations.[17] An important tool in this research was his theory of divisibility of modules.[18][19]

In 1972 Nazarova and Roiter[20] introduced representations of partially ordered sets, an important class of matrix problems with many applications in mathematics, such as the representation theory of finite dimensional algebras. (In 2005 they with M. N. Smirnova proved a theorem about antimonotone quadratic forms and partially ordered sets.[21]) Also in the 1970s Roiter in three papers, two of which were joint work with Mark Kleiner,[22][23][24][25] introduced representations of bocses, a very large class of matrix problems.[3]

The monograph by Roiter and P. Gabriel (with a contribution by Bernhard Keller), published by Springer in 1992 in English translation, is important for its influence on the theory of representations of finite-dimensional algebras and the theory of matrix problems.[26][3][27] There is a 1997 reprint of the English translation.[28]

In the years shortly before his death, Roiter did research on representations in Hilbert spaces.[29] In two papers,[30][31] he with his wife and Stanislav A. Kruglyak introduced the notion of locally scalar representations of quivers (i.e. directed multigraphs) in Hilbert spaces. In their 2006 paper they constructed for such representations Coxeter functors analogous to Bernstein-Gelfand-Ponomarev functors[32] and applied the new functors to the study of locally scalar representations. In particular, they proved that a graph has only finitely many indecomposable locally scalar representations (up to unitary isomorphism) if and only if it is a Dynkin graph. Their result is analogous to that of Gabriel[33] for the “usual” representations of quivers.[3]

In 1961 Roiter started in Kyiv a seminar on the theory of representations. The seminar became the foundation of the highly esteemed Kyiv school of the representation theory. He was the supervisor for 13 Candidate of Sciences degrees (PhDs). In 2007 A. V. Roiter was posthumously awarded the State Prize of Ukraine in Science and Technology for his research on representation theory.[3]

References

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  1. ^ Yakovlev, A. V. (2007). "To the memory of Andrei Vladimirovich Roiter". Journal of Mathematical Sciences. 145 (1): 4831–4835. doi:10.1007/s10958-007-0316-x. S2CID 123095732. (with list of Roiter's publications; 67 titles)
  2. ^ V. A. Roiter, Selected Works [in Russian]. Kiev: Naukova Dumka. 1976.
  3. ^ a b c d e f g h i j Drozd, Yu.; Kirichenko, V.; Krugliak, S.; Kleiner, M.; Bondarenko, V.; Ovsienko, S. (2012). "Andrei Vladimirovich Roiter. To the 75th anniversary". Algebra Discrete Math. 14 (2): C–H. "In Memory of Andrei Vladimirovich Roiter"
  4. ^ Andrei V. Roiter at the Mathematics Genealogy Project
  5. ^ Dmitry Konstantinovich Faddeev at the Mathematics Genealogy Project
  6. ^ Roiter, A. V. "Matrix problems". Proceedings of the International Congress of Mathematicians, 1978, Helsinki. Vol. 1. pp. 319–322.
  7. ^ Roiter, A. V. (1960). "On the representations of the cyclic group of fourth order by integral matrices". Vestnik Leningrad. Univ. 15: 65–74.
  8. ^ Isaacs, M.; Lichtman, A.; Passman, D.; Sehgal, S.; Sloane, N. J. A.; Zassenhaus, Hans (1989). "SD Herman's Contribution to the Theory of Integral Representations of Finite Groups by Alexander I. Lichtman". Representation Theory, Group Rings, and Coding Theory: Papers in Honor of SD Berman (1922-1987). Vol. 93. American Mathematical Soc. p. 27. ISBN 9780821850985.
  9. ^ Roiter, A. V. (1966). "Integer-valued representations belonging to one genus". Izv. Akad. Nauk SSSR Ser. Mat. 30: 1315–1324.
  10. ^ a b c "A. V. Roiter, "Unbounded dimensionality of indecomposable representations of an algebra with an infinite number of indecomposable representations", Izv. Akad. Nauk SSSR Ser. Mat., 32:6 (1968), 1275–1282; Math. USSR-Izv., 2:6 (1968), 1223–1230". {{cite journal}}: Cite journal requires |journal= (help)
  11. ^ Krause, Henning (2011). "Notes on the Gabriel-Roiter measure". arXiv:1107.2631 [math.RT].
  12. ^ a b Auslander, M.; Smalø, S. O. (1980). "Preprojective modules over Artin algebras" (PDF). Journal of Algebra. 66 (1): 61–122. doi:10.1016/0021-8693(80)90113-1. MR 0591246. (Note: the word "technic" is a jargon term sometimes used by algebraists working in Auslander–Reiten theory.)
  13. ^ Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (21 August 1997). Representation Theory of Artin Algebras. Cambridge University Press. ISBN 978-0-521-59923-8.
  14. ^ Roiter, A. V. (1966). "An analogue of the theorem of Bass for modules of representations of non-commutative orders". Dokl. Akad. Nauk SSSR. 168: 1261–1264.
  15. ^ Drozd, Y. A.; Kirichenko, V. V.; Roiter, V. A. (1967). "Hereditary and Bass orders". Izv. Akad. Nauk SSSR Ser. Mat. 31 (6): 1415–1436. Bibcode:1967IzMat...1.1357D. doi:10.1070/IM1967v001n06ABEH000625. AMS copyrighted: Drozd, Ju A.; Kiričenko, V. V.; Roĭter, A. V. (1967). "On hereditary and Bass orders". Math. USSR Izvestiya. 1 (6): 1357–1376. Bibcode:1967IzMat...1.1357D. doi:10.1070/IM1967v001n06ABEH000625.
  16. ^ Yang, Tse-Chung; Yu, Chia-Fu (2013). "Monomial, Gorenstein and Bass Orders". arXiv:1308.6017 [math.RA].
  17. ^ Drozd, Yu. A.; Roĭter, A V (1967). "Commutative rings with a finite number of indecomposable integral representations". Mathematics of the USSR-Izvestiya. 1 (4): 757–772. Bibcode:1967IzMat...1..757D. doi:10.1070/IM1967v001n04ABEH000588. ISSN 0025-5726.
  18. ^ Roiter, A. V. (1963). "Categories with division and integral representations". Dokl. Akad. Nauk SSSR. 153: 46–48.
  19. ^ Roiter, A. V. (1965). "Divisibility in the category of representations over a complete local Dedekind ring". Ukrain. Mat. J. 17 (4): 124–129.
  20. ^ Nazarova, L. A.; Roiter, A. V. (1972). "Representations of the partially ordered sets". Zapiski Nauchnykh Seminarov POMI. 28: 5–31.
  21. ^ Nazarova, L. A.; Roĭter, A. V.; Smirnova, M. N. (2006). "Antimonotone quadratic forms and partially ordered sets". St. Petersburg Mathematical Journal. 17 (6): 1015–1030. doi:10.1090/S1061-0022-06-00938-1. ISSN 1061-0022.
  22. ^ "Mark Kleiner, Professor of Mathematics". Faculty, Syracuse University.
  23. ^ Roiter, A. V.; Kleiner, M. M. (1975). "Representations of differential graded categories". Representations of algebras (Proc. Internat. Conf., Carleton Univ., Ottawa, Ont., 1974). Lecture Notes in Math. Vol. 488. Berlin: Springer. pp. 316–339.
  24. ^ Kleiner, M. M.; Roiter, A. V. (1977). "Representations of differential graded categories. (Russian)". Matrix problems (Russian). Akad. Nauk Ukrain. SSR Inst. Mat., Kiev. pp. 5–70.
  25. ^ Roiter, A. V. (1979). "Matrix problems and representations of BOCSs. (Russian)". Representations and quadratic forms (Russian). Vol. 154. Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev. pp. 3–38.
  26. ^ Gabriel, Peter; Roiter, Andrei V. (8 October 1992). Representations of Finite-Dimensional Algebras. Encyclopedia of Mathematical Sciences. Vol. 73. Springer Science & Business Media. ISBN 978-3-540-53732-8.
  27. ^ Denton, Brian H. (1993). "Reviewed work: Algebra VIII. Representations of Finite-Dimensional Algebras". The Mathematical Gazette. 77 (480): 386–387. doi:10.2307/3619799. JSTOR 3619799.
  28. ^ Gabriel, Peter; Roiter, Andrei V. (12 September 1997). Representations of Finite-Dimensional Algebras. Springer. ISBN 9783540629900.
  29. ^ Roiter, A. V.; Kruglyak, S. A.; Nazarova, L. A. (2006). "Matrix Problems in Hilbert Spaces". arXiv:math/0605728.
  30. ^ Kruglyak, S. A.; Nazarova, L. A.; Roiter, A. V. (2006). "Orthoscalar representations of quivers in the category of Hilbert spaces". Zap. Nauchn. Semin. POMI. 338: 180–201.
  31. ^ Kruglyak, S.A.; Roiter, A. V. (2005). "Locally scalar graph representations in the category of Hilbert spaces". Funkts. Anal. Prilozh. 39 (2): 13–30. Eng. translation: Kruglyak, S. A.; Roiter, A. V. (2005). "Locally Scalar Graph Representations in the Category of Hilbert Spaces". Functional Analysis and Its Applications. 39 (2): 91–105. doi:10.1007/s10688-005-0022-8. ISSN 0016-2663. S2CID 121930940.
  32. ^ Bernstein, I. N.; Gelfand, I. M.; Ponomarev, V. A. (1973). "Coxeter functors and a theorem of Gabriel". Uspekhi Mat. Nauk. 28: 19–33.
  33. ^ Gabriel, P. (1972). "Unzerlegbare Darstellungen I.". Manuscripta Math. 6: 71–103. doi:10.1007/BF01298413. S2CID 119425731.
  • Conde, Teresa (2020). "Gabriel–Roiter Measure, Representation Dimension and Rejective Chains". The Quarterly Journal of Mathematics. 71 (2): 619–635. arXiv:1903.05555. doi:10.1093/qmathj/haz062.
  • Külshammer, Julian (2016). "In the bocs seat: Quasi-hereditary algebras and representation type". arXiv:1601.03899 [math.RT].
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