Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

Long root tori in Chevalley groups


Authors: N. A. Vavilov and A. A. Semenov
Translated by: N. A. Vavilov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 387-430
MSC (2010): Primary 20G15, 20G40, 20G35
DOI: https://doi.org/10.1090/S1061-0022-2013-01245-3
Published electronically: March 21, 2013
MathSciNet review: 3014127
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study some remarkable semisimple elements of an (extended) Chevalley group that are diagonalizable over the ground field -- the `weight elements'. In particular, we calculate the Bruhat decomposition of microweight elements. Results of the present paper are crucial for the description of overgroups of split maximal tori in Chevalley groups.


References [Enhancements On Off] (What's this?)

  • 1. A. Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (Inst. Adv. Study, Princeton, NJ, 1968/1969), Lecture Notes in Math., vol. 131, Springer-Verlag, Berlin-New York, 1970, pp. 1-55. MR 0258838 (41:3484)
  • 2. N. Bourbaki, Lie groups and Lie algebras. Ch. 4-6, Springer-Verlag, Berlin, 2002. MR 1890629 (2003a:17001)
  • 3. -, Lie groups and Lie algebras. Ch. 7-9, Springer-Verlag, Berlin, 2005. MR 2109105 (2005h:17001)
  • 4. N. A. Vavilov, Subgroups of the special linear group which contain the group of diagonal matrices. I-V, Vestnik Leningrad. Univ. Ser. 1 1985, vyp. 4, 3-7; 1986, vyp. 1, 10-15; 1987, vyp. 2, 3-8; 1988, vyp. 3, 10-15; 1993, vyp. 2, 10-15; English transl. in Vestnik Leningrad Univ. Math. 18 (1985), no. 4; 19 (1986), no. 1; 20 (1987), no. 2; 21 (1988), no. 3, 7-15; 26 (1993), no. 2, 6-9. MR 0827955 (87e:20081); MR 0841489 (87j:20074); MR 0926257 (89b:20097); MR 0974786 (90a:20088); MR 1370226 (96k:20089)
  • 5. -, Bruhat decomposition of one-dimensional transformations, Vestnik Leningrad. Univ. Ser. 1 1986, vyp. 3, 14-20; English transl. in Vestnik Leningrad Univ. Math. 19 (1986), no. 3. MR 0867389 (88g:20094)
  • 6. -, Weight elements of Chevalley groups, Dokl. Akad. Nauk SSSR 298 (1988), no. 3, 524-527; English transl., Soviet Math. Dokl. 37 (1988), no. 1, 92-95. MR 0925952 (88m:20093)
  • 7. -, Conjugacy theorems for subgroups of extended Chevalley groups that contain split maximal tori, Dokl. Akad. Nauk SSSR 299 (1988), no. 2, 269-272; English transl., Soviet Math. Dokl. 37 (1988), no. 2, 360-363. MR 0943230 (89e:20081)
  • 8. -, Bruhat decomposition for long root semisimple elements in Chevalley groups, Rings and Modules. Limit Theorems of Probability Theory, No. 2, Leningrad. Univ., Leningrad, 1988, pp. 18-39. (Russian) MR 0974130 (89k:20057)
  • 9. -, Bruhat decomposition of two-dimensional transformations, Vestnik Leningrad. Univ. Ser. 1 1989, vyp. 3, 3-7; English transl., Vestnik Leningrad Univ. Math. 22 (1989), no. 3, 1-6. MR 1055331 (91c:20057)
  • 10. -, Root semisimple elements and triples of root unipotent subgroups in Chevalley groups, Problems in Algebra, No. 4, Universitetskoe, Minsk, 1989, pp. 162-173. (Russian) MR 1011925 (90i:20046)
  • 11. -, Subgroups of Chevalley groups containing a maximal torus, Trudy Leningrad. Mat. Obshch. 1 (1990), 64-109; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 155, Amer. Math. Soc., Providence, RI, 1993, pp. 59-100. MR 1104207 (92f:20046)
  • 12. -, Unipotent elements in subgroups of extended Chevalley groups that contain a split maximal torus, Dokl. Akad. Nauk 328 (1993), no. 5, 536-539; English transl., Russian Acad. Sci. Dokl. Math. 47 (1993), no. 1, 112-116. MR 1218959 (94k:20083)
  • 13. -, Subgroups of the group $ \mathrm {SL}_n$ over a semilocal ring, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 343 (2007), 33-53; English transl., J. Math. Sci. (N. Y.) 147 (2007), no. 5, 6995-7004. MR 2469412 (2009k:20112)
  • 14. -, Geometry of $ 1$-tori in $ \mathrm {GL}_n$, Algebra i Analiz 19 (2007), no. 3, 119-150; English transl., St. Petersburg Math. J. 19 (2008), no. 3, 407-429. MR 2340708 (2008g:20115)
  • 15. -, Can one see the signs of structure constants? Algebra i Analiz 19 (2007), no. 4, 34-68; English transl., St. Petersburg Math. J. 19 (2008), no. 4, 519-543. MR 2381932 (2009b:20087)
  • 16. -, On subgroups of a symplectic group containing a subsystem subgroup, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 349 (2007), 5-29; English transl., J. Math. Sci. (N. Y.) 151 (2008), no. 3, 2937-2948. MR 2742852 (2011k:20103)
  • 17. -, Weight elements of Chevalley groups, Algebra i Analiz 20 (2008), no. 1, 34-85; English transl., St. Petersburg Math. J. 20 (2009), no. 1, 23-57. MR 2411968 (2009c:20086)
  • 18. N. A. Vavilov and E. V. Dybkova, Subgroups of the general symplectic group containing the group of diagonal matrices. I, II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 103 (1980), 31-47; 132 (1983), 44-56; English transl. in J. Soviet Math. 24 (1984), no. 4; 30 (1985), no. 1. MR 0618492 (82h:20054); MR 0717571 (85i:20049)
  • 19. N. A. Vavilov and M. Yu. Mitrofanov, Intersection of two Bruhat cells, Dokl. Akad. Nauk 377 (2001), no. 1, 7-10; English transl., Dokl. Math. 63 (2001), no. 2, 149-151. MR 1833978 (2003b:20065)
  • 20. N. A. Vavilov and V. V. Nesterov, Geometry of microweight tori, Vladikavkaz. Mat. Zh. 10 (2008), no 1, 10-23. (Russian) MR 2434648 (2009g:20105)
  • 21. -, Geometry of $ 2$-tori in $ \mathrm {GL}_n$ (to appear).
  • 22. -, Geometry of $ \varpi _1$-tori in $ \mathrm {SO}_{2l}$ (to appear).
  • 23. -, Pairs of microweight tori in Chevalley group of type $ \mathrm E_6$ (to appear).
  • 24. -, Pairs of microweight tori in Chevalley group of type $ \mathrm E_7$ (to appear).
  • 25. -, Pairs of long root tori in Chevalley groups (to appear).
  • 26. N. A. Vavilov and I. M. Pevzner, Triples of long root subgroups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 343 (2007), 54-83; English transl., J. Math. Sci. (N. Y.) 147 (2007), no. 5, 7005-7020. MR 2469413 (2009j:20067)
  • 27. N. A. Vavilov and A. A. Semenov, Bruhat decomposition for long root tori in Chevalley groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 175 (1989), 12-23; English transl., J. Soviet Math. 57 (1991), no. 6, 3453-3458. MR 1047233 (91b:20060)
  • 28. -, Long root semisimple elements in Chevalley groups, Dokl. Akad. Nauk 338 (1994), no. 6, 725-727; English transl., Russian Acad. Sci. Dokl. Math. 50 (1995), no. 2, 325-329. MR 1311310 (96c:20084)
  • 29. E. V. Dybkova, Form nets and the lattice of overdiagonal subgroups of the symplectic group over a field of characteristic two, Algebra i Analiz 10 (1998), no. 4, 113-129; English transl., St. Petersburg Math. J. 10 (1999), no. 4, 651-661. MR 1654075 (2000a:20107)
  • 30. -, Subgroups of hyperbolic unipotent groups, Dokt. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 2006, c. 1-182. (Russian)
  • 31. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. (N. S.) 30 (72) (1952), no 2, 349-462. (Russian) MR 0047629 (13:904c)
  • 32. A. E. Zalesskiĭ, Semisimple root elements of algebraic groups, Preprint no. 13, Inst. Mat. Akad. Nauk BSSR, Minsk, 1980, pp. 1-24. (Russian)
  • 33. -, Linear groups, Itogi Nauki i Tekhniki. Algebra. Topology. Geometry, vol. 21, VINITI, Moscow, 1983, pp. 135-182; English transl., J. Soviet Math. 31 (1985), no. 3, 2974-3004. MR 0724616 (86b:20051)
  • 34. V. V. Kashin, Orbits of an adjoint and co-adjoint action of Borel subgroups of a semisimple algebraic group, Problems in Group Theory and Homological Algebra, Yaroslav. Gos. Univ., Yaroslavl', 1990, pp. 141-158. (Russian) MR 1169975 (93m:20058)
  • 35. A. S. Kondrat'ev, Subgroups of finite Chevalley groups, Uspekhi Mat. Nauk 41 (1986), no. 1, 57-96; English transl., Russian Math. Surveys 41 (1986), no. 1, 65-118. MR 0832410 (87g:20078)
  • 36. V. V. Nesterov, Pairs of short root subgroups in Chevalley group, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 1995, pp. 1-72. (Russian)
  • 37. -, Pairs of short root subgroups in Chevalley group, Dokl. Akad. Nauk 357 (1997), no. 3, 302-305; English transl., Dokl. Math. 56 (1997), no. 3, 870-872. MR 1606437 (2000c:20068)
  • 38. -, Arrangement of long and short root subgroups in a Chevalley group of type $ \mathrm G_2$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), 273-285; English transl., J. Math. Sci. (N. Y.) 116 (2003), no. 1, 3035-3041. MR 1811807 (2002m:20080)
  • 39. -, Pairs of short root subgroups in a Chevalley group of type $ \mathrm G_2$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), 253-273; English transl., J. Math. Sci. (N. Y.) 120 (2004), no. 4, 1630-1641. MR 1875729 (2002i:20024)
  • 40. -, Generation of pairs of short root subgroups in a Chevalley group, Algebra i Analiz 16 (2004), no. 6, 172-208; English transl., St. Petersburg Math. J. 16 (2005), no. 6, 1051-1077. MR 2117453 (2006b:20070)
  • 41. I. M. Pevzner, The geometry of root elements in groups of type $ \mathrm E_6$, Algebra i Analiz 23 (2011), no. 3, 261-309; English transl., St. Petersburg Math. J. 23 (2012), no. 3, 603-635. MR 2896171
  • 42. -, The width of groups of type $ \mathrm E_6$ with respect to the set of root elements. I, Algebra i Analiz 23 (2011), no. 5, 155-198; English transl. in St. Petersburg Math. J. 23 (2012), no. 5. MR 2918427
  • 43. -, The width of groups of type $ \mathrm E_6$ with respect to the set of root elements. II, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), 242-264; English transl., J. Math. Sci. (N. Y.) 180 (2012), no. 3, 338-350. MR 2784137 (2012f:20144)
  • 44. A. A. Semenov, Bruhat decomposition of root semisimple subgroups in special linear group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160 (1987), 239-246; English transl., J. Soviet Math. 52 (1990), no. 3, 3178-3185. MR 0906861 (88g:20093)
  • 45. -, Bruhat decomposition of long root tori in Chevalley groups, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 1991, pp. 1-143. (Russian)
  • 46. T. A. Springer and R. Steinberg, Conjugacy classes, 1970 Seminar on Algebraic Groups and Related Finite Groups (Inst. Adv. Study, Princeton, NJ, 1968/1969), Lecture Notes in Math., vol. 131, Springer, Berlin, 1970, pp. 167-266. MR 0268192 (42:3091)
  • 47. R. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, Conn., 1968. MR 0466335 (57:6215)
  • 48. J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., No. 9, Springer-Verlag, New York-Berlin, 1978. MR 0499562 (81b:17007)
  • 49. C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. (2) 7 (1955), 14-66. MR 0073602 (17:457c)
  • 50. H. Azad, M. Barry, G. M. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), 551-562. MR 1047327 (91d:20048)
  • 51. M. Brion, Représentations exceptionnelles des groupes semi-simples, Ann. Sci. École Norm. Sup. (4) 18 (1985), 345-387. MR 0816368 (87e:14043)
  • 52. H. Bürgstein, W. H. Hesselink, Algorithmic orbit classification for some Borel group actions, Compositio Math. 61 (1987), no. 1, 3-41. MR 0879187 (88k:20069)
  • 53. N. Cantarini, G. Carnovale, M. Costantini, Spherical orbits and representations of $ {\mathcal U}_{\varepsilon }({\mathfrak{g}})$, Transform. Groups 10 (2005), no. 1, 29-62. MR 2127340 (2005m:17020)
  • 54. G. Carnovale, Spherical conjugacy classes and the Bruhat decomposition, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2329-2357. MR 2640922 (2011c:20094)
  • 55. R. W. Carter, Simple groups of Lie type, 2nd ed., John Wiley and Sons, Inc., London etc., 1989. MR 1013112 (90g:20001)
  • 56. -, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), no. 1, 1-59. MR 0318337 (47:6884)
  • 57. Chan Kei Yuen, Lu Jiang-Hua, To Kai-Ming, On intersections of conjugacy classes and Bruhat cells, Transform. Groups 15 (2010), no. 2, 243-260. MR 2657442 (2011e:20068)
  • 58. J.-L. Clerc, Special prehomogeneous vector spaces associated to $ \mathrm F_4$, $ \mathrm E_6$, $ \mathrm E_7$, $ \mathrm E_8$ and simple Jordan algebras of rank $ 3$, J. Algebra 264 (2003), no. 1, 98-128. MR 1980688 (2004h:17008)
  • 59. A. M. Cohen, H. Cuypers, H. Sterk, Linear groups generated by reflection tori, Canad. J. Math. 51 (1999), no. 6, 1149-1174. MR 1756876 (2001e:20038)
  • 60. M. Costantini, On the coordinate ring of spherical conjugacy classes, Math. Z. 264 (2010), 327-359. MR 2574980 (2010k:20071)
  • 61. E. W. Ellers, N. L. Gordeev, Intersection of conjugacy classes with Bruhat cells in Chevalley groups, Pacific J. Math. 214 (2004), no. 2, 245-261. MR 2042932 (2004m:20091)
  • 62. -, Intersection of conjugacy classes with Bruhat cells in Chevalley groups: the cases $ \mathrm {SL}_n(K)$, $ \mathrm {GL}_n(K)$, J. Pure Appl. Algebra 209 (2007), 703-723. MR 2298850 (2007m:20071)
  • 63. -, Big and small elements in Chevalley groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), 203-296; English transl., J. Math. Sci. (N. Y.) 180 (2012), no. 3, 315-329. MR 2784135 (2012c:20126)
  • 64. A. L. Harebov, N. A. Vavilov, On the lattice of subgroups of Chevalley groups containing a split maximal torus, Comm. Algebra 24 (1996), no. 1, 109-133. MR 1370526 (97a:20077)
  • 65. S. Haris, Some irreducible representations of exceptional algebraic groups, Amer. J. Math. 93 (1971), no. 1, 75-106. MR 0279103 (43:4829)
  • 66. W. Hesselink, A classification of the nilpotent triangular matrices, Compositio Math. 55 (1985), no. 1, 89-133. MR 0791648 (87a:20049)
  • 67. N. Kawanaka, Unipotent elements and characters of finite Chevalley groups, Osaka J. Math. 12 (1975), no. 2, 523-554. MR 0384914 (52:5784)
  • 68. S. Krutelevich, On a canonical form of a $ 3\times 3$ Hermitian matrix over the ring of integral split octonions, J. Algebra 253 (2002), no. 2, 276-295. MR 1929190 (2003g:17044)
  • 69. -, Jordan algebras, exceptional groups, and Bhargava composition, J. Algebra 314 (2007), 924-977. MR 2344592 (2008k:20103)
  • 70. M. W. Liebeck, G. M. Seitz, Subgroups generated by root elements in groups of Lie type, Ann. of Math. (2) 139 (1994), no. 2, 293-361. MR 1274094 (95d:20078)
  • 71. -, Subgroups of simple algebraic groups containing elements of fundamental subgroups, Math. Proc. Cambridge Philos. Soc. 126 (1999), 461-479. MR 1684243 (2000d:20064)
  • 72. G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes. I-III arXiv:1003. 0412v5 [math.RT] (23 Aug. 2010), 1-41; arXiv:1104.0196v2 [math.RT] (30 Apr. 2011), 1-26; arXiv:1104.3112v1 [math.RT] (15 Apr. 2011), 1-22.
  • 73. -On $ C$-small conjugacy classes in a reductive group, arXiv:1005.4313v2 [math.RT] (10 Dec. 2010), 1-19.
  • 74. -Bruhat decomposition and applications, arXiv:1006.5004v1 [math.RT] (25 Jun. 2010), 1-4.
  • 75. -From unipotent classes to conjugacy classes in the Weyl group, arXiv:1008.2692v1 [math.RT] (16 Aug. 2010), 1-10.
  • 76. J. G. M. Mars, Les nombres de Tamagawa de certains groupes exceptionnels, Bull. Soc. Math. France 94 (1966), 97-140. MR 0213363 (35:4227)
  • 77. H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm.Sup.(4) 2 (1969), 1-62. MR 0240214 (39:1566)
  • 78. E. B. Plotkin, A. A. Semenov, N. A. Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), 61-95. MR 1492062 (98m:17010)
  • 79. R. Richardson, G. E. Röhrle, R. Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649-671. MR 1189494 (93j:20092)
  • 80. G. E. Röhrle, Orbits in internal Chevalley modules, Groups, Combinatorics and Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 311-315. MR 1200268
  • 81. -, On the structure of parabolic subgroups in algebraic groups, J. Algebra 157 (1993), no. 1, 80-115. MR 1219660 (94d:20053)
  • 82. -, On extraspecial parabolic subgroups, Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 143-155. MR 1247502 (94k:20082)
  • 83. A. A. Semenov, N. A. Vavilov, Halbeinfache Wurzelelemente in Chevalley-Gruppen, Preprint Universität Bielefeld, 1994, no. 2, 1-11.
  • 84. T. A. Springer, Linear algebraic groups, 2nd ed., Progr. in Math., vol. 9, Birkhäuser, Boston, 1981. MR 0632835 (84i:20002)
  • 85. M. R. Stein, Stability theorems for $ \mathrm K_{1}$, $ \mathrm K_{2}$ and related functors modeled on Chevalley groups, Japan J. Math. (N.S.) 4 (1978), no. 1, 77-108. MR 0528869 (81c:20031)
  • 86. R. Steinberg, Regular elements of semi-simple algebraic groups, Inst. Hautes Études Sci. Publ. Math. No 25 (1965), 49-80. MR 0180554 (31:4788)
  • 87. N. A. Vavilov, Structure of Chevalley groups over commutative rings, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., Singapore etc., 1991, pp. 219-335. MR 1150262 (92k:20090)
  • 88. -, Intermediate subgroups in Chevalley groups, Groups of Lie Type and their Geometries (Como, 1993), London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 233-280. MR 1320525 (96c:20085)
  • 89. -, Unipotent elements in subgroups which contain a split maximal torus, J. Algebra 176 (1995), 356-367. MR 1351614 (96i:20060)
  • 90. -, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201-250. MR 1809357 (2001i:20099)
  • 91. N. A. Vavilov, E. B. Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), 73-113. MR 1409655 (97h:20056)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 20G15, 20G40, 20G35

Retrieve articles in all journals with MSC (2010): 20G15, 20G40, 20G35


Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: nikolai-vavilov@yandex.ru

A. A. Semenov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: semenov@math.spbu.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01245-3
Keywords: Chevalley groups, semisimple root elements, Bruhat decomposition, Borel orbits, parabolic subgroups with extraspecial unipotent radical
Received by editor(s): September 9, 2011
Published electronically: March 21, 2013
Additional Notes: For the first author, the main motivation to finalize the present paper came from the research within the framework of the RFBR project 10-01-90016 “The study of the structure of forms of reductive groups, and behavior of small unipotent elements in representations of algebraic groups” (SPbGU). Apart from that, at the final stage his work was supported by the RFBR projects 09-01-00762 (Siberian Federal University), 09-01-00784 (POMI RAS), 09-01-00878 (SPbGU), 09-01-91333 (POMI RAS), 09-01-90304 (SPbGU), 10-01-92651 (SPbGU), and 11-01-00756 (RGPU). The work of both authors was supported by the Presidential Grant NSh-5282.2010.1 “Motives, cohomologies, algebraic groups, representations, reciprocity laws, lower and upper bounds of scheme complexity of Boolean functions” and by the State Financed Research Task 6.38.74.2011 at the Saint Petersburg State University “Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic $K$-theory”
Dedicated: To Nikolaĭ Gordeev, a remarkable mathematician, a dear friend, and a generous colleague
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society