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Algebraic K-theory, morita theory, and the classical groups

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Bibliography

  1. E. Artin, Geometric Algebra, Wiley Interscience, New York, 1957.

    MATH  Google Scholar 

  2. A. Bak, On modules with quadratic forms, pp. 55–66, in Lecture Notes in Mathematics 108, Springer Verlag, Berlin 1969.

    MATH  Google Scholar 

  3. A. Bak, K-Theory of Forms, Annals of Mathematics Studies 98, Princeton University Press, 1981.

    Google Scholar 

  4. A. Bak, Le probleme des sous-groupes de congruence et le probleme metaplectique pour les groupes classiques de rang >1, C.R. Acad. Sc. Paris, 292, 307–310 (1981).

    MathSciNet  MATH  Google Scholar 

  5. A. Bak and U. Rehann, The congruence subgroup and metaplectic problems for SLN>2 of division algebras, J. Algebra, 78 (1982), 475–547.

    Article  MathSciNet  Google Scholar 

  6. H. Bass, Algebraic K-Theory, Benjamin, New York, 1968.

    MATH  Google Scholar 

  7. H. Bass, Unitary Algebraic K-Theory, pp. 57–265, in Lecture Notes in Mathematics 343, Springer Verlag, Berlin 1973.

    MATH  Google Scholar 

  8. H. Bass, Introduction to some Methods of Algebraic K-Theory, American Mathematical Soc., Providence, R.I., 1974.

    Book  MATH  Google Scholar 

  9. H. Bass, Clifford algebras and spinor norms over a commutative ring, Amer. J. Math. 96 (1974), 156–206.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Bass, J. Milnor, and J.P. Serre, Solution of the Congruence Subgroup Problem for SLn and Sp2n. Publ. Math. IHES 33 (1967), 59–137.

    Article  MATH  Google Scholar 

  11. D. Callan, The isomorphisms of unitary groups over non-commutative domains, J. Algebra 52 (1978), 475–503.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Connolly, Linking numbers and surgery, Topology 12, 1973, 389–409.

    Article  MathSciNet  MATH  Google Scholar 

  13. V.V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math. 100 (1978), 303–386.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Dieudonne, La Geometrie des Groupes Classiques, 3rd ed., Springer Verlag, Berlin-New York, 1971.

    MATH  Google Scholar 

  15. P. Draxl, Skew Fields, Cambridge University Press, 1982.

    Google Scholar 

  16. A. Fröhlich and E.M. McEvett, Forms over rings with involution, J. Algebra, 12 (1969), 79–104.

    Article  MathSciNet  MATH  Google Scholar 

  17. I. Golubchik, On the general linear group over an associative ring, Uspekhi Mat. Nauk, 28:3 (1973), 179–180 (Russian).

    MathSciNet  Google Scholar 

  18. A. Hahn, Isomorphism theory for orthogonal groups over arbitrary integral domains, J. Algebra, 51 (1978), 233–287.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Hahn, Category equivalences and linear groups over rings, J. Algebra, 77 (1982), 505–543.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Hahn, A hermitian Morita theorem for algebras with anti-structure, J. Algebra, 93 (1985), 215–235.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Hahn and Z.X. Li, Hermitian Morita theory and hyperbolic unitary groups, to appear in J. Algebra.

    Google Scholar 

  22. A. Hahn, D. James and B. Weisfeiler, Homomorphisms of algebraic and classical groups: a survey, in Canadian Mathematical Society Conference Proceedings, Volume 4 (1984), 249–296.

    MathSciNet  MATH  Google Scholar 

  23. I. Hambleton, L. Taylor and B. Williams, An introduction to maps between surgery obstruction groups, pp. 49–127, in Lecture Notes in Mathematics 1051, Springer Verlag 1982.

    Google Scholar 

  24. J. Humphreys, Arithmetic Groups, Lecture Notes in Mathematics 789, Springer Verlag, Berlin, 1980.

    MATH  Google Scholar 

  25. D. James, W. Waterhouse and B. Weisfeiler, Abstract homomorphisms of algebraic groups: problems and bibliography, Comm. Algebra, 9 (1981), 95–114.

    Article  MathSciNet  MATH  Google Scholar 

  26. W. van der Kallen, Generators and relations in Algebraic K-Theory, pp. 305–210, in Proceedings of the International Conference of Mathematicians, Helsinki, 1978.

    Google Scholar 

  27. W. van der Kallen, Stability for K2 in Dedekind rings of arithmetic type, pp. 217–248, in Lecture Notes in Mathematics 854, Springer-Verlag, Berlin, 1980.

    Google Scholar 

  28. M. Kolster, Surjective stability for unitary K-groups, preprint, 1975.

    Google Scholar 

  29. M. Kolster, General symbols and presentation of elementary linear groups, J. für reine u. angew. Math. 353 (1984), 132–164.

    MathSciNet  MATH  Google Scholar 

  30. T.Y. Lam, The Algebraic Theory of Quadratic Forms, 2nd ed., Benjamin, New York, 1980.

    MATH  Google Scholar 

  31. K. Leung, The isomorphism theory of projective pseudo-orthogonal groups, J. Algebra, 61 (1979), 367–387.

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Mennicke, Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965), 115–129.

    Article  MathSciNet  MATH  Google Scholar 

  33. H. Matsumoto, Sur les sousgroupes arithmetiques des groupes semisimple deployes, Ann. Sci. Ecole Norm. Sup. (4) 2(1969), 1–62.

    MATH  Google Scholar 

  34. A. Merkurjev and A. Suslin, K-Cohomologies of Severi-Brauer varieties and norm residue homomorphism, Izv. Akad. Nauk SSSR, 16 (1982), 1011–1046.

    Google Scholar 

  35. J. Milnor, Introduction to algebraic K-Theory, Annals of Mathematical Studies 72, Princeton University Press, 1971.

    Google Scholar 

  36. N.M. Mustafa-Zade, On epimorphic stability of a unitary K2-functor, Russian Math. Surveys (1980), 99–100.

    Google Scholar 

  37. O.T. O'Meara, Lectures on Linear groups, Amer. Math. Society, Providence, R.I., 1974.

    Google Scholar 

  38. O.T. O'Meara, A general isomorphism theory for linear groups, J. Algebra, 44 (1977), 93–142.

    Article  MathSciNet  MATH  Google Scholar 

  39. O.T. O'Meara, Symplectic groups, Math. Surveys, Amer. Math. Soc. Providence, R.I., 1978.

    Google Scholar 

  40. O.T. O'Meara, A survey of the isomorphism theory of the classical groups, pp. 225–242, in "Ring theory and Algebra III", Dekker, New York, 1980.

    Google Scholar 

  41. W. Pender, Automorphisms and Isomorphisms of the indefinite modular classical groups, Ph.D. Thesis, Sydney University (1972).

    Google Scholar 

  42. W. Pender, Classical groups over division rings of characteristic 2, Bull. Aust. Math. Soc. 7 (1972), 191–226.

    Article  MathSciNet  MATH  Google Scholar 

  43. G. Prasad and M.S. Ragunathan, On the congruence subgroup problem: determination of the "metaplectic kernel", Invent. math. 71, (1983), 21–42.

    Article  MathSciNet  MATH  Google Scholar 

  44. H.G. Quebbemann, W. Scharlau, and M. Schulte, Quaratic and hermitian forms in additive and abelian categories, J. Algebra, 59 (1979), 264–289.

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Ranicki, The algebraic theory of surgery I, Foundations, Proc. London Math. Soc. (3) 40 (1980), 87–192.

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery, Princeton Mathematical Notes, Princeton University Press, 1981.

    Google Scholar 

  47. H.S. Ren and Z.X. Wan, Automorphisms of PSL+2 (K) over any skew field K, Acta. Math. Sinica, 25, (1982), 484–492.

    MathSciNet  MATH  Google Scholar 

  48. J.P. Serre, Trees, Springer-Verlag, Berlin, New York, 1980.

    Book  MATH  Google Scholar 

  49. R. Sharpe, On the structure of the unitary Steinberg group, Ann. Math. 96 (1972), 444–479.

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Silvester, Introduction to Algebraic K-Theory, Chapman and Hall, London, 1981.

    MATH  Google Scholar 

  51. G. Soule, K2 et le groupe de Brauer [d'apres A.S. Merkurjev et A.A. Suslin]. Seminare Bourbaki, 1982/83, No. 601 (1982).

    Google Scholar 

  52. R. Steinberg, Generateurs, relations et revetements de groups algebriques, Colloque de Bruxelles, 1962, 113–127.

    Google Scholar 

  53. R. Steinberg, Lecture Notes on Chevalley Groups, Yale University, 1967.

    Google Scholar 

  54. A. Suslin, On the structure of the special linear group over polynomial rings, Math. USSR Izvestija, Vol. II (1972), No. 2, 221–328.

    MATH  Google Scholar 

  55. A. Suslin and V. Kopeiko, Quadratic modules and the orthogonal group over polynomial rings, Zap. Naucn. Sem. Leningrad. Otdel. Math. Inst. Steklov. (LOMI) 71 (1977), 216–250.

    MathSciNet  Google Scholar 

  56. A. Suslin, Reciprocity laws and the stable rank of polynomial rings, Math. USSR Izvestija, Vol 15(1980), No. 3, 589–623.

    Article  MATH  Google Scholar 

  57. R. Swan, K-Theory of Finite Groups and Orders, Lecture Notes in Mathematics 149, Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

  58. J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics 383, Springer-Verlag, Berlin, 1974.

    MATH  Google Scholar 

  59. L. Vaserstein, Stabilization of unitary and orthogonal groups over a ring with involution, Math. USSR Sbornik, Vol. 10 (1970), 307–326.

    Article  MATH  Google Scholar 

  60. L. Vaserstein, The stabilization for classical groups over rings, Math. USSR Sbornik 22, (1974), 271–303.

    Article  MATH  Google Scholar 

  61. L. Vaserstein, Foundations of algebraic K-theory, Russian Math. Surveys, 31:4 (1976), 89–156.

    Article  MathSciNet  MATH  Google Scholar 

  62. L. Vaserstein, On the normal subgroups of GLn over a ring, pp. 456–465, in Lecture Notes in Mathematics 854, Springer-Verlag, Berlin, 1981.

    Google Scholar 

  63. L. Vaserstein, On full subgroups in the sense of O'Meara, J. Algebra, 75 (1982), 437–44.

    Article  MathSciNet  MATH  Google Scholar 

  64. L. Vaserstein, Classical groups over rings, in Canadian Mathematical Society Conference Proceedings, Volume 4 (1984).

    Google Scholar 

  65. L. Vaserstein and A. Suslin, Serre's problem on projective modules over polynomial rings and algebraic K-theory, Math. USSR Izvestija, Vol. 10 (1976), No. 5, 937–1001.

    Article  MathSciNet  MATH  Google Scholar 

  66. A. Wadsworth, Merkurjev's elementary proof of Merkurjev's theorem, Boulder Conference in Algebraic K-theory, to appear.

    Google Scholar 

  67. C.T.C. Wall, Surgery on Compact Manifolds, Academic Press, 1970.

    Google Scholar 

  68. C.T.C. Wall, On the axiomatic foundation of the theory of Hermitian forms, Proc. Camb. Phil. Soc., 67 (1970), 243–250.

    Article  MathSciNet  MATH  Google Scholar 

  69. C.T.C. Wall, Foundations of algebraic L-Theory, pp. 266–300, in Lecture Notes in Mathematics 343, Springer Verlag, Berlin, 1973.

    MATH  Google Scholar 

  70. C.T.C. Wall, On the classification of Hermitian Forms III, semisiple rings, Invent. Math., 18 (1972), 119–141.

    Article  MathSciNet  MATH  Google Scholar 

  71. G.E. Wall, The Structure of a unitary factor group, Publ. Math., IHES, No. 1, (1959), 7–23.

    Google Scholar 

  72. Z.X. Wan, The Classical Groups, Shanghai University Press, 1981.

    Google Scholar 

  73. Z.X. Wan and J.G. Yang, Automorphisms of the projective quaternion unimodular group in dimension 2, Chinese Annals of Math., 3(1982), 395–402.

    MATH  Google Scholar 

  74. B. Weisfeiler, Abstract homomorphisms of big subgroups of algebraic groups, pp. 135–181, in Topics in the theory of Algebraic Groups, Notre Dame Mathematical Lectures, No. 10, University of Notre Dame Press, 1982.

    Google Scholar 

  75. J.S. Wilson, The normal and subnormal structure of general linear groups, Proc. Camb. Phil. Soc. 71(1972), 163–177.

    Article  MathSciNet  MATH  Google Scholar 

  76. Zalesky, Linear groups, Russian Math. Surveys, 36, No. 5, (1981), 63–128.

    Google Scholar 

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Hsio-Fu Tuan

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Hahn, A.J. (1986). Algebraic K-theory, morita theory, and the classical groups. In: Tuan, HF. (eds) Group Theory, Beijing 1984. Lecture Notes in Mathematics, vol 1185. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076172

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