Equality of elementary linear and symplectic orbits with respect to an alternating form. (English) Zbl 1400.14126
Summary: An elementary symplectic group w.r.t. an invertible alternating matrix is defined. It is shown that the group of symplectic transvections of a symplectic module coincides with this elementary symplectic group in the free case. Equality of orbit spaces of a unimodular element under the action of the linear group, symplectic group, and symplectic group w.r.t. an invertible alternating matrix is established.
MSC:
| 14L35 | Classical groups (algebro-geometric aspects) |
| 20G35 | Linear algebraic groups over adèles and other rings and schemes |
References:
| [1] | Bass, H., Algebraic \(K\)-Theory (1968), W.A. Benjamin, Inc. · Zbl 0174.30302 |
| [2] | Bass, H., Unitary algebraic \(K\)-theory, Lecture Notes in Math., 343, 57-265 (1973) · Zbl 0299.18005 |
| [3] | Chattopadhyay, P.; Rao, R. A., Elementary symplectic orbits and improved \(K_1\)-stability, J. K-Theory, 7, 389-403 (2011) · Zbl 1218.19001 |
| [4] | Chattopadhyay, P.; Rao, R. A., Equality of elementary and symplectic orbits, J. Pure Appl. Algebra, 219, 12, 5363-5386 (2015) · Zbl 1334.13006 |
| [5] | van der Kallen, W., A group structure on certain orbit sets of unimodular rows, J. Algebra, 82, 2, 363-397 (1983) · Zbl 0518.20035 |
| [6] | Kopeĭko, V. I., The stabilization of symplectic groups over a polynomial ring, Mathematics of the USSR. Sbornik, 34, 655-669 (1978) · Zbl 0414.20037 |
| [7] | Quillen, D., Projective modules over polynomial rings, Invent. Math., 36, 167-171 (1976) · Zbl 0337.13011 |
| [8] | Rao, R. A.; Swan, R. G., On some actions of stably elementary matrices on alternating matrices, see excerpts on the homepage of R.G. Swan at |
| [9] | Suslin, A. A.; Vaserstein, L. N., Serre’s problem on projective modules over polynomial rings and algebraic K-theory, Math. USSR, Izv., 10, 937-1001 (1976) · Zbl 0379.13009 |
| [10] | Suslin, A. A., On the structure of the special linear group over polynomial rings, Math. USSR, Izv., 11, 221-238 (1977) · Zbl 0378.13002 |
| [11] | Swan, R. G., Serre’s Problem, Queen’s Paper in Pure and Applied Mathematics, vol. 42, 1-60 (1975), Queen’s University: Queen’s University Kingston, Ontario · Zbl 0332.13002 |
| [12] | Vaserstein, L. N., On the normal subgroups of \(GL_n\) over a ring, (Algebraic \(K\)-Theory, Evanston 1980. Algebraic \(K\)-Theory, Evanston 1980, Proc. Conf., Northwestern Univ., Evanston, IL, 1980. Algebraic \(K\)-Theory, Evanston 1980. Algebraic \(K\)-Theory, Evanston 1980, Proc. Conf., Northwestern Univ., Evanston, IL, 1980, Lecture Notes in Math., vol. 854 (1981), Springer: Springer Berlin-New York), 456-465 · Zbl 0464.20030 |
| [13] | Vaserstein, L. N., Normal subgroups of symplectic groups over rings, K-Theory, 2, 647-673 (1989) · Zbl 0669.20040 |
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