Representations of anisotropic unitary groups

James, Donald G.

doi:10.1090/S0002-9947-1988-0933318-3

Representations of anisotropic unitary groups
HTML articles powered by AMS MathViewer

by Donald G. James PDF

Trans. Amer. Math. Soc. 306 (1988), 791-804 Request permission

Abstract:

Let $SU(f)$ be the special unitary group of an anisotropic hermitian form $f$ over a field $k$. Assume $f$ represents only one norm class in $k$. The representations $\alpha : SU(f) \to SL(n, R)$ are characterized when $R$ is a commutative ring with $2$ not a zero divisor and $n = \dim f \geqslant 3$ with $n \ne 4, 6$. In particular, a partial classification of the normal subgroups of $SU(f)$ is given when $k$ is the function field ${\mathbf {C}}(X)$.

References

E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Armand Borel and Jacques Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571 (French). MR 316587, DOI 10.2307/1970833
Ludwig Bröcker, Zur orthogonalen Geometrie über pythagoreischen Körpern, J. Reine Angew. Math. 268(269) (1974), 68–77 (German). MR 387191, DOI 10.1515/crll.1974.268-269.68
Chan Nan Chang, Integral orthogonal groups over $\textbf {r}((\pi _{1}))((\pi _{2}))\ldots ((\pi _{n}))$, J. Algebra 39 (1976), no. 1, 308–327. MR 424689, DOI 10.1016/0021-8693(76)90079-X
Jean Dieudonné, On the automorphisms of the classical groups. With a supplement by Loo-Keng Hua, Mem. Amer. Math. Soc. 2 (1951), vi+122. MR 45125

La ǵeometrié des groupes classiques

Alexander J. Hahn, Donald G. James, and Boris Weisfeiler, Homomorphisms of algebraic and classical groups: a survey, Quadratic and Hermitian forms (Hamilton, Ont., 1983) CMS Conf. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 1984, pp. 249–296. MR 776458
D. G. James, Projective geometry for orthogonal groups, J. Reine Angew. Math. 319 (1980), 104–117. MR 586117, DOI 10.1515/crll.1980.319.104
Donald G. James, Homomorphisms of unitary groups, Math. Z. 178 (1981), no. 3, 343–352. MR 635203, DOI 10.1007/BF01214871
D. James, W. Waterhouse, and B. Weisfeiler, Abstract homomorphisms of algebraic groups: problems and bibliography, Comm. Algebra 9 (1981), no. 1, 95–114. MR 599074, DOI 10.1080/00927878108822565
T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR 0396410
Bernard R. McDonald, Geometric algebra over local rings, Pure and Applied Mathematics, No. 36, Marcel Dekker, Inc., New York-Basel, 1976. MR 0476639
Barth Pollak, Orthogonal groups over $R((\pi ))$, Amer. J. Math. 90 (1968), 214–230. MR 223462, DOI 10.2307/2373433
J. Tits, Homorphismes “abstraits” de groupes de Lie, Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 479–499 (French). MR 0379749
B. Weisfeiler, On abstract homomorphisms of anisotropic algebraic groups over real-closed fields, J. Algebra 60 (1979), no. 2, 485–519. MR 549942, DOI 10.1016/0021-8693(79)90095-4
B. Weisfeiler, Abstract isomorphisms of simple algebraic groups split by quadratic extensions, J. Algebra 68 (1981), no. 2, 335–368. MR 608539, DOI 10.1016/0021-8693(81)90268-4