Homomorphisms of algebraic and classical groups: A survey.

*(English)*Zbl 0547.20036
Quadratic and Hermitian forms, Conf. Hamilton/Ont. 1983, CMS Conf. Proc. 4, 249-296 (1984).

[For the entire collection see Zbl 0542.00004.]

From the authors’ introduction: ”We have attempted to survey the whole area of homomorphisms and isomorphisms between algebraic, classical and Lie groups, and their subgroups, with particular emphasis on the last fifteen years during which there has been great advancement in many directions. One striking feature is that the final results almost always have essentially the same formulation, namely, the group isomorphism is constructed from an isomorphism \(\phi\) of the underlying fields or rings, an \(\phi\) -semialgebraic isomorphism and a radial homomorphism. For homomorphisms of groups over fields, the kernel is central for isotropic groups and a congruence subgroup with respect to an induced integral structure for anisotropic groups. The usual underlying idea is to characterize in group terms certain kinds of subgroups. One way to proceed is to use generators and relations to identify the given isomorphism with a ”standard” one. Alternatively, and this is now more common, one identifies the subgroups in question with some geometry and then uses the fundamental theorem of projective geometry, or a suitable analogue, to obtain a description of the isomorphism.”

The contents of the article is following. 1. Introduction. 2. Matrix methods and involutions. 3. Isomorphisms of full subgroups and their geometries (Tits geometry, non-Tits geometry). 4. The results of Borel and Tits. 5. Homomorphisms between Lie groups. 6. Isomorphisms of lattices: strong rigidity. 7. The results of Margulis. 8. Homomorphisms of anisotropic groups. 9. Automorphisms of classical groups over commutative rings. 10. Isomorphisms of classical groups over general rings. 11. Automorphisms of \(GL_ 2\) and \(SL_ 2\). 12. Exceptional groups and associated geometries. 13. Miscellaneous results (Aut \(\Omega {}_ 3(f,k)\), Aut \(\prod GL_{n_ i}\), Aut S\(L_ 3(R))\). 14. Related areas (geometry, arithmeticity, normal subgroups). Bibliography (174 items).

From the authors’ introduction: ”We have attempted to survey the whole area of homomorphisms and isomorphisms between algebraic, classical and Lie groups, and their subgroups, with particular emphasis on the last fifteen years during which there has been great advancement in many directions. One striking feature is that the final results almost always have essentially the same formulation, namely, the group isomorphism is constructed from an isomorphism \(\phi\) of the underlying fields or rings, an \(\phi\) -semialgebraic isomorphism and a radial homomorphism. For homomorphisms of groups over fields, the kernel is central for isotropic groups and a congruence subgroup with respect to an induced integral structure for anisotropic groups. The usual underlying idea is to characterize in group terms certain kinds of subgroups. One way to proceed is to use generators and relations to identify the given isomorphism with a ”standard” one. Alternatively, and this is now more common, one identifies the subgroups in question with some geometry and then uses the fundamental theorem of projective geometry, or a suitable analogue, to obtain a description of the isomorphism.”

The contents of the article is following. 1. Introduction. 2. Matrix methods and involutions. 3. Isomorphisms of full subgroups and their geometries (Tits geometry, non-Tits geometry). 4. The results of Borel and Tits. 5. Homomorphisms between Lie groups. 6. Isomorphisms of lattices: strong rigidity. 7. The results of Margulis. 8. Homomorphisms of anisotropic groups. 9. Automorphisms of classical groups over commutative rings. 10. Isomorphisms of classical groups over general rings. 11. Automorphisms of \(GL_ 2\) and \(SL_ 2\). 12. Exceptional groups and associated geometries. 13. Miscellaneous results (Aut \(\Omega {}_ 3(f,k)\), Aut \(\prod GL_{n_ i}\), Aut S\(L_ 3(R))\). 14. Related areas (geometry, arithmeticity, normal subgroups). Bibliography (174 items).

Reviewer: Yu.I.Merzlyakov

##### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

20E36 | Automorphisms of infinite groups |

20G35 | Linear algebraic groups over adĂ¨les and other rings and schemes |