Isomorphism Theory Research Articles

Isomorphism theory for orthogonal groups over arbitrary integral domains

The isomorphism theory of matrix groups, which began over fields, has in recent years turned to rings. Developments over rings have been primarily in two separate directions: the first is concerned with the classical groups and their congruence groups over integral domains, while the second is concerned with algebraic groups and their arithmetic subgroups. So the first direction considers a very general underlying domain, while the second considers more general groups of matrices. For a historical account of these matters refer to the books of DieudonnC [12] and O’Meara [27] and the survey article by Mostow [22]. For more recent results consult the references of the present article. Regarding the first direction, which is considered here, the theory is complete (with isolated exceptions) for the linear groups in general, the symplectic groups in general, and the unitary groups under additional assumptions of isotropy. As in the classical situation over fields, every isomorphism between such groups is determined in a natural way by a “metric preserving” semilinear isomorphism of the underlying spaces. In sharp contrast, results for the orthogonal groups exist only for arithmetic and local domains, but not for arbitrary integral domains. Refer to O’Meara [26], Hahn [14] and Prasad [31] for example. It is the purpose of this paper to extend the isomorphism theory of the orthogonal groups from these special domains to arbitrary integral domains. One insurmountable obstruction becomes apparent quickly: in any dimension there exist free b-modules with anisotropic quadratic structure, the orthogonal groups of which are trivial (for instance, see 3.1 of [26]); take two such situations in different dimensions; these orthogonal groups are obviously isomorphic since they are trivial; but their spaces cannot be connected with a semilinear isomorphism since their dimensions are not equal. So isotropy assumptions are inevitable in the isomorphism theory of the orthogonal groups over arbitrary integral domains. It is the goal of this paper to prove that isotropy assumptions are also sufficient.

Some recent discoveries in the isomorphic theory of Banach spaces

Some recent discoveries in the isomorphic theory of Banach spaces

A general isomorphism theory for linear groups

A general isomorphism theory for linear groups

Isomorphism theory of congruence groups

This note is an announcement of four theorems I proved in [5][9] on the isomorphisms of the linear, symplectic, and unitary congruence groups. A sketch of the proofs is given. NOTATION. Let V be an w-dimensional vector space over the field F, n*z2. For crGGLw(F), let cr denote the contragredient of cr (inverse of the transpose). Let ~ denote the natural map of GLn( V) onto PGLn( V) ; for any subset 5 of GLn(V), S is the image of S in PGLn(V) under the map. A transvection r is a linear transformation of determinant one which fixes all vectors of some hyperplane, called the proper hyperplane of r. If r 5 1 then (r — 1) V is a line called the proper line of r. An element f of PGLn(V) is called a (projective) transvection if r is a transvection. The proper line and proper hyperplane of f are defined as the proper line and hyperplane of r.

Free states of the canonical anticommutation relations

Each gauge invariant generalized free state ω A of the anticommutation relation algebra over a complex Hilbert spaceK is characterized by an operatorA onK. It is shown that the cyclic representations induced by two gauge invariant generalized free states ω A and ω B are quasi-equivalent if and only if the operatorsA 1/2−B 1/2 and (I−A)1/2−(I−B)1/2 are of Hilbert-Schmidt class. The combination of this result with results from the theory of isomorphisms of von Neumann algebras yield necessary and sufficient conditions for the unitary equivalence of the cyclic representations induced by gauge invariant generalized free states.

CXIV.—Contributions to the theory of isomorphism based on experiments on the regular growths of crystals of one substance on those of another

CXIV.—Contributions to the theory of isomorphism based on experiments on the regular growths of crystals of one substance on those of another

3. Examination of certain Objections to the Theory of Isomorphism

The object of this paper was the refutation of certain arguments brought forward against the doctrine of Isomorphism, more especially by those who allow the existence of an approach to, although they deny absolute identity of, form.The author sets out by shewing, that most of the arguments are deduced from observations made on crystallized minerals; and he then proceeds to state the many important circumstances which tend to influence the form of these bodies, but which can be easily avoided in those salts which admit of being formed artificially; such are, the intense temperature to which they are subjected, the pressure of surrounding matter, the admixture of foreign bodies, &c. each and all of which he proves to exert a powerful influence over the forms of crystals.

Researches on electro-chemical equivalents, and on a supposed discrepancy between some of them and the atomic weight of the same bodies, as deduced from the theory of isomorphism

The author describes various experiments made with a view to determine the electro-chemical equivalents of sodium and potassium. Three experiments gave, respectively, 22.3, 22.9, and 25, as the equivalent of the former; and two other experiments gave, respectively, 45 and 41.7, as the equivalent of the latter of these substances. He then inquires what would be the result of the electro-lyzation of the aqueous solutions of soda and potash, on the hypothesis of these bodies being composed of two equivalents, or atoms, of metal, and one of oxygen. To determine this question he employs a solution of dichloride of copper in muriatic acid, as being a substance composed of two atoms of metal and one of an electro-negative element. Its electrolysis gave as the equivalent of copper, 52.8, 59.4, and 61.6, numbers approximating closely to 63.2, or double the atomic weight of copper. After a long train of investigation, he concludes that there is no reason deducible from the theory of isomorphism for doubting the correctness of the received atomic weights of silver, sodium, &c., but that the difficulty, or anomaly, if it may be so called, should be considered as attaching itself to the di-compounds of copper; and that Faraday’s propositions on this subject remain unimpeached.