Elementary equivalence of infinite-dimensional classical groups

In this chapter we discuss the infinite-dimensional analogues of the classical groups. We denote by U(∞), O(∞), GL(∞, ℝ), and GL(∞, ℂ) the full unitary, full orthogonal, and full linear groups of real and complex Hilbert space, respectively. Let G(∞): ⊃ K(∞) be two groups of this type. We denote by (G(∞), K(∞)) the group of operators A ∈ G(∞) that are representable in the form A= B(l + T), where B ∈ K(∞) and T is a Hilbert-Schmidt operator. We have already encountered many groups of this type. We are not aiming at a full discussion of the theory of representations of such groups in this form. Here there are at present just two typical models, namely, (GL(∞, ℝ), O(∞)) and (U(∞), O(∞)), chiefly the latter. In §9.5 we shall make several remarks concerning the other groups. In §§9.6, 9.7 we give several applications of our theory. In §§9.1-9.5 we shall, in the main, follow the papers (1983), (1990) of Ol’shanskii.

Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

Structural Properties of Models of N1-Categorical Theories

Chapter 2 - Projective and Affine Geometry over Division Rings

Let A be an indiscrete locally compact primitive ring having minimal left ideals.Algebraically, A may be regarded as a dense ring of linear operators containing nonzero linear operators of finite rank on a vector space E over a division ring K, and furthermore if A is simple, then A contains only linear operators of finite rank.The topology of A induces topologies on F and A" in a natural way, so that K becomes a locally compact division ring and E a locally compact vector space over K. Theorems about locally compact division rings and locally compact vector spaces imply that if K is indiscrete, then E is necessarily finite dimensional over K, and thus A is the ring of all linear operators on a finite-dimensional vector space over a locally compact division ring.Kaplansky proved that if A has characteristic zero, then K is indeed indiscrete, but he showed by an example that E could be infinite dimensional.Kaplansky remarked [9, p. 459], however, that if A is simple, it seemed unlikely that F could be infinite dimensional, since otherwise "completeness appears to require the presence of linear transformations with infinitedimensional range."Our principal result in 1 is the verification of this conjecture; however, our argument is based not on the fact that a locally compact ring is complete, but rather on the fact that a locally compact space is a Baire space and on the symmetry between minimal left and right ideals in a simple ring.In 2 we shall present certain results about locally compact vector spaces over discrete division rings; these enable us to prove that a locally compact, indiscrete, central primitive algebra having minimal left ideals is finite dimensional provided its scalar field either is indiscrete, has characteristic zero, or is uncountably infinite.However, there is a very natural example of a locally compact, indiscrete, infinitedimensional central primitive algebra having minimal left ideals over a discrete countably infinite field of prime characteristic.1. Locally compact simple rings having minimal left ideals.We begin with a summary of the structure theory of a primitive ring A having minimal left ideals.Let e be a minimal idempotent of A, that is, an idempotent such that Ae is a minimal left ideal, or equivalently, such that eA is a minimal right ideal.Then eAe is a division ring, the additive group Ae becomes a right eAe-\tctor space by defining scalar multiplication to be the restriction of multiplication on A to

We investigate some connections between the Fraiss? theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ?-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.

We consider Abelian p-groups (p ≥ 3) A1 and A2 with nonzero divisible parts. In this paper, we prove that the automorphism groups AutA1 and AutA2 are elementarily equivalent if and only if the groups A1 and A2 are equivalent in second-order logic.

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout’s theorem, we obtain the sufficient and necessary conditions on general curves where sharp L^p - L^r extension estimates hold. Our main result can be considered as a nice generalization of works by Mockenhaupt and Tao in [17] and Iosevich and Koh in [10]. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.

Let (D, Λ) and (D′, Λ′) be form rings over division rings D and D′. Let V and V′ be non-degenerate, finite dimensional (right) quadratic spaces over (D, Λ) and (D′, Λ′) respectively. Assume that there exists a quadratic semilinear isomorphism α from V onto V′.

Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N= 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of singular vector. As a consequence we prove the conjecture of Gato-Rivera and Rosado on the possible existing types of topological singular vectors (4 in chiral Verma modules and 29 in complete Verma modules). Interestingly, we have found two-dimensional spaces of singular vectors at level 1. Finally, by using the topological twists and the spectral flows, we also obtain the maximal dimensions of the singular vector spaces for the Neveu–Schwarz N= 2 algebra (0, 1 or 2) and for the Ramond N= 2 algebra (0, 1, 2 or 3).

Let S = (S,J) be a Riemann surface of genus g ≥ 1. Three vector spaces of equal real dimension are attached to the surface S, namely the \( \mathbb{C} \)-vector spaces of complex dimension g of J-holomorphic, resp.

Let V be an n-dimensional left vector space over a division ring R and n ≥ 3. Denote by ## add figure 'advg.5.3.455_01.gif'## k the Grassmann space of k-dimensional subspaces of V and write ## add figure 'advg.5.3.455_02.gif'## k for the set of all pairs (S, U ) ∈ ## add figure 'advg.5.3.455_01.gif'## k ｘ ## add figure 'advg.5.3.455_01.gif'## n - k such that S + U = V. We study bijective transformations of ## add figure 'advg.5.3.455_02.gif'## k preserving the class of base subsets and show that these mappings are induced by semilinear isomorphisms of V to itself or to the dual space V* if n ≠ 2k ; for n = 2k this fails. This result can be formulated as the following: if n ≠ 2k and the characteristic of R is not equal to 2 then any commutativity preserving transformation of the set of (k, n - k )-involutions can be extended to an automorphism of the group GL(V ).

Let D be a division ring. The aim of this paper is to explore the problem of decomposing an infinite matrix over D into a product of involutions and a product of commutators of involutions within the context of covering numbers. Specifically, we focus on decomposing matrices in the commutator subgroup SLV K,∞(D) of the Vershik–Kerov group and in the subgroup SL∞(D) of the stable general linear group GL∞(D).

-In a previous paper [2] we developed the basic features of the vector and tensor analysis under the assumption that there is an n -dimensional vector space at every point of an m -dimensional parameter manifold. Here we make the final step to connect the vector spaces X(P), P ∈ Ω with the parameter manifold as far as possible. Such a manifold with a structure determined by the fundamental tensor and by the connection coefficients may be called a space. We therefore require the dimension m of the parameter manifold and the dimension n of the vector spaces X(P), P ∈ Ω to be equal, m = n Naturally, the results obtained for the case m not necessarily equal to n [2], when adapted to the case m = n remain conserved, while some new results and relations, characteristic of the case m = n, will be derived.

In this communication the design of an array consisting of dipoles having an arbitrary orientation is effected; the dipoles are specified by five coordinates, three positional and two directional. The intensity of the electric field is considered as both i) a vector in Euclidian space and ii) a vector of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional vector space. Some examples follow.