Vector Space Isomorphism Research Articles

О конструкции канонической формы на расслоении реперов

The detailed description of the construction of the canonical form on the higher order frame bundle over an n-dimensional smooth manifold is given. In particular, it is shown that some vector space isomorphism play­ing the key role in this construction is defined correctly, i. e. it depends only on the frame of order p + 1 and does not depend on the choice of its representative, i. e. a local diffeomorphism which (p + 1)-jet is exactly this frame. This isomorphism acts from the direct sum of n-dimensional arithmetic space and the Lie algebra of the p-th order differential group to the tangent space to the p-th order frame bundle over the manifold at the p-th order frame lying “below”. The action of this isomorphism can be splitted into two its restrictions. The first one acts from the first direct sum­mand, and the second one acts from the second direct summand. It is shown that the first restriction depends only on the choice of the (p + 1)-fra­me, while the second one is closely related to fundamental vector fields and therefore does not depend of this frame at all.

Mathcal{N} = 4 SYM, Argyres-Douglas theories, and an exact graded vector space isomorphism

In this first of two papers, we explain in detail the simplest example of a broader set of relations between apparently very different theories. Our example relates mathfrak{su}(2) mathcal{N} = 4 super Yang-Mills (SYM) to a theory we call “(3, 2)”. This latter theory is an exactly marginal diagonal SU(2) gauging of three D3(SU(2)) Argyres-Douglas (AD) theories. We begin by observing that the Schur indices of these two theories are related by an algebraic transformation that is surprisingly reminiscent of index transformations describing spontaneous symmetry breaking on the Higgs branch. However, this transformation breaks half the supersymmetry of the SYM theory as well as its full mathcal{N} = 2 SU(2)F flavor symmetry. Moreover, it does so in an interesting way when viewed through the lens of the corresponding 2D vertex operator algebras (VOAs): affine currents of the small mathcal{N} = 4 super-Virasoro algebra at c = −9 get mapped to the mathcal{A}(6) stress tensor and some of its conformal descendants, while the extra supersymmetry currents on the mathcal{N} = 4 side get mapped to higher-dimensional fermionic currents and their descendants on the mathcal{A}(6) side. We prove these relations are facets of an exact graded vector space isomorphism (GVSI) between these two VOAs. This GVSI respects the U(1)r charge of the parent 4D theories. We briefly sketch how more general mathfrak{su}(N) mathcal{N} = 4 SYM theories are related to an infinite class of AD theories via generalizations of our example. We conclude by showing that, in this class of theories, the mathcal{A}(6) VOA saturates a new inequality on the number of strong generators.

The q-Onsager algebra and the positive part of [formula omitted

The positive part Uq+ of Uq(slˆ2) has a presentation by two generators X,Y that satisfy the q-Serre relations. The q-Onsager algebra Oq has a presentation by two generators A,B that satisfy the q-Dolan/Grady relations. We give two results that describe how Uq+ and Oq are related. First, we consider the filtration of Oq whose nth component is spanned by the products of at most n generators. We show that the associated graded algebra is isomorphic to Uq+. Second, we introduce an algebra □q and show how it is related to both Uq+ and Oq. The algebra □q is defined by generators and relations. The generators are {xi}i∈Z4 where Z4 is the cyclic group of order 4. For i∈Z4 the generators xi,xi+1 satisfy a q-Weyl relation, and xi,xi+2 satisfy the q-Serre relations. We show that □q is related to Uq+ in the following way. Let □qeven (resp. □qodd) denote the subalgebra of □q generated by x0,x2 (resp. x1,x3). We show that (i) there exists an algebra isomorphism Uq+→□qeven that sends X↦x0 and Y↦x2; (ii) there exists an algebra isomorphism Uq+→□qodd that sends X↦x1 and Y↦x3; (iii) the multiplication map □qeven⊗□qodd→□q, u⊗v↦uv is an isomorphism of vector spaces. We show that □q is related to Oq in the following way. For nonzero scalars a,b there exists an injective algebra homomorphism Oq→□q that sends A↦ax0+a−1x1 and B↦bx2+b−1x3.

Nonzero Component Graph of a Finite Dimensional Vector Space

In this article, we introduce a graph structure, called a nonzero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms, and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite.

Vector space isomorphisms of non-unital reduced Banach *-algebras

Let A and B be two non-unital reduced Banach *-algebras and φ: A → B be a vector space isomorphism. The two following statement holds: If φ is a *-isomorphism, then φ is isometric (with respect to the C<sup>*</sup>-norms), bipositive and φ maps some approximate identity of A onto an approximate identity of B. Conversely, any two of the later three properties imply that φ is a *-isomorphism. Finally, we show that a unital and self-adjoint spectral isometry between semi-simple Hermitian Banach algebras is an *-isomorphism.

Dual-depth Adapted Irreducible Formal Multizeta Values

Let $\mathfrak{ds}$ denote the double shuffle Lie algebra, equipped with the standard weight grading and depth filtration; we write $\mathfrak{ds}=\oplus_{n\ge 3} \mathfrak{ds}_n$ and denote the filtration by $\mathfrak{ds}^1\supset \mathfrak{ds}^2\supset \cdots$. The double shuffle Lie algebra is dual to the new formal multizeta space $\mathfrak{nfz}=\oplus_{n\ge 3} \mathfrak{nfz}_n$, which is equipped with the dual depth filtration $\mathfrak{nfz}^1\subset \mathfrak{nfz}^2\subset\cdots$ Via an explicit canonical isomorphism $\mathfrak{ds}\buildrel \sim\over\rightarrow\mathfrak{nfz}$, we define the "dual" in $\mathfrak{nfz}$ of an element in $\mathfrak{ds}$. For each weight $n\ge 3$ and depth $d\ge 1$, we then define the vector subspace $\mathfrak{ds}_{n,d}$ of $\mathfrak{ds}$ as the space of elements in $\mathfrak{ds}_n^d-\mathfrak{ds}_n^{d+1}$ whose duals lie in $\mathfrak{nfz}_n^d$. We prove the direct sum decomposition \[ \mathfrak{ds}=\bigoplus_{n\ge 3}\bigoplus_{d\ge 1} \mathfrak{ds}_{n,d}, \] \noindent which yields a canonical vector space isomorphism between $\mathfrak{ds}$ and its associated graded for the depth filtration, $\mathfrak{ds}_{n,d}\simeq \mathfrak{ds}_n^d/ \mathfrak{ds}_n^{d+1}$. A basis of $\mathfrak{ds}$ respecting this decomposition is dual-depth adapted, which means that it is adapted to the depth filtration on $\mathfrak{ds}$, and the basis of dual elements is adapted to the dual depth filtration on $\mathfrak{nfz}$. We use this notion to give a canonical depth 1 generator $f_n$ for $\mathfrak{ds}$ in each odd weight $n\ge 3$, namely the dual of the new formal single zeta value $\zeta(n)\in\mathfrak{nfz}_n$. At the end, we also apply the result to give canonical irreducibles for the formal multizeta algebra in weights up to 12.

Biseparating maps on generalized Lipschitz spaces

Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F-valued functions on Y is said to be biseparating if f and g are disjoint if and only if T f and T g are disjoint. We intro- duce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continu- ous functions. Linear biseparating maps between generalized Lipschitz spaces are characterized as weighted composition operators, i.e., of the form T f(y) = Sy(f(h −1 (y)) for a family of vector space isomorphisms Sy : E ! F and a homeomorphism h : X ! Y. We also investigate the continuity of T and related questions. Here the functions involved (as well as the metric spaces X and Y ) may be unbounded. Also, the arguments do not require the use of compactification of the spaces X and Y.

The Weyl algebra, spherical harmonics, and Hahn polynomials

In this article we apply the duality technique of R. Howe to study the structure of the Weyl algebra. We introduce a one-parameter family of “ordering maps”, where by an ordering map we understand a vector space isomorphism of the polynomial algebra on $\

On the cohomology algebra of a fiber

Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces between H^*(F;F_p) and Tor^{C^*(B)}(C^*(E),F_p). Generalizing the rational case proved by Sullivan, Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417--453] proved that if X is a finite r-connected CW-complex of dimension < rp+1 then the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative differential graded algebra A(X) with the same cohomology. Therefore if we suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29 (1979) 17--37] and [S Halperin, Lectures on minimal models, Soc. Math. France 9-10 (1983)] we prove that this isomorphism is in fact an isomorphism of algebras. In particular, $H^*(F;F_p) is a divided powers algebra and p-th powers vanish in the reduced cohomology \mathaccent "707E {H}^*(F;F_p).

The non-commutative Weil algebra

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo map extends to a linear map from compactly supported distributions on the Lie algebra g to compactly supported distributions on the Lie group G, which is a ring homomorphism for G-invariant distributions. In this paper we obtain analogues of the Duflo map and of Duflo's theorem in the context of equivariant cohomology of G-manifolds. Our result involves a non-commutative version of the Weil algebra and of the de Rham model of equivariant cohomology.

Enlarged geometries of gauge bundles

The geometrical picture of gauge theories must be enlarged when a gauge potentialceases to behave like a connection, as it does in electroweak interactions. Whenthe gauge group has dimension four, the vector space isomorphism betweenspacetime and the gauge algebra is realized by a tetrad-like field. The objectmeasuring the deviation from a strict bundle structure has the formal behaviorof a spacetime connection, of which the deformed gauge field strength is thetorsion. A generalized derivative emerges in terms of which the two Bianchiidentities are formally recovered. Effects of gravitational type turn up. Thedynamical equations obtained correspond to a broken gauge model on acurved spacetime.

New conformal gauging and the electromagnetic theory of Weyl

A new eight-dimensional conformal gauging solves the auxiliary field problem and eliminates unphysical size change from Weyl’s electromagnetic theory. We derive the Maurer–Cartan structure equations and find the zero curvature solutions for the conformal connection. By showing that every one-particle Hamiltonian generates the structure equations we establish a correspondence between phase space and the eight-dimensional base space, and between the action and the integral of the Weyl vector. Applying the correspondence to generic flat solutions yields the Lorentz force law, the form and gauge dependence of the electromagnetic vector potential and minimal coupling. The dynamics found for these flat solutions applies locally in generic spaces. We then provide necessary and sufficient curvature constraints for general curved eight-dimensional geometries to be in 1–1 correspondence with four-dimensional Einstein–Maxwell space–times, based on a vector space isomorphism between the extra four dimensions and the Riemannian tangent space. Despite part of the Weyl vector serving as the electromagnetic vector potential, the entire class of geometries has vanishing dilation, thereby providing a consistent unified geometric theory of gravitation and electromagnetism. In concluding, we give a concise discussion of observability of the extra dimensions.

Congruence subgroups, elliptic cohomology, and the Eichler—Shimura map

In 1986 Landweber [7] introduced the connective and periodic elliptic cohomology theories whose coefficient rings can be interpreted as a ring of modular functions for certain congruence subgroups of SL 2( Z ). One of the open questions in the subject has been to produce a geometric definition of these theories. Nishida [8] defines a spectrum X Γ based on the congruence subgroup Γ, which is related to the connective elliptic cohomology theory when Γ = Γ 0(2). X Γ has a stable summand X Γ − , and he proposes that the Eichler-Shimura map gives a real vector space isomorphism from the modular forms of Γ of weight 2 k + 2 to the real cohomology of X Γ − in dimension 4 k + 1 for Γ = Γ 0(2). One of our main results is a proof of this claim when k > 0 for Γ = Γ 0( p) and when k ≥ 0 for Γ = Γ 0(2) or Γ = SL 2( Z ). Using obstruction theory, we are able to construct a non-trivial geometric map from ∑ 3 X Γ − to the 3-connected cover of the spectrum representing the connective theory which is an equivalence through dimension 4. We also produce a stable splitting of X Γ and of the spectrum representing the periodic theory introducd by Baker [2].

Divided Differences and Linearly Recursive Sequences

We show that the theory of divided differences is a natural tool for the study of linearly recurrent sequences. The divided differences functional associated with a monic polynomial w on degree n + 1 yields a vector space isomorphism between the space of polynomials of degree at most equal to n and the space of linearly recurrent sequences f that satisfy the difference equation w(E)f=0 where E is the usual shift operator. Using such isomorphisms, we can translate problems about recurrent sequences into simple problems about polynomials. We present here a new approach to the theory of divided differences, using only generating functions and elementary linear algebra, which clarifies the connections of divided differences with rational functions, polynomial interpolation, residues, and partial fractions decompositions.

Information sets in quadratic-residue codes

In a linear (n, k) code, a set of k coordinates is said to be an information set for the code if the projection map onto those k coordinates is a vector space isomorphism. Extended (cyclic) quadratic-residue codes are linear (p+1,12(p+1)) codes, defined whenever p is prime. The automorphism group of this code contains PSL(2, p) (the Gleason-Prange theorem). This group includes elements with order 12(p+1) and it is conceivable that one or both orbits of such an element will serve as an information set for the code. When this happens, simplified decoding procedures are possible.These codes can be defined over a ring of algebraic integers in such a way that the QR code with entries in a finite field is just the image of this global code under a residue-class map. It is shown that the global code includes a codeword of the form (z(p) 0 0 … 0) on a given orbit, where z(p) is a rational integer. A series of theorems show that knowing this codeword is enough to answer the question (for either orbit) for QR codes over fields of any characteristic. (Briefly an orbit can fail to be an information set for the characteristic q case if and only if q devides z(p). A method for finding this codeword is given which is based on a connection between z(p) and the determinant of a certain 0, 1-matrix. The results of this technique are given which answer the question for 5 ⩽ p ⩽ 83.

Operational Rules

The concept of operational rules in the two-sided operational calculus is formalized and extensively illustrated. The paper continues the conceptual evelopment, begun in an earlier one, which is based upon the Fourier transform from an operator ring K onto the arithmetical ring $\mathcal{K}$ of all functions which are measurable and finite-valued almost everywhere on the real line. The Fourier transform can be further extended so that it is a vector space isomorphism from the vector (nondirect) sum $\mathcal{S} = K + \mathcal{K}$ onto itself. This is an extension of the classical $L_1 $ and $L_2 $ Fourier transforms which is also compatible with the distributional Fourier transform. The concept of operational rules is extended $\mathcal{S}$.

Canonically Isomorphic Spaces of Bounded Solutions of △u = Pu

Let R be a hyperbolic Riemann surface and P, Q nonnegative C1 2-forms on R. The space of bounded solutions of △u = Pu (△u = Qu, respectively) on R is denoted by PB(R) (QB(R), respectively). A vector space isomorphism S between PB(R) and QB(R) is called canonical if for each u ϵ PB(R), there is a potential pu on R with \u — Su\ ≦ pu. The canonical isomorphism theme in the study of the equation △u = Pu was introduced in H. Royden's paper [9] on the order comparison condition.

On the topology of tangent bundles

is a vector space isomorphism of Vn onto T(m). The tangent bundle 3(M) of the manifold M consists of the ordered pairs (m, v) where meM and vE T(m). Therefore, as a point set only, 3(M) is M x V1. We shall assume that the reader is familiar with the fibre space topology which is customarily assigned to 3(M). (For a description of this topology and the facts concerning fibre bundles which we shall need in other parts of this paper see N. Steenrod, The topology of fibre bundles, Princeton, 1951.) For many manifolds M this topology differs from the topology of the product space M X Vn. This fact indicates that it is the assignment of this topology which makes 3(M) an interesting mathematical object. It is the purpose of this paper to present an intrinsic characterization of this topology.