Left Vector Space Research Articles

Structural Convergence Results for Approximation of Dominant Subspaces from Block Krylov Spaces

This paper is concerned with approximating the dominant left singular vector space of a real matrix $A$ of arbitrary dimension, from block Krylov spaces generated by the matrix ${A}{A}^T$ and the b...

Characterization of isometric embeddings of Grassmann graphs

Abstract Let V be an n-dimensional left vector space over a division ring R. We write Gk(V ) for the Grassmannian formed by k-dimensional subspaces of V and denote by Γk(V ) the associated Grassmann graph. Let also V′ be an n0-dimensional left vector space over a division ring R0. Isometric embeddings of Γk(V ) in Γk′(V′) are classified in [13]. A classification of J(n; k)-subsets in Gk′(V′), i.e. the images of isometric embeddings of the Johnson graph J(n; k) in Γk′(V′), is presented in [12]. We characterize isometric embeddings of Γk(V ) in Γk′(V′) as mappings which transfer apartments of Gk(V) to J(n; k)-subsets of Gk′(V′). This is a generalization of the earlier result concerning apartment preserving mappings [11, Theorem 3.10].

A quaternionic construction of 𝐸₇

We give an explicit construction of the simply-connected compact real form of the Lie group of type E 7 E_7 , as a group of 28 × 28 28\times 28 matrices over quaternions, acting on a 28 28 -dimensional left quaternion vector space. This leads to a description of the simply-connected split real form, acting on a 56 56 -dimensional real vector space, and thence to the finite quasi-simple groups of type E 7 E_7 . The sign problems usually associated with constructing exceptional Lie groups are almost entirely absent from this approach.

Embedding semilattices of subspaces of vector spaces

Let V be a left vector space over the division ring D and let dim DV = κ. Let (SubV; ∩) and (Sub V; +) denote the meet-semilattice and join-semilattice of the subspaces of V, respectively. We prove that for κ finite (Sub V; ∩) embeds into (Sub V; +) if and only if D embeds into Dop. We show that the similar statement holds for every infinite κ if and only if it holds for κ = ℵ0. For vector spaces over fields we obtain that (Sub V; ∩) always embeds into (Sub V; +).

Density of Polynomial Maps

AbstractLet R be a dense subring of End(DV), where V is a left vector space over a division ring D. If dimDV = ∞, then the range of any nonzero polynomial ƒ (X1, … , Xm) on R is dense in End(DV). As an application, let R be a prime ring without nonzero nil one-sided ideals and 0 ≠ a ∈ R. If a f (x1, … , xm)n(xi) = 0 for all x1, … , xm ∈ R, where n(xi ) is a positive integer depending on x1, … , xm, then ƒ (X1, … , Xm) is a polynomial identity of R unless R is a finite matrix ring over a finite field.

Surjections on Grassmannians preserving pairs of elements with bounded distance

Let m and k be two fixed positive integers such that m > k ⩾ 2 . Let V be a left vector space over a division ring with dimension at least m + k + 1 . Let G m ( V ) be the Grassmannian consisting of all m -dimensional subspaces of V . We characterize surjective mappings T from G m ( V ) onto itself such that for any A , B in G m ( V ) , the distance between A and B is not greater than k if and only if the distance between T ( A ) and T ( B ) is not greater than k .

A note on additive mappings decreasing rank one

Kuzma characterized additive mappings on the space of all finite rank bounded linear operators on a real or complex Banach space that decreases operators of rank one. In this note, we give a short proof of his result in a slightly more general setting of the tensor product of a right and a left vector space over a division ring.

On the geometry of linear involutions

Abstract 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 ).

Completely Reducible Infinite-Dimensional Skew Linear Groups

Let V be a left vector space over a division ring D and \(\) the group of all D-automorphisms of V. A subgroup G of \(\) is completely reducible of V is completely reducible as D–G bimodule. Our aim in this brief note is to point out that in a sense the very useful notion of a local marker extends from V finite-dimensional to V infinite-dimensional. (A local marker of a subgroup G of \(\) is any finitely generated subgroup X of G such that row n-space \(\) has least composition length as D–X bimodule. A local marker of G controls to a considerable extent the local behaviour of G.)

A Jordan-Hölder Theorem for Finitary Groups

AbstractLet V be any left vector space over any division ring D and let G be any group of finitary linear maps of V. Then the D — G bimodule V satisfies a Jordan- Hölder theorem. Specifically, there is a bijection between the G-nontrivial factors in two composition series of V such that corresponding factors are isomorphic as D — G bimodules. This cannot be extended to cover the G-trivial factors.

Irreducible locally nilpotent finitary skew linear groups

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

Locally soluble primitive finitary skew linear groups

Throughout this paper D denotes a division ring and V a left vector space over D. The finitary general linear group FGL(V) or FA AutDV over V is the subgroup of AutDV of D-automorphisms g of V such that [V,g] = V(g-l) has finite (left) dimension over D. By a finitary skew linear group we mean any subgroup G of FGL(V) for any D and V. Such a G is irreducible if V is irreducible as D-G (bi)module and is primitive if whenever V = ⊕ω ∊ ΩVomega as D-module, where for all g∊G and ω∊Ω, Vωg = Vω for some ω∊Ω, we have |Ω| = 1. In [4] we showed that a primitive irreducible finitary skew linear group is finite dimensional if it is hyper locally nilpotent (that is radical in the sense of Kuros) and sometimes if it is locally soluble. Here we complete the locally soluble case and, in fact, we can be a little more general.

Supersoluble and Related Cofinitary Groups

AbstractLet D be a division ring (possibly commutative) and V an infinite‐dimensional left vector space over D. We consider irreducible subgroups G of GL(V) containing an element whose fixed‐point set in V is non‐zero but finite dimensional (over D). We then derive conclusions about cofinitary groups, an element of GL(V) being cofinitary if its fixed‐point set is finite dimensional and a subgroup G of GL(V) being cofinitary if all its non‐identity elements are confinitary. In particular we show that an irreducible cofinitary subgroup G of GL(V) is usually imprimitive if G is supersoluble and is frequently imprimitive if G is hypercyclic. The latter includes the case where G is hypercentral, which apparently is also new.

Primitive finitary skew linear groups

1. Introduction. Throughout this paper D denotes a division ring and V a left vector space over D. The finitary general linear group FGL (V) or FAutoV over V is the subgroup of Aut o V of D-automorphisms 9 of V such that [V, 9] = V (9 - 1) has finite (left) dimension over D. By afinitary skew linear group we mean any subgroup G of FGL(V) for any D and V. Such a G is irreducible if V is irreducible as D-G (bi)module and is primitive if whenever V = �9 V~, as D-module, where for all 9 ~ G and co ~ s V~, 9 = Vo), coco for some ~'~ s we have 1s = I. Normally we only consider primitivity for irreducible groups.

Geradenmodelle der M�bius- und Burau-Geometrien

Let (L, K) be an arbitrary (not necessary abelian) field extension, let V be a (left) vector space over L (and therefore over the subfield K) and let μ be the set of all sub-spaces of the projective space II(V, K). Then the canonical map ϰ: P ≔ V*/K* → U given by ϰ: K*a → L*a/K* ≔ {K*(λa)∣λ ∈ L*{ induces a spreadS in Π(V, K). Furthermore ϰ induces a chain structure by applying ϰ on certain subspaces of U. Our purpose is to give a synthetic description of these chain structures. For a regular SpreadS, i.e. K is commutative, the chain structure is a generalization of the Burau geometries [11]. In this paper we represent the Staudt chains by Segre manifolds.

Commutative semifields, two dimensional over their middle nuclei

Then we say that S is a (finite) semzj?eld. The subset N, = (n E S: (xn)y = x(ny), Vx, y E S} is known as the middle nucleus of S. N, is a field, and S may be regarded as a (left or right) vector space over N,. A semifield in which multiplication is not associative (i.e., N, # S) is called proper. A finite semifield necessarily has prime power order and proper finite semifields exist for all orders q = p’, p prime, where Y > 3 if p is odd and r>4ifp=2. There is a considerable link between finite semifields and finite projective planes of Lenz-Barlotti class V.1. Indeed, any finite proper semilield gives rise to such a plane, and conversely any plane of the above type yields many isotopic, but not necessarily isomorphic, semifields. The paper by Knuth [4] is an excellent survey of finite semifields and their connections with projective planes. We mention one of the examples given in 141‘

On characterizing the standard quantum logics

Let L \mathcal {L} be a complete projective logic. Then L \mathcal {L} has a natural representation as the lattice of ⟨ ⋅ , ⋅ ⟩ \langle { \cdot , \cdot } \rangle -closed subspaces of a left vector space V over a division ring D, where ⟨ ⋅ , ⋅ ⟩ \langle {\cdot ,\cdot } \rangle is a definite θ \theta -bilinear symmetric form on V, θ \theta being some involutive antiautomorphism of D. Now a well-known theorem of Piron states that if D is isomorphic to the real field, the complex field or the sfield of quaternions, if θ \theta is continuous, and if the dimension of L \mathcal {L} is properly restricted, then L \mathcal {L} is just one of the standard Hilbert space logics. Here we also assume L \mathcal {L} is a complete projective logic. Then if every θ \theta -fixed element of D is in the center of D and can be written as ± d θ ( d ) \pm \,d\theta (d) , some d ∈ D d \in D , and if the dimension of L \mathcal {L} is properly restricted, we show that L \mathcal {L} is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron’s theorem to discontinuous θ \theta . Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.

Products of Transvections

This paper is concerned with the presentation of certain elements of the groupSL(n,K) as products of a minimal number of transvections. To explain the terminology, letVbe ann-dimensional left vector space over a (not necessarily commutative) fieldK. The group of all non-singular linear transformations ofVontoV(i.e. the group of all collineations ofV) is the groupGL(n,K). This group is generated by collineations leaving a hyperplane pointwise fixed. Whenn= 2 these collineations are called axial collineations and the invariant hyperplane (line) is then called an axis.

The centralizer of a matrix with real quaternion elements

Let T be a Q-endomorphism on a finite dimensional left vector space V over the division ring of real quaternions Q. In this paper we show that the centralizer of T can be regarded as an algebra over the complex numbers C and the dimension of this algebra is computed in terms of the invariant factors of T. Thus the number of C-linearly independent matrices which commute with a given matrix representing T is determined.

A characterization of the definiteness of a Hermitian matrix

We denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,…, Ak} and UBU* = diag{B1…, Bk}, where, for each i = 1, …, k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.).