Invariant Lattice Research Articles

Invariant subspace lattices that complement every subspace

Invariant subspace lattices that complement every subspace

A finite size scaling analysis of the O(4)-invariant lattice δ4 theory

The critical behaviour of the three- and four-dimensional N=4 vector model is investigated by means of a Monte Carlo simulation on lattices with size between 4 3 and 16 3, and between 4 4 and 12 4, respectively. For obtaining information about some critical properties of the model, we use a method due to Binder which is based on the theory of finite size scaling. For the three-dimensional model we get estimates of the critical exponents ν and η which are compatible with estimates obtained from the ϵ-expansion. In four dimensions we study for two different values of the bare self-coupling λ ( λ=1 in our normalization, and λ=∞) the scaling behaviour of some Green function ratios at the phase boundary. In both cases we find compatibility with the “predicted” scaling behaviour at the gaussian fixed point. This is another independent numerical hint that the continuum limit of the four-dimensional O(4)-invariant lattice δ 4-model is a free field theory.

A chiral invariant decoupling of replica lattice fermions

A chiral invariant lattice fermion action which solves the species-doubling problem of Dirac fermions on a lattice for (even) dimensions of space-time less than eight is presented. The decoupling of replica fermions requires to give up rotational and gauge symmetry in the regularization procedure. The corresponding lattice Schwinger model reproduces all the results of the continuum theory.

The U(1) anomaly in a chiral invariant lattice action

A chiral invariant lattice fermion action which violates reflection positivity and avoids the doubling of fermion species is studied in perturbation theory in two-dimensional spacetime. The absence of anomalies in the lattice regularization is shown to be compatible with the identification of current operators which reproduce the correct (axial) vector current Ward identities.

Invertible composition operators on Hp

The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H 2 is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.

INVARIANT LATTICES, THE LEECH LATTICE AND ITS EVEN UNIMODULAR ANALOGUES IN THE LIE ALGEBRAS $ A_{p-1}$

For any prime 2$ SRC=http://ej.iop.org/images/0025-5734/58/2/A09/tex_sm_3113_img2.gif/> a classification (up to similarity) is given of all invariant integral lattices that correspond to an orthogonal decomposition of the Lie algebra . Even unimodular lattices without roots are distinguished. For they contain the Leech lattice. For some of the resulting lattices the automorphism groups are studied, and lower bounds for the minimal length of vectors are obtained.Figures: 2. Bibliography: 17 titles.

Self-similar and translationally invariant structures in a bond-diluted Ising model

A bond-diluted Ising model is used to simulate a process from a Cantor set to a translationally invariant lattice. The free energy function shows that there is no singularity, which seem to imply that no transition exists in this process.

Einführung in eine anorganische Strukturchemie

Abstract A systematic description and classification of inorganic structure types is proposed on the basis of homogeneous or heterogeneous point configurations (Bauverbände) described by invariant lattice complexes and coordination polyhedra; subscripts or matrices explain the transformation of the complexes in respect (M) to their standard setting; the value of the determinant of the transformation matrix defines the order of the complex. The Bauverbände (frameworks) may be described by three-dimensional networks or two-dimensional nets explicitely shown with structures types of the I- and F-family. Examples of structure types of different orders are listed for the homogeneous or pseudohomogeneous I-, P-, F-, Y**-, S-, + Y-, Q- and C-families and for the heterogeneous [I + W]-, [D + T]- and [Y(31) + Dx]-families. Homeotypic are those structure types, which replace their points of the configurations by polyhedra like dumb-bell (21), triangle (31), tetrahedron (4t), octahedron (6o), cuboctahedron (12co), icosahedron (12i) etc. forming the I[polyhedra]- P[polyhedra]- etc. families. Furthermore, layer descriptions are applied with two-dimensional nets of H, G, N, N[43243] and N[6434] with different stacking sequences and in combination with other nets. The system is proposed after the study of the cubic inorganic structure types and the structures of intermetallic compounds with tetragonal symmetry.

Translation invariant forms on $L^p(G)(1<p<\infty)$

It is shown that if G is a connected metrizable compact Abelian group and 1<p<∞, any (possibly discontinuous) translation invariant linear form on L p (G) is a scalar multiple of the Haar measure. This result extends the theorem of G.H. Meisters and W.M. Schmidt (J. Funct. Anal. 13 (1972), 407-424) on L 2 (G). Our method permits in fact to consider any superreflexive translation invariant Banach lattice on G, which is the adopted point of view. We study the representation of an element f of this invariant lattice X as a sum of a bounded number of elements of the form g-τ(a)g, where g in X, a in G and τ(a) the corresponding translation operator. Our approach consists in proving the boundedness of certain random convolution operators using interpolation techniques.

Is a self-adjoint operator determined by its invariant subspace lattice?

If A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ( A) and σ( B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ( A). If A is any self-adjoint operator, there is an associated function κ A : N ∪ {∞} → ( N ∪ {0, ∞}) × {0,1} defined in this paper. If F denotes the collection of all functions from N ∪ {∞} into ( N ∪ {0,∞}) × {0,1}, then F is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κ A = κ B . The paper ends with some comments on the corresponding problem for normal operators.

Character table for the 1080-element point-group-like subgroup of SU(3)

The character table for S(1080), the largest point-group-like subgroup of SU(3), is presented. So is the reduction of the first 28 SU(3) characters to S(1080) characters. These tables are needed in a recently proposed systematic description of SU(3)-invariant lattice gauge theories by effective S(1080)-invariant theories. Problems encountered in an alternative systematics, using only local S(1080)-invariant theories, are discussed.

A lattice model for the dynamical Higgs mechanism

We present an SU(4) gauge invariant lattice model with elementary fermions which dynamically develops a Higgs phase in its strong coupling region. The Higgs fields are the fermion bilinears for which the effective action is computed in the infinite coupling limit. The achieved breaking is SU(4) to Sp(4).

A Characterization of the Invariant Subspaces of Direct Sums of Strictly Cyclic Algebras

Two characterizations of the invariant subspace lattice of ${A^{(n)}}$ for a strictly cyclic operator algebra $A$ on a separable Hilbert space are proven.

Internal space decimation for lattice gauge theories

By a systematic decimation of internal space lattice gauge theories with continuous symmetry groups are mapped into effective lattice gauge theories with finite symmetry groups. The decimation of internal space makes a larger lattice tractable with the same computational resources. In this sense the method is an alternative to Wilson's and Symanzik's programs of improved actions. As an illustrative test of the method U(1) is decimated to Z( N) and the results compared with Monte Carlo data for Z(4)- and Z(5)-invariant lattice gauge theories. The result of decimating SU(3) to its 1080-element crystal-group-like subgroup is given and discussed.

Invariant lattices and modular decomposition of irreducible representations

Let p be a prime. Let H be a finite group, and let c be an (absolutely) irreducible character of H. Any thorough study of the p-modular decom- position of c requires passage to some integral representation. If S is a (finite) group acting on H and leaving [ invariant, such an integral represen tation will not be S-invariant in general. It is important to have invariant integral representations for purposes of Clifford theory. In fact, modular decomposition often gives new insights in such “classical” questions like extendability of characters. For this reason we consider the more general situation that H is a normal subgroup of some group G, and let S = G/H. (S still acts on the characters of H and the isomorphism types of H-modules.) We also fix a p-modular system (K, R, k), where

Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ∗-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg( L 1 ⊗ L 2) = Alg L 1 ⊗ Alg L 2, where each L i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each L i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if L is a finite-width commutative subspace lattice and K is the set of compact operators then the quasitriangular algebra Alg L + K is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.

A note on rank-one operators in reflexive algebras

It is shown that if the invariant subspace lattice of a reflexive algebra A \mathcal {A} , acting on a separable Hilbert space, is both commutative and completely distributive, then the algebra generated by the rank-one operators of A \mathcal {A} is dense in A \mathcal {A} is any of the strong, weak, ultrastrong or ultraweak topologies. Some related density results are also obtained.

Invariant subspace lattices of Lambert's weighted shifts

AbstractLet B(H) be the Banach algebra of all (bounded linear) operators on an infinite-dimensional separable complex Hilbert space H and let be a bounded sequence of positive real numbers. For a given injective operator A in B(H) and a non-zero vector f in H, we put We define a weighted shift Tw with the weight sequence on the Hilbert space 12 of all square-summable complex sequences by . The main object of this paper is to characterize the invariant subspace lattice of Tw under various nice conditions on the operator A and the sequence .

On a Characterization of Invariant Subspace Lattices of Weighted Shifts

The paper concerns itself with the characterization of invariant subspace lattices of weighted shift operators on the Hilbert space ${l^2}$ with suitable conditions on their weights. This characterization is also extended to the case of Banach spaces ${l^p}$, $1 < p < \infty$.

On projective equivalence of invariant subspace lattices

For i = 1,2, let A i be a linear transformation on a complex vector space and let σ be a lattice isomorphism from the invariant subspace lattice of A 1 onto the invariant subspace lattice of A 2. We determine the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds).