Which O-commutative Basic Algebras Are Effect Algebras

By a basic algebra is meant an MV-like algebra (A,⊕,?,0) of type 〈2,1,0〉 derived in a natural way from bounded lattices having antitone involutions on their principal filters. We show that (i) atomic Archimedean basic algebras for which the operation ⊕ is o-commutative are effect algebras and (ii) atomic Archimedean commutative basic algebras are MV-algebras. This generalizes the results by Botur and Halaš on finite commutative basic algebras and complete commutative basic algebras.     

Anti-BZ-Structure in Effect Algebras

The definitions of sharply approximating effect algebras, anti-BZ-effect algebras, central approximating effect algebras, and S-anti-BZ-effect algebras are given, the relationships between sharply approximating effect algebras and anti-BZ-effect algebras, between central approximating effect algebras and anti-BZ-effect algebras are established, and the set of anti-BZ-sharp elements in S-anti-BZ-effect algebras is proved to be an orthomodular lattice.     

Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras

We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.     

Unital Groups and General Comparability Property

Pseudo-effect algebras are partial algebras (E;+,0,1) with a partially defined addition + which is not necessarily commutative and therefore with two complements, left and right. If they satisfy a special kind of the Riesz decomposition property, they are intervals in unital po-groups. The general comparability property in unital po-groups with strong unit (G,u), allows to compare elements of G in some intervals with Boolean ends. Such a po-group is always an -group admitting a state. We prove that every such (G,u) is a subdirect product of linearly ordered unital po-groups.     

The cis-effect using the topology of the electronic charge density

We provide a physics-inspired coupling mechanism explaining the cis-effect in terms of electronic and nuclear degrees of freedom and explore the implications for three families of molecules. The cis- or trans-effect is related to the tendency of electronic charge density to move away from the bond critical point (BCP) and towards the associated nuclear attractors. A quantitative measure of this effect is given by the λ ₃ eigenvalue of the Hessian matrix of the electronic charge density. The physical origin of the cis-effect is tied to the observation that the central X=X, X=C or N bond-paths of the cis-isomers are more bent (they are up to 1.5% longer than the internuclear distance) than the bond-paths of the corresponding trans-isomers. Greater bond-path bending is associated with a stronger cis-effect; the direction of bond deformation can in all cases be predicted by the most facile (least compressible) mode of the electronic stress tensor. Further to this, the ellipticity ε of the X=X BCPs of a molecule displaying the cis-effect is lower in the cis-isomer than for the corresponding trans-isomer, suggesting that the cis-effect is less counterintuitive than previously thought. The molecules that exhibit the greatest cis-effect are those with fluorinated double bonds; this is because the most facile modes of the C–F bond couple with the highest-symmetry normal mode of vibration. Qualitative agreement is found with existing experimental data and predictions are made where experimental data is lacking.     

Generalization of Blocks for D-Lattices and Lattice-Ordered Effect Algebras

We show that everyD-lattice (lattice-ordered effect algebra)P is a set-theoreticunion of maximal subsets of mutually compatible elements, called blocks.Moreover, blocks are sub-D-lattices and sub-effect-algebras ofP which areMV-algebras closed with respect to all suprema and infima existing inP.     

Euclidean Majorana and Weyl Spinors

The complex form algebra of Schwinger functions of a Dirac field on a Euclidean R ^d with arbitrary dimension d is decomposed into the form algebras of Majorana spinors and of Weyl spinors. The existence of real form algebras is investigated. The reality condition leads to severe restrictions in the case of Majorana forms which do not agree with the results of classical field theory. For all real form algebras Euclidean spinors are constructed as elements of a measure space.     

Certain Extensions¶of Vertex Operator Algebras of Affine Type

We generalize Feigin and Miwa's construction of extended vertex operator (super)algebras A _k (sl(2)) for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type. Received: 7 March 2000 / Accepted: 10 November 2000     

Local atomic structures of amorphous Pd80Si20 alloys and their configuration heredity in the rapid solidification

The local atomic structures of amorphous Pd₈₀Si₂₀ alloys and their configuration heredity in the rapid solidification are investigated by a molecular dynamics simulation with the help of cluster-type index method based on Honeycutt–Anderson bond-type index and an inversely tracking technique of atomic trajectories. Their short-range orders are found to be various Kasper clusters as well as their distorted configurations, and among which (10 2/1441 8/1551) bi-capped square Archimedean anti-prism (BSAP) clusters are dominated, e.g. Si-centred Pd₁₀Si₁ clusters. These Kasper clusters mainly exist in the form of isolated basic clusters. Few medium-range orders can be detected, especially for Si-centred Kasper clusters. Similarly to icosahedrons of Cu–Zr amorphous alloys, their sustainable configuration heredity also occurs firstly in the super-cooled liquid region, and BSAP clusters have higher onset temperature T_onset and bigger descendible fraction F than other Kasper clusters in the rapid solidification of Pd₈₀Si₂₀ alloys.     

Atomic Effect Algebras with the Riesz Decomposition Property

We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any σ-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states.     

Vertex Operators – From a Toy Model to Lattice Algebras

Within the framework of the discrete Wess–Zumino–Novikov–Witten theory we analyze the structure of vertex operators on a lattice. In particular, the lattice analogues of operator product expansions and braid relations are discussed. As the main physical application, a rigorous construction for the discrete counterpart g _n $ of the group valued field g(x) is provided. We study several automorphisms of the lattice algebras including discretizations of the evolution in the WZNW model. Our analysis is based on the theory of modular Hopf algebras and its formulation in terms of universal elements. Algebras of vertex operators and their structure constants are obtained for the deformed universal enveloping algebras . Throughout the whole paper, the abelian WZNW model is used as a simple example to illustrate the steps of our construction. Received: 16 December 1996 / Accepted: 5 May 1997     

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions

In (Rie?anová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set $\mathcal{M}$ of states can by embedded into a Hilbert space effect algebra $\mathcal{E}(l_{2}(\mathcal{M}))$ . We consider the problem when its effect algebraic MacNeille completion $\hat{E}$ can be also embedded into the same Hilbert space effect algebra $\mathcal {E}(l_{2}(\mathcal{M}))$ . That is when the ordering set $\mathcal{M}$ of states on E can be extended to an ordering set of states on $\hat{E}$ . We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.     

Unification Theory of Different Causal Algebras and Its Applications to Theoretical Physics

This paper gives a generalization of group theory, i.e. a unification theory of different causal algebras, and its applications to theoretical physics. We propose left and right causal algebras, left and right causal decomposition algebras, causal algebra and causal decomposition algebras in terms of quantitative causal principle. The causal algebraic system of containing left (or right) identity I _jL (or I _jR ) is called as the left (or right) causal algebra, and associative law is deduced. Furthermore the applications of the new algebraic systems are given in theoretical physics, specially in the reactions of containing supersymmetric particles, we generally obtain the invariance of supersymmetric parity of multiplying property. In the reactions of particles of high energy, there may be no identity, but there are special inverse elements, which make that the relative algebra be not group, however, the causal algebra given in this paper is just a tool of severely and directly describing the real reactions of particle physics. And it is deduced that the causal decomposition algebra is equivalent to group.     

Classification of Cylindrically Symmetric Static Spacetimes According to Their Ricci Collineations

A complete classification of cylindrically symmetric static Lorentzian manifolds according to their Ricci collineations (RCs) is provided. The Lie algebras of RCs for the non-degenerate Ricci tensor have dimensions 3 to 10, excluding 8 and 9. For the degenerate tensor the algebra is mostly but not always infinite dimensional; there are cases of 10-, 5-, 4- and 3-dimensional algebras. The RCs are compared with the Killing vectors (KVs) and homothetic motions (HMs). The (non-linear) constraints corresponding to the Lie algebras are solved to construct examples which include some exact solutions admitting proper RCs. Their physical interpretation is given. The classification of plane symmetric static spacetimes emerges as a special case of this classification when the cylinder is unfolded.     

Compact linearly ordered effect algebras

Compact linearly ordered effect algebras are shown to be Archimedean MV-algebras with unique maximal ideal. Consequently they are isomorphic to the unit interval [0, 1] equipped with restricted addition.     

OnE 10 and the DDF construction

An attempt is made to understand the root spaces of Kac Moody algebras of hyperbolic type, and in particularE ₁₀, in terms of a DDF construction appropriate to a subcritical compactified bosonic string. While the level-one root spaces can be completely characterized in terms of transversal DDF states (the level-zero elements just span the affine subalgebra), longitudinal DDF states are shown to appear beyond level one. In contrast to previous treatments of such algebras, we find it necessary to make use of a rational extension of the self-dual root lattice as an auxiliary device, and to admit non-summable operators (in the sense of the vertex algebra formalism). We demonstrate the utility of the method by completely analyzing a non-trivial level-two root space, obtaining an explicit and comparatively simple representation for it. We also emphasize the occurrence of several Virasoro algebras, whose interrelation is expected to be crucial for a better understanding of the complete structure of the Kac Moody algebra.Supported by Konrad-Adenauer-Stiftung e.V.This article was processed by the author using the Latex style filepljour1 from Springer-Verlag.     

Simple algebras and Drinfeld doubles

The Drinfeld double structure underlying all the Cartan series of simple Lie algebras is discussed. The two solvable algebras that allow its definition are constructed enlarging each simple algebra of rank n with a central Abelian algebra of dimension n. In these solvable algebras, isomorphic to the two Borel subalgebras, a pairing can be built. The complete machinery of Drinfeld doubles is described in all details. This offers a new approach to the explicit construction of canonical quantum deformation of simple algebras and fixes uniquely, independently and differently from known conventions, canonical bases for all of them. The Drinfeld doubles for A _n and C _n are explicitly written. The full quantization of su(3) is discussed in terms of standard commutators as the A ₂ Drinfeld double requires. The text was submitted by the authors in English.     

Casimir Elements for Some Graded Lie Algebras and Superalgebras

We consider a class of Lie algebras L such that L admits a grading by a finite Abelian group so that each nontrivial homogeneous component is one-dimensional. In particular, this class contains simple Lie algebras of types A, C and D where in C and D cases the rank of L is a power of 2. We give a simple construction of a family of central elements of the universal enveloping algebra U(L). We show that for the A-type Lie algebras the elements coincide with the Gelfand invariants and thus generate the center of U(L). The construction can be extended to Lie superalgebras with the additional assumption that the group grading is compatible with the parity grading.     

Some Quantum-like Hopf Algebras which Remain Noncommutative when q = 1

Starting with only three of the six relations defining the standard (Manin) GL _q (2), we try to construct a quantum group. The antipode condition requires some new relations, but the process stops at a Hopf algebra with a Birkhoff–Witt basis of irreducible monomials. The quantum determinant is group-like but not central, even when q = 1. So, the two Hopf algebras constructed in this way are not isomorphic to the Manin GL _q (2), all of whose group-like elements are central. Analogous constructions can be made starting with the Dipper–Donkin version of GL _q (2), but these turn out to be included in the two classes of Hopf algebras described above.     

Fine Gradings on Non-simple Lie Algebras: Example of o(4,C)

On any Lie algebra L, it is of significant convenience to have at one's disposal all the possible fine gradings of L, since they reflect the basic structural properties of the Lie algebra. They also provide useful bases of the representations of the algebra -- namely such bases that are preserved by the commutator.We list all the six fine gradings on the non-simple Lie algebra o(4,C) and we explain their relation to the fine gradings of the Lie algebra sl(2,C) where relevant. The existence of such relation is not surprising, since o(4,C) is in fact a product of two specimen of sl(2,C). The example of o(4,C) is especially important due to the fact that one of its fine gradings is not generated by any MAD-group. This proves that, unlike in the case of classical simple Lie algebras over C, on the non-simple classical Lie algebras over C there can exist a fine grading that is not generated by any MAD-group on the Lie algebra.