From Effect Algebras to Algebras of Effects (Sum Brouwer—Zadeh Algebras)

The algebraic structures arising in the axiomatic framework of unsharp quantummechanics based on effect operators on a Hilbert space are investigated. It isstressed that usually considered effect algebras neglect the unitary Brouwerianmap of complementation, and the main results based on this complementationare collected, showing the enrichment produced into the theory by its introduction.In particular, in these structures two notions of sharpness can be considered: K-sharpness induced by the usual complementation of effect algebrasand B-sharpness induced by this new complementation. Quantum (resp., classical) SBZalgebras are then characterized by the condition of B-coherence (resp., B-coherence plusB-compatibility), showing that in this case the poset of all B-sharp elements is orthomodular (resp., Boolean algebra). In the unsharp contextof effect operators, the finite dimensionality of the Hilbert space or the finitenessof a von Neumann algebra are both characterized by a de Morgan property ofthe Brouwer complementation. Moreover, since effect operators on a pre-Hilbertspace give rise to a standard model of effect algebras, a characterization ofcompleteness of pre-Hilbert spaces is given making use of the Brouwercomplement.     

Number Operator Algebras and Deformations of ε-Poisson Algebras

It is well known that the Lie algebra structure on quantum algebras gives rise to a Poisson algebra structure on classical algebras as the Planck constant goes to 0. We show that this correspondence still holds in the generalization of superalgebra introduced by Scheunert, called -algebra. We illustrate this with the example of Number Operator Algebras, a new kind of object that we have defined and classified under some assumptions.     

From Quantum Quasi-Shuffle Algebras to Braided Rota–Baxter Algebras

In this letter, we use quantum quasi-shuffle algebras to construct Rota–Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota–Baxter algebras, the relevant object of Rota–Baxter algebras in a braided tensor category. Examples of such new algebras are provided using quantum multi-brace algebras in a category of Yetter–Drinfeld modules.     

Polyvector Super-Poincaré Algebras

A class of ₂-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form where the algebra of generalized translations W=W₀+W₁ is the maximal solvable ideal of W₀ is generated by W₁ and commutes with W. Choosing W₁ to be a spinorial module (a sum of an arbitrary number of spinors and semispinors), we prove that W₀ consists of polyvectors, i.e.all the irreducible submodules of W₀ are submodules of We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of invariant valued bilinear forms on the spinor module S.     

Generalized Quantum Current Algebras

Two general families of new quantum-deformed current algebras are proposed and identified both as infinite Hopf family of algebras ,a structure which enables one to define “Tensor products“ of these algebras.The standard quantum affine algebras turn out to be a very special case of the two algebra families,in which case the infinite Hopt family structure degenerates into a standard Hopf algebra.The relationship between the two algebraic families as well as their various special examples are discussed,and the free boson representation is also considered.     

Courant–Dorfman Algebras and their Cohomology

We introduce a new type of algebra, the Courant–Dorfman algebra. These are to Courant algebroids what Lie–Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant–Dorfman algebra ${(\mathcal{R}, \mathcal{E})}$ we associate a differential graded algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ in a functorial way by means of explicit formulas. We describe two canonical filtrations on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ , and derive an analogue of the Cartan relations for derivations of ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; we classify central extensions of ${\mathcal{E}}$ in terms of ${H^2(\mathcal{E}, \mathcal{R})}$ and study the canonical cocycle ${\Theta \in \mathcal{C}^3(\mathcal{E}, \mathcal{R})}$ whose class ${[\Theta]}$ obstructs re-scalings of the Courant–Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ ; for Courant–Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra ${\mathcal{C}(\mathcal{E}, \mathcal{R})}$ is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.     

Rota–Baxter Algebras and New Combinatorial Identities

The word problem for an arbitrary associative Rota–Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects are indicated.      

Kochen–Specker Theorem for von Neumann Algebras

The Kochen–Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type I_n factor as algebra of observables, including I_∞. Afterwards, we give a proof of the Kochen–Specker theorem for an arbitrary von Neumann algebra without summands of types I₁ and I₂, using a known result on two-valued measures on the projection lattice . Some connections with presheaf formulations as proposed by Isham and Butterfield are made.     

Factoriality of Boejko–Speicher von Neumann Algebras

We study the von Neumann algebra generated by q–deformed Gaussian elements l_i+l^*_i, where operators l_i fulfill the q–deformed canonical commutation relations l_il^*_j–ql^*_jli=_ij for –1<q<1. We show that if the number of generators is finite, greater than some constant depending on q, it is a II₁ factor which does not have the property . Our technique can be used for proving factoriality of many examples of von Neumann algebras arising from some generalized Brownian motions, both for the type II₁ and type III case.Research supported by State Committee for Scientific Research (KBN) grant 2 P03A 007 23.     

Twisted Poincaré Duality for some Quadratic Poisson Algebras

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier. This research of the first author was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The second author was supported by EPSRC Grant EP/D034167/1.     

Extended Diffeomorphism Algebras¶and Trajectories in Jet Space

Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra?DRO(N) be the extension of diff(N)⊕ diff(1) by its four inequivalent Virasoro-like cocycles. Here diff(N) is the diffeomorphism algebra in N-dimensional spacetime and diff(1) describes reparametrizations of trajectories in the space of tensor-valued p-jets. DRO(N) has a Fock module for each p and each representation of gl(N). Analogous representations for gauge algebras (higher-dimensional Kac–Moody algebras) are also given. The reparametrization symmetry can be eliminated by a gauge fixing procedure, resulting in previously discovered modules. In this process, two DRO(N) cocycles transmute into anisotropic cocycles for diff(N). Thus the Fock modules of toroidal Lie algebras and their derivation algebras are geometrically explained. Received: 29 October 1998 / Accepted: 2 May 2000     

Diassociative Algebras and Milnor’s Invariants for Tangles

We extend Milnor’s μ-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for μ-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday’s diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.     

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

Ultraweak Continuity of σ-derivations on von Neumann Algebras

Let σ be a surjective ultraweakly continuous ∗-linear mapping and d be a σ-derivation on a von Neumann algebra . We show that there are a surjective ultraweakly continuous ∗-homomorphism and a Σ-derivation such that D is ultraweakly continuous if and only if so is d. We use this fact to show that the σ-derivation d is automatically ultraweakly continuous. We also prove the converse in the sense that if σ is a linear mapping and d is an ultraweakly continuous ∗-σ-derivation on , then there is an ultraweakly continuous linear mapping such that d is a ∗-Σ-derivation.      

A Representation of Quantum Measurement in Nonassociative Algebras

Starting from an abstract setting for the Lüders-von Neumann quantum measurement process and its interpretation as a probability conditionalization rule in a non-Boolean event structure, the author derived a certain generalization of operator algebras in a preceding paper. This is an order-unit space with some specific properties. It becomes a Jordan operator algebra under a certain set of additional conditions, but does not own a multiplication operation in the most general case. A major objective of the present paper is the search for such examples of the structure mentioned above that do not stem from Jordan operator algebras; first natural candidates are matrix algebras over the octonions and other nonassociative rings. Therefore, the case when a nonassociative commutative multiplication exists is studied without assuming that it satisfies the Jordan condition. The characteristics of the resulting algebra are analyzed. This includes the uniqueness of the spectral resolution as well as a criterion for its existence, subalgebras that are Jordan algebras, associative subalgebras, and more different levels of compatibility than occurring in standard quantum mechanics. However, the paper cannot provide the desired example, but contribute to the search by the identification of some typical differences between the potential examples and the Jordan operator algebras and by negative results concerning some first natural candidates. The possibility that no such example exists cannot be ruled out. However, this would result in an unexpected new characterization of Jordan operator algebras, which would have a significant impact on quantum axiomatics since some customary axioms (e.g., power-associativity or the sum postulate for observables) might turn out to be redundant then.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

The Matrix Algebras of Continuous Groups with Antilinear Operations

The corepresentation theory of continuous groups is presented without the assumption that the subgroup G of the group with antilinear operations is unitary. Continuous groups of the form: G+a ₀ G are defined, where G denotes a linear Lie group and a ₀ denotes an antilinear operation which fulfils the condition a²₀=±1a^{2}_{0}=\pm1. The matrix algebras connected with the groups G+a ₀ G are defined. The structural constants of these algebras fulfill the conditions following from the Jacobi identities. Applications are presented to the groups G=SU(d), d=1,2,… , for a ₀=K, the complex conjugation operation, and to the group SL(2,C) for a ₀=K or Θ, the time-reversal operation.     

States on Effect Algebras That Have the ϕ-Symmetry Property

The relationship between the property of havinga full set of states and the archimedean property in thecase of -symmetric effect algebra is explored, andequivalent conditions are obtained.     

Some Remarks on the Cohomology of Krichever–Novikov Algebras

We calculate the continuous cohomology of the Lie algebra of meromorphic vector fields on a compact Riemann surface from the cohomology of the holomorphic vector fields on the open Riemann surface pointed in the poles. This cohomology has been given by Kawazumi. Our result shows the Feigin–Novikov conjecture.