Chain tensor products and interval effect algebras

Weak and strongn-doublings (n∈N) are defined for an effect algebraP and the concept of a normal interval algebra is introduced. It is shown that the following statements are equivalent: (1) There is a morphism fromP into an interval algebra. (2)P admits a tensor product with every finite chain. (3)P has a weakn-doubling for everyn∈N. Moreover, the following are equivalent: (4)P is a normal interval algebra. (5)P admits a strong tensor product with every chain of length 2 ⁿ ,n∈N. (6)P has a strongn-doubling for everyn∈N. Finally, it is shown that ifP possesses an order-determining set of states, thenP is a normal interval algebra.     

Complex quantum enveloping algebras as twisted tensor products

We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universalR-matrices and give its relation to subgroups in the dual Hopf algebra.     

Sum logics and tensor products

A notion of factorizability for vector-valued measures on a quantum logic L enables us to pass from abstract logics to Hilbert space logics and thereby to construct tensor products. A claim by Kruszynski that, in effect, every orthogonally scattered measure is factorizable is shown to be false. Some criteria for factorizability are found.Work done partly when the second author visited the University of Nottingham, supported by SERC grant GR/G3/376.     

Sums and products of interval algebras

Aninterval algebra is an interval from zero to some positive element in a partially ordered Abelian group, which, under the restriction of the group operation to the interval, is a partial algebra. In this paper we study interval algebras from a categorical point of view, and show that Cartesian products and horizontal sums are effective as categorical products and coproducts, respectively. We show that the category of interval algebras admits a tensor product, and introduce a new class of interval algebras, which are in fact orthoalgebras, called-algebras.     

Quantum forms of tensor products

We describe the skew tensor-product structure inherent to Yang-Mills procedures pertaining to non-commutative spaces obtained by (skew) tensorization.Dedicated to Huzihiro Araki     

Graded tensor products of quantum logics

Two notions of grading of a quantum logic by a product of copies of the group ₂ are introduced and used to define graded tensor products of quantum logics.     

Bell-type inequalities on tensor products of quantumlogics

The problem of Bell-type inequalities on tensor products of quantum logics is studied. To formulate the problem the definition of tensor product used for sum logics by Hudson and Pulmannová is generalized to arbitrary orthomodular lattices and the existence of such product for lattices describing the simplest EPR-Bohm experiment is shown. It is proved that Bell-type inequalities hold for all product states.     

Tensor products of Hilbert space effect algebras

A definition of a tensor product in the category of Hilbert space effect algebras is introduced such that the tensor product reflects as much as possible of the physically important properties of the components. It is shown that in the complex case, there are two candidates to the tensor product, which are not equivalent. The situation is similar to the tensor product in the category of projection lattices, but also the two dimensional case is included. The case of tensor product of two classical systems is also investigated.     

On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

Letters in Mathematical Physics - In this paper, we study the asymptotics of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple...     

Effect test spaces and effect algebras

The concept of an effect test space, which is equivalent to a D-test space of Dvurečenskij and Pulmannová, is introduced. Connections between effect test space. (E-test space, for short) morphisms, and event-morphisms as well as between algebraic E-test spaces and effect algebras, are studied. Bimorphisms and E-test space tensor products are considered. It is shown that any E-test space admits a unique (up to an isomorphism) universal group and that this group, considered as a test group, determines the E-test space uniquely (up to an isomorphism).     

Effect algebras and unsharp quantum logics

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.     

On compatibility of tensor products on solutions to commutativity and WDVV equations

We show that topological conformal theory contains solutions for WDVV equations for genus zero amplitudes and solutions to Commutativity equations from its quantum mechanics on the annulus. We explain behavior of these solutions under the tensor product of the theories. We make a conjecture that it is compatible with the construction of solutions to WDVV equations from a solution to the Commutativity equation and explicitly check it in the first nontrivial case.     

On the reduction of tensor products of representations of pseudo-orthogonal groups

We verify and generalize a conjecture of Fulling [1] that the Kronecker product of a finite number of unitary representations, not all of which possess an invariant vector, of the Lorentz group SO ₀(1, n), any n2, does not contain the trivial representation discretely.     

On modular invariant partition functions for tensor products of conformal field theories

We give two results concerning the construction of modular invariant partition functions for conformal field theories constructed by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type SU(2)_KASU(2)_KB, finding all consistent theories for K_A, K_B odd. Second we show how known diagonal modular invariants can be used to construct some inherently asymmetric ones where the holomorphic and antiholomorphic theories do not share the same chiral algebra. Some explicit examples are given.     

Covering and gluing of algebras and differential algebras

Extending work of Budzyński and Kondracki, we investigate coverings and gluings of algebras and differential algebras. We describe in detail the gluing of two quantum discs along their classical subspace, giving a C^*-algebra isomorphic to a certain Podleś sphere, as well as the gluing of U_q^1/2(sl₂)-covariant differential calculi on the discs.     

Bicovariant quantum algebras and quantum Lie algebras

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun toU _q g, given by elements of the pure braid group. These operators—the reflection matrixYL ⁺ SL ^– being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO _q (N).This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139     

Relative tensor calculus and the tensor time derivative

A relative tensor calculus is formulated for expressing equations of mathematical physics. A tensor time derivative operator ▽ _b ^a is defined which operates on tensors λ^ia...ib. Equations are written in a rigid, flat, inertial or other coordinate system a, altered to relative tensor notation, and are thereby expressed in general flowing coordinate systems or materials b, c, d, .... Mirror tensor expressions for ▽ _b ^a λ^ic...id and ▽ _b ^a λ_ic...id exist in a relative geometry G if and only if a rigid coordinate system a exists in G, where ▽ _b ^a λ^ic = λ _,0c ^ic + λ^kev _ckc ^aic + λ _kc ^ic v _b ^ckc , ▽_jcλ^ic = λ _,jc ^ic + λ^kcΓ _{jc kc} ^ie , and v _b ^aic is the velocity of b relative to a with components in c. These operators are convenient in theoretical analyses and can be incorporated into machine programs for the numerical solution of physical problems.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

Effect of breakup on the deuteron nucleus spin orbit and tensor potentials

A previous treatment of breakup in terms of a coupled channel formulation is generalized so as to include the spin of the nucleus. The break up spectrum is described in terms of the eigenstates of the free neutron proton Hamiltonian, based on the Reid soft core potential. Nearly spherically symmetric as well as nearly quadrupole deformed break up states are included. The latter give large contributions to the correction to the spin orbit and tensor parts of the deuteron-nucleus elastic optical model. A numerical application for 13 MeV and 21.6 MeV deuterons incident on Ni is presented, in which virtual breakup is included to second order in the breakup transition matrix element. The breakup correction to the deuteron potential is strongly dependent on the orbital angular momentum and the energy of the deuteron-nucleus motion. The spin orbit potential is rendered less diffuse and acquires an imaginary part. The tensor potential correction is comparable in radial shape to the Watanabe static folding model result, and tends to increase its magnitude by nearly 100%.