Total extensions of effect algebras

It is shown that if the natural order on a total extension of an effect algebra coincides with the order on, then is unique. The structure of is called a QI-algebra. It is shown that a QI-algebra is less general than a QMV-algebra, but that a QI-algebra is equivalent to a quasi-linear QMV-algebra. Some examples are given and the properties of these structures are studied.     

Tensor Product of Distributive Sequential Effect Algebras and Product Effect Algebras

A distributive sequential effect algebra (DSEA) is an effect algebra on which a distributive sequential product with natural properties is defined. We define the tensor product of two arbitrary DSEA’s and we give a necessary and sufficient condition for it to exist. As a corollary we obtain the result (see Gudder, S. in Math. Slovaca 54:1–11, 2004, to appear) that the tensor product of a pair of commutative sequential effect algebras exists if and only if they admit a bimorphism. We further obtain a similar result for the tensor product of a pair of product effect algebras.     

Open Problems for Sequential Effect Algebras

A sequential effect algebra (SEA) is an effect algebra on which a sequential product with certain natural properties is defined. In such structures, we can study combinations of simple measurements that are series as well as parallel. This article presents some open problems for SEAs together with background material, comments and partial results. Two examples of open problems are the following: is A° B = A ^1/2 BA ^1/2 the only sequential product on a Hilbert space SEA? It is known that the sharp elements of a SEA form an orthomodular poset. Is every orthomodular poset isomorphic to the set of sharp elements for some SEA?     

Graded quasi-Lie algebras

This paper is concerned with a new class of graded algebras naturally uniting within a single framework various deformations of the Witt, Virasoro and other Lie algebras based on twisted and deformed derivations, as well as color Lie algebras and Lie superalgebras. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

Projective geometry from Poisson algebras

By analogy with the Poisson algebra of quadratic forms on the symplectic plane and with the concept of duality in the projective plane introduced by Arnold (2005) [1], where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, on the pseudo-sphere and on the hyperboloid, to obtain analogous duality concepts and similar results for spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic either to the Lie algebra of the vectors in R³${\mathbb{R}}^3$, with the vector product, or to algebra s_l2(R)$s{l}_2\left(\mathbb{R}\right)$. The Tomihisa identity, introduced in (Tomihisa, 2009) [3] for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relationships between the different definitions of duality in projective geometry inherited by these structures are shown here.     

On Drinfeld''s realization of quantum affine algebras

We show how to obtain Drinfeld's realization of quantum nontwisted affine algebras from the quantized Cartan-Weyl basis. The formulae for comultiplications in this realization are discussed.     

Novikov algebras carrying an invariant Lorentzian symmetric bilinear form

In this note, we shall classify Novikov algebras that admit an invariant Lorentzian symmetric bilinear form.     

Infinitesimal deformations of filiform Lie algebras of order 3

The Lie algebras of order$F$ have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order$F$ which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014).     

Entropy algebras and Birkhoff factorization

We develop notions of Rota–Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin’s renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.     

Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

Using deformation theory, Braverman and Joseph constructed certain primitive ideals in the enveloping algebras of the simple Lie algebras. Except in the case sl(2,C)$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$, there is a special value of the deformation parameter giving an ideal of infinite codimension. For the classical Lie algebras, the uniqueness of the special value is equivalent to the existence of tensors with very particular properties. The existence of these tensors was concluded abstractly by Braverman and Joseph but here we present explicit formulae. This allows a rather direct computation of the special value of the deformation parameter.     

SUSY QM, symmetries and spectrum generating algebras for two-dimensional systems

We show in a systematic and clear way how factorization methods can be used to construct the generators for hidden and dynamical symmetries. This is shown by studying the 2D problems of hydrogen atom, the isotropic harmonic oscillator and the radial potential Aρ^2ζ−2 − Bρ^ζ−2. We show that in these cases the non-compact (compact) algebra corresponds to so(2, 1) (su(2)).     

Dynamical algebras for Pöschl-Teller Hamiltonian hierarchies

The dynamical algebras of the trigonometric and hyperbolic symmetric Pöschl-Teller Hamiltonian hierarchies are obtained. A kind of discrete-differential realizations of these algebras are found which are isomorphic to so(3, 2) Lie algebras. In order to get them, first the relation between ladder and factor operators is investigated. In particular, the action of the ladder operators on normalized eigenfunctions is found explicitly. Then, the whole dynamical algebras are generated in a straightforward way.     

Faux-boolean algebras and classical models

A certain substructure of the lattice of projections on a Hilbert space is defined, and it is shown under what conditions such a structure can be said to have a classical model (in a sense made precise). The results have implications for the interpretation of quantum mechanics.     

Classification of Lie algebras with naturally graded quasi-filiform nilradicals

The whole class of complex Lie algebras g$\mathfrak{g}$ having a naturally graded nilradical with characteristic sequence c(g)=(dimg−2,1,1)$c\left(\mathfrak{g}\right)=(dim\mathfrak{g}-2,1,1)$ is classified. It is shown that up to one exception, such Lie algebras are solvable.     

Noncommutative *-product continual Lie algebras and solvable models

Noncommutative counterparts of exactly solvable models are proposed on the basis of *-product continual Lie algebras. Examples of noncommutative Liouville and sine/sh-Gordon equations are discussed.