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.     

On Root Multiplicities of Some Hyperbolic Kac–Moody Algebras

Using the coset construction, we compute the root multiplicities at level-3 for some hyperbolic Kac–Moody algebras including the basic hyperbolic extension of A₁ ¹ and E₁₀.     

Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

We systematically study noncommutative and nonassociative algebras $A$ and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of $A$-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras $A$ the full subcategory of symmetric $A$-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.     

A(SLq(2)) at Roots of Unity is a Free Module over A(SL(2))

It is shown that when q is a primitive root of unity of order not equal to 2 mod 4, A(SL_q(2)) is a free module of finite rank over the coordinate ring of the classical group SL(2). An explicit set of generators is provided.     

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.     

Cost Efficient Lie Group Integrators in the RKMK Class

In this work a systematic procedure is implemented in order to minimise the computational cost of the Runge—Kutta—Munthe-Kaas (RKMK) class of Lie-group solvers. The process consists of the application of a linear transformation to the stages of the method and the analysis of a graded free Lie algebra to reduce the number of commutators involved. We consider here RKMK integration methods up to order seven based on some of the most popular Runge—Kutta schemes.This revised version was published online in October 2005 with corrections to the Cover Date.     

On a Plus-Construction for Algebras over an Operad

We prove a analogous to Quillen's plus-construction in the category of algebras over an operad. For that purpose we prove that this category is a closed model category and prove the existence of an obstruction theory. We apply further this plus-construction for the specific cases of Lie algebras and Leibniz algebras which are a noncommutative version of Lie algebras: let sl(A) be the kernel of the trace map gl(A)A/[A,A], where A is an associative algebra with unit and gl(A) is the Lie algebra of matrices over A. Then the homotopy of slA)⁺ in the category of Lie algebras is the cyclic homology of A whereas it is the Hochschild homology of A in the category of Leibniz algebras.     

强可约极大三角代数的空间一秩元和谱一秩元

本文研究了强可约极大三角代数R中的空间一秩元和谱一秩元 ,得到R中的算子 T为空间一 秩元的充要条件和R中的紧算子为谱一秩元的充要条件 .