Journal of Generalized Lie Theory and Applications

A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a series of papers which show that such a structure allows to do non-commutative analysis, in the sense of differential calculus, on a large class of metric spaces, some of them fractals. We also describe a formal, universal calculus with binary decorated planar trees, which underlies any dilatation structure.

Starting from a Hecke R-matrix, Jing and Zhang constructed a new deformation Uq(sl2) of U(sl2) and studied its finite dimensional representations in [Pacific J. Math., 171 (1995), 437-454]. In this note, more irreducible representations for this algebra are constructed. At first, by using methods in noncommutative algebraic geometry the points of the spectrum of the category of representations over this new deformation are studied. The construction recovers all finite dimensional irreducible representations classified by Jing and Zhang, and yields new families of infinite dimensional irreducible weight representations.

An analogue of the Livernet–Loday operad for two compatible brackets, which is a flat deformation of the bi-Hamiltonian operad is constructed. The Livernet–Loday operad can be used to define ?-products and deformation quantization for Poisson structures. The constructed operad is used in the same way, introducing a definition of operadic deformation quantization of compatible Poisson structures.

We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controlling the deformation theory of associative algebras. We prove that there is a canonical solution (choice independent) within the class of graphs without circuits, i.e. at the level of the free operad, without imposing the Jacobi identity. The proof is a consequence of the unique factorization property of the pre-Lie algebra of graphs (tree operad), where composition is the insertion of graphs. The restriction to graphs without circuits, i.e. at “tree level”, accounts for the interpretation as a semi-classical solution. The fact that this solution is canonical should not be surprising, in view of the Hausdorff series, which lies at the core of almost all quantization prescriptions.

An operator constraint for a slave-boson decomposition in t-J model of high temperature superconductivity is considered. It is shown that the constraint can be resolved by introducing a non-associative operator. In this case the constraint is an antiassociative generating relation of a new algebra. Similar constraint is offered for splitting the gluon.