Realism about Structure

This chapter rebuts objections to the account, dispelling concerns about taking the mathematical structures of our best physical theories seriously. It outlines further aspects of the book’s realism about structure and explains what this realism is, and what it is not, committed to. It discusses the ideas of reasonably straightforward interpretations of physics; of perspicuous formulations of physics; and of direct and indirect representations. It argues against a different kind of realism about structure in physics, which is at once both stronger and weaker than the realism about structure defended in the book. It ends by arguing against focusing on the structure of a theory’s models in our physical theorizing in general, and the model isomorphism criterion for theoretical equivalence in particular.

Simple modules over the Lie algebras of divergence zero vector fields on a torus

Let {n\geq 2} be an integer, {\mathbb{S}_{n}} the Lie algebra of divergence zero vector fields on an n-dimensional torus, and {\mathcal{K}_{n}} the Weyl algebra over the Laurent polynomial algebra {A_{n}=\mathbb{C}[x_{1}^{\pm 1},x_{2}^{\pm 1},\dots,x_{n}^{\pm 1}]} . For any {\mathfrak{sl}_{n}} -module V and any module P over {\mathcal{K}_{n}} , we define an {\mathbb{S}_{n}} -module structure on the tensor product {P\otimes V} . In this paper, necessary and sufficient conditions for the {\mathbb{S}_{n}} -modules {P\otimes V} to be simple are given, and an isomorphism criterion for nonminuscule {\mathbb{S}_{n}} -modules is provided. More precisely, all nonminuscule {\mathbb{S}_{n}} -modules are simple, and pairwise nonisomorphic. For minuscule {\mathbb{S}_{n}} -modules, minimal and maximal submodules are concretely determined.

On isomorphism criterion for a subclass of complex filiform Leibniz algebras

In this paper, we present an algorithm to give the isomorphism criterion for a subclass of complex filiform Leibniz algebras arising from naturally graded filiform Lie algebras. This subclass appeared as a Leibniz central extension of a linear deformation of filiform Lie algebra. We give the table of multiplication choosing appropriate adapted basis, identify the elementary base changes and describe the behavior of structure constants under these base changes, then combining them the isomorphism criterion is given. The final result of calculations for one particular case also is provided.