Deformation of expressions for elements of an algebra

Summary

1. Introduction

We introduce a notion of deformation of expressions for elements of an algebra. Deformation quantization [Bayen et al. 1978a; 1978b] deforms the commutative world into a noncommutative world. In contrast, our formalism involves the deformation of expressions of elements of algebras from one commutative world into another.

2. Definition of *-functions and intertwiners

Let ℂ [ω] be the space of polynomials in one variable ω. For a complex parameter τ we define a new product on this space:

We see easily that *_τ makes ℂ [ω] into a commutative and associative algebra, which we denote by ℂ [ω], *_τIf, τ=0 then . ℂ [ω], *₀ is the usual polynomial algebra, and τ ∈ ℂ is called a deformation parameter. What is deformed is not the algebraic structure, but the expression of elements.

Intertwiners and infinitesimal intertwiners. It is not hard to verify that the mapping.