Idempotent (Asymptotic) Mathematics and the Representation Theory

Abstract

1. Introduction. Idempotent Mathematics is based on replacing the usual arithmetic operations by a new set of basic operations (e.g., such as maximum or minimum), that is on replacing numerical fields by idempotent semirings and semifields. Typical (and the most common) examples are given by the so-called (max, +) algebra R max and (min, +) algebra R min . Let R be the field of real numbers. Then R max = R ∪ {−∞} with operations x ⊕ y = max{x, y} and x ⊙ y = x + y. Similarly R min = R ∪ {+∞} with the operations ⊕ = min, ⊙ = +. The new addition ⊕ is idempotent, i.e., x ⊕ x = x for all elements x. Idempotent Mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking pure imaginary values. Some problems that are nonlinear in the traditional sense turn out to be linear over a suitable idempotent semiring (idempotent superposition principle [1]). For example, the Hamilton-Jacobi equation (which is an idempotent version of the Schrödinger equation) is linear over R min and R max .The basic paradigm is expressed in terms of an idempotent correspondence principle [2].This principle is similar to the well-known correspondence principle of N. Bohr in quantum theory (and closely related to it). Actually, there exists a heuristic correspondence between important, interesting and useful constructions and results of the traditional mathematics over fields and analogous constructions and results over idempotent semirings and semifields (i.e., semirings and semifields with idempotent addition).A systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. As a result, in parallel with the traditional mathematics over rings, its "shadow", the Idempotent Mathematics, appears. This "shadow" stands approximately in the same relation to the traditional mathematics as classical physics to quantum theory. In many respects Idempotent Mathematics is simpler than the traditional one. However, the transition from traditional concepts and results to their idempotent analogs is often nontrivial.There is an idempotent version of the theory of linear representations of groups. We shall present some basic concepts and results of the idempotent representation theory. In the framework of this theory the well-known Legendre transform can be treated as an R max -version of the traditional Fourier transform (this observation

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

“…This point of view was presented by G. L. Litvinov and V. P. Maslov [102][103][104], see also [110,111]. In other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the fields of real and complex numbers.…”

“…By misuse of language, we shall also call this passage the Maslov dequantization. Connections with physics and the meaning of imaginary values of the Planck constant are discussed below (Section 6) and in [110,111]. The idempotent semiring R∪{−∞}∪{+∞} with the operations ⊕ = max, ⊙ = min can be obtained as a result of a "second dequantization" of C, R or R + .…”

The Maslov Dequantization, idempotent and tropical mathematics: a very brief introduction

“…This point of view was presented by G. L. Litvinov and V. P. Maslov [70][71][72], see also [78,79]. In other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the fields of real and complex numbers.…”

“…Thus all elements of S are nonnegative: 0 a for all a ∈ S. Due to the existence of this order, idempotent analysis is closely related to the lattice theory, theory of vector lattices, and theory of ordered spaces. Moreover, this partial order allows to model a number of basic "topological" concepts and results of idempotent analysis at the purely algebraic level; this line of reasoning was examined systematically in [76][77][78][79][80] and [17].…”

“…There is an "idempotent" version of the theory of linear representations of groups and semigroups and the abstract harmonic analysis, see, e.g., [79]. In the framework of this theory the well-known Legendre transform can be treated as an idempotent version of the traditional Fourier transform (this observation is due to V. P. Maslov).…”