Topical on Green Process Engineering

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Topical on Green Process Engineering

Martine Poux, Patrick Cognet

To cite this version:

Martine Poux, Patrick Cognet. Topical on Green Process Engineering. Green Processing and Synthe-sis, De Gruyter, 2019, 8 (1), pp.786-786. �10.1515/gps-2019-0047�. �hal-02297701�

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To cite this version:

Poux, Martine

and Cognet, Patrick

Topical on Green Process Engineering.

(2019) Green Processing and Synthesis, 8 (1). 786. ISSN 2191-9542

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Green Process Synth 2019; 8: 786

Martine Poux* and Patrick Cognet*

Topical on Green Process Engineering

https://doi.org/10.1515/gps-2019-0047

This Topical on Green Process Engineering collects some selected contributions presented at the 6th Inter-national Congress on Green Process Engineering (GPE 2018) held from 3 to 6 June 2018 in Toulouse (France). 10 years ago, we created the first International Congress on Green Process Engineering also in Toulouse and it was the beginning of the GPE congress series. 10 years is a milestone, after 5 other editions in different countries around the world. 10 years has seen green process engi-neering (GPE) concepts in both education and research be vastly implemented. Education programs dedicated to GPE have appeared in many universities with courses covering numerous aspects, ranging from chemistry to new process technologies. In research, many groups have re-oriented their investigation paths in order to propose new solutions, including both environmentally friendly processes and economical aspects. Indeed, green process engineering is now an essential topic in most of general chemical engineering conferences and

Open Access. © 2019 Poux and Cognet, published by De Gruyter. Open Access. © 2019 Kazumasa Nomura and Paul Terwilliger, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License.

Spec. Matrices 2019; 7:1–19

Research Article

Open Access

Kazumasa Nomura* and Paul Terwilliger

Self-dual Leonard pairs

https://doi.org/10.1515/spma-2019-0001

Received May 8, 2018; accepted September 22, 2018

Abstract: Let F denote a field and let V denote a vector space over F with finite positive dimension. Consider

a pair A, A∗_{of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an}

irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of V that swaps A and A∗_{. Such an automorphism}

is unique, and called the duality A↔ A∗. In the present paper we give a comprehensive description of this

duality. In particular, we display an invertible F-linear map T on V such that the map X→TXT−1is the duality

A↔ A∗. We express T as a polynomial in A and A∗. We describe how T acts on 4 flags, 12 decompositions,

and 24 bases for V.

Keywords: Leonard pair, tridiagonal matrix, self-dual Classification: 17B37, 15A21

1 Introduction

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair A, A∗_{of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an}

irreducible tridiagonal fashion. Such a pair is called a Leonard pair (see [13, Definition 1.1]). The Leonard pair

A, A∗_{is said to be self-dual whenever there exists an automorphism of the endomorphism algebra of V that}

swaps A and A∗_{. In this case such an automorphism is unique, and called the duality A}_↔_A∗_.

The literature contains many examples of self-dual Leonard pairs. For instance (i) the Leonard pair associ-ated with an irreducible module for the Terwilliger algebra of the hypercube (see [4, Corollaries 6.8, 8.5]); (ii) a Leonard pair of Krawtchouk type (see [10, Definition 6.1]); (iii) the Leonard pair associated with an irreducible module for the Terwilliger algebra of a distance-regular graph that has a spin model in the Bose-Mesner alge-bra (see [1, Theorem], [3, Theorems 4.1, 5.5]); (iv) an appropriately normalized totally bipartite Leonard pair (see [11, Lemma 14.8]); (v) the Leonard pair consisting of any two of a modular Leonard triple A, B, C (see [2, Definition 1.4]); (vi) the Leonard pair consisting of a pair of opposite generators for the q-tetrahedron alge-bra, acting on an evaluation module (see [5, Proposition 9.2]). The example (i) is a special case of (ii), and the examples (iii), (iv) are special cases of (v).

Let A, A∗_{denote a Leonard pair on V. We can determine whether A, A}∗_{is self-dual in the following way.}

By [13, Lemma 1.3] each eigenspace of A, A∗_{has dimension one. Let}_{_θ_i_}d

i=0denote an ordering of the

eigen-values of A. For 0 ≤ i ≤ d let vi denote a θi-eigenvector for A. The ordering{θi}di=0is said to be standard

whenever A∗_{acts on the basis}_{_v_i_}d

i=0in an irreducible tridiagonal fashion. If the ordering{θi}d_i=0is standard

then the ordering{θd−i}di=0is also standard, and no further ordering is standard. Similar comments apply to A∗_{. Let}_{_θ_i_}d

i=0denote a standard ordering of the eigenvalues of A. Then A, A∗is self-dual if and only if{θi}d_i=0

is a standard ordering of the eigenvalues of A∗_{(see [7, Proposition 8.7]).}

*Corresponding Author: Kazumasa Nomura: Tokyo Medical and Dental University, Ichikawa, 272-0827, Japan,

E-mail: knomura@pop11.odn.ne.jp

Paul Terwilliger: Department of Mathematics, University of Wisconsin, Madison, WI53706, USA, E-mail:

terwilli@math.wisc.edu

This work is licensed under the Creative Commons Attribution alone 4.0 License.

* Corresponding authors: Martine Poux and Patrick Cognet,

Laboratoire de Génie Chimique, University of Toulouse, CNRS, INPT, UPS, Toulouse, France, e-mail: martine.poux@ensiacet.fr (Martine Poux); patrick.cognet@ensiacet.fr (Patrick Cognet)

specific international journals have appeared, compi-ling all articles relevant to this area. 10 years has also been sufficient to evaluate the progress carried out in industries due to the implementation of new process technologies.

More than 250 participants attended this conference coming from 28 countries all around the world, displaying a very active participation into the sessions and other conference activities, as was highlighted by the great attendance to the plenary lectures and oral presentations, as well as by the lively discussions around the poster presentations.

We had the pleasure to welcome 4 plenary speakers, 16 keynotes speakers, 71 oral communications and a hundred poster presentations. Two prizes for the best posters were awarded sponsored by PROSIM S.A.

A great success which shows that the International Congress on Green Process Engineering is always an excellent opportunity to present and discuss the progress and latest advances in the area, as well as to network in order to build the new industry, the factory of the future!

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