Spec. Matrices 2019; 7:1–19 Research Article Open Access Kazumasa Nomura* and Paul Terwilliger e-Polymers 2019; 19: 290–296 Self-dual Leonard pairs Communication Open Access https://doi.org/10.1515/spma-2019-0001 Received May 8, 2018; accepted September 22, 2018 Zhen Wang#, Fu-Hou Lei#*, Wen Li#*, Peng-Fei Li#, Ting Wang, Li-Ting Qin, Ge-Ge Cheng, Yao Sun, Yu-Yu E, Wen-Bo Xie andAbstract: Jie DengLet F denote a eld and let V denote a vector space over F with nite 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 Preparation of rosin-basedirreducible tridiagonal polymer fashion. Such a pair microspheres 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 as a stationary phaseis unique, andin called high-performance the duality A A∗. In the present liquid 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− is the duality → chromatography Ato Aseparate∗. We express T as apolycyclic polynomial in A and aromaticA∗. We describe how T acts on ags, decompositions, ↔ hydrocarbons andand alkaloids 24 bases for V. Keywords: Leonard pair, tridiagonal matrix, self-dual https://doi.org/10.1515/epoly-2019-0029 theophylline–caffeine and camptothecin–10-hydroxy- 17B37,15A21 Received November 12, 2018; accepted DecemberClassication: 17, 2018. camptothecin, and the good results were obtained with the Rs of 4.617 ± 0.005 and 2.245 ± 0.049, respectively. Abstract: Acrylpimaric acid ethylene glycol acrylate was used as the cross-linking agent to prepare rosin-based Keywords: rosin-based stationary phase; alkaloids; poly- polymer microspheres through membrane emulsifica- Introductioncyclic aromatic hydrocarbons; polymer; microspheres tion–microsuspension polymerization. The prepared microspheres were used as the stationaryLet phaseF denote in high- a eld and let V denote a vector space over F with nite positive dimension. We consider a performance liquid chromatography for the separation of pair A, A∗ of diagonalizable1 IntroductionF-linear maps on V, each of which acts on an eigenbasis for the other one in an polycyclic aromatic hydrocarbons (PAHs)irreducible and two tridiagonal ana- fashion. Such a pair is called a Leonard pair (see [13, Denition 1.1]). The Leonard pair logues of alkaloids. Five PAHs (benzene, naphthalene, A, A∗ is said to be self-dualPolymer whenevermicrospheres there are exists polymeric an automorphism materials ofor the endomorphism algebra of V that fluorene, pyrene, and benzo(e)pyrene) were well sepa- swaps A and A∗. Inpolymer this case composites such an automorphism with sizes ranging is unique, from nanometer and called the duality A A∗. rated in the rosin-based polymer column with the reso- ↔ to micrometer scale and spherical geometrical shapes lution (Rs) of 6.088 ± 0.006, 5.759 ± 0.017, The7.019 literature ± 0.020, contains many examples of self-dual Leonard pairs. For instance (i) the Leonard pair associ- (1–3). Polymer microspheres have wide applications and 8.584 ± 0.063. A linear relationshipated was with observed an irreducible for module for the Terwilliger algebra of the hypercube (see [4, Corollaries 6.8, 8.5]); (ii) a in stationary phases, biological medicine and many the carbon numbers of PAHs and lnKLeonard as obtained pair with of Krawtchouk type (see [10, Denition 6.1]); (iii) the Leonard pair associated with an irreducible other fields (4–7). Organic polymer stationary phases the equation of ln k = 0.22n – 2.45, R2 =module 0.996 . The for therosin- Terwilliger algebra of a distance-regular graph that has a spin model in the Bose-Mesner alge- c mainly include polysaccharide polymers, polystyrenes/ based polymer columns were also employedbra (see to [1,separate Theorem], [3, Theorems 4.1, 5.5]); (iv) an appropriately normalized totally bipartite Leonard pair divinylbenzene polymers and polymethacrylate polymers (see [11, Lemma 14.8]);(8–10). (v) theDue Leonard to the pairpoor consisting mechanical of anystability, two of when a modular Leonard triple A, B, C (see [2, * Corresponding author: Fu-Hou Lei and Wen Li,Denition College of 1.4]); (vi)polysaccharide the Leonard pair polymers consisting and polymethacrylate of a pair of opposite polymers generators for the q-tetrahedron alge- Chemistry and Chemical Engineering, Guangxibra, University acting for on an evaluationare used in module high performance (see [5, Proposition liquid phase, 9.2]). the The particles example (i) is a special case of (ii), and the nationalities, Nanning 530006, China; Guangxiexamples Key Laboratory (iii), of (iv) areare specialeasy to collapse cases of under (v). high pressure (11). Polystyrenes/ Chemistry and Engineering of Forest Products, Guangxi University Let A, A∗ denotedivinylbenzene a Leonard pair polymers on V. We can can bear determine the high pressure whether andA, A∗ is self-dual in the following way. for nationalities, Nanning 530006, China; College of Light Industry d By [13, Lemma 1.3] eachhave eigenspacea good adaptability of A, A∗ forhas chromatographic dimension one. purposes Let θ denote an ordering of the eigen- and Food Engineering, Guangxi University, Nanning 530004, China, { i}i= e-mail: [email protected] (Fu-Hou Lei); [email protected] but may release residuals, which are harmful to health, d values of A. For ≤ i ≤ d let vi denote a θi-eigenvector for A. The ordering θi i= is said to be standard (Wen Li). Tel.: +86 0771 3267071. when they are dused for the separation and analysis { } d whenever A∗ acts on the basis vi in an irreducible tridiagonal fashion. If the ordering θi is standard College of i= i= Zhen Wang, Peng-Fei Li, Ting Wang and Li-Ting Qin, of drugsd (12,13).{ } As the ternary ring structure of rosin { } Chemistry and Chemical Engineering, Guangxithen University the for ordering θd−i i= is also standard, and no further ordering is standard. Similar comments apply to d { can }be introduced into high molecular structure, rosin d nationalities, Nanning 530006, China; GuangxiA Key∗. LetLaboratoryθ of denote a standard ordering of the eigenvalues of A. Then A, A∗ is self-dual if and only if θ { i}i= can enhance the mechanical strength of polymer and { i}i= Chemistry and Engineering of Forest Products, Guangxi University is a standard ordering of the eigenvalues of A∗ (see [7, Proposition 8.7]). for nationalities, Nanning 530006, China. is an important resource to synthesize high-molecular Ge-Ge Cheng, Yao Sun, Yu-Yu E and Wen-Bo Xie, College of stationary phase instead of divinylbenzene or styrene. Chemistry and Chemical Engineering, Guangxi University for In addition, rosin is biodegradable, non-toxic, and thus nationalities, Nanning 530006, China. can be used as a raw material to separate and purify the College of chemistry and Environmental Engineering, Jie Deng, *Corresponding Author:effective Kazumasa ingredients Nomura: of TokyoChinese Medical herbs and (14). Dental University, Ichikawa, 272-0827,Japan, Hubei University for Nationalities, Enshi Tujia andE-mail: Miao [email protected] Autonomous Polycyclic aromatic hydrocarbons (PAHs) are Prefecture 445000, China. Paul Terwilliger: Department of Mathematics, University of Wisconsin, Madison, WI53706, USA, E-mail: # carcinogenic pollutants that can be absorbed by crops Zhen Wang, Fu-Hou Lei, Wen Li and Peng-Fei [email protected] contributed equally to this work. via polluted water, air, or, soil (15). PAHs could be Open Access. © 2019 Wang et al., published byOpen De Gruyter. Access. ©This2019 work Kazumasa is licensed Nomura under and the Paul Creative Terwilliger, Commons published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Attribution alone 4.0 License. Z. Wang et al.: Preparation of rosin-based polymer microspheres as a stationary 291 generated during the general extraction process of natural In this study, rosin-based polymer microspheres medicinal active components in a high temperature (15). were prepared and used as the stationary phase in high- Hence, detecting PAHs and removing them from natural performance liquid chromatography (HPLC) to separate medicines are important. Theophylline (THO)–caffeine PAHs and alkaloids. The main objectives of this work are (CAF) and camptothecin (CPT)–10-hydroxycamptothecin as follows: (1) preparation of polymer microspheres using (HCPT) belong to the two sets of alkaloids with similar acrylpimaric acid ethylene glycol acrylate (AAEGA) as the structures and are widely applied in clinical drugs cross-linking agent through membrane emulsification– (16,17). Slight modification of the molecule configuration microsuspension polymerization; (2) preparation of the could lead to different pharmacology activities and rosin-based polymer columns using the microspheres to applications. THO is used extensively to treat asthma, and separate PAHs, THO–CAF, and CPT–HCPT. CAF is often added to pain medications. Compared with CPT, HCPT holds up better in anticancer activity and has lower toxicity. Detecting and separating these two sets of 2 Materials and methods alkaloids is of importance. In our previous work (13), paclitaxel imprinted 2.1 Experimental principles polymer microspheres (using ethylene glycol maleic rosinate acrylate (EGMRA) as cross-linking agent) were Figure 1 illustrates the principles and schematic of the prepared by microsuspension