Lie superalgebras and the multiplet structure of the genetic code. II. Branching schemes

Forger, Michael; Sachse, Sebastian

doi:10.1063/1.533418

Cited by 11 publications

(10 citation statements)

References 15 publications

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“…Using the representation theory developed by Kac,60,61 we have recently derived a complete classification of all typical codon representations of basic classical Lie superalgebras, 62 toghether with a systematic investigation of their symmetry breaking patterns through chains of subalgebras. 63 The subject has also been adressed by various other authors. 64,65 Another set of questions arises when one wonders what might be the origin of symmetry in the genetic code and of the mechanism that triggers its breakdown.…”

Section: Discussionmentioning

confidence: 96%

Symmetry and Symmetry Breaking: An Algebraic Approach to the Genetic Code

Hornos

Forger

1999

Int. J. Mod. Phys. B

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We give a comprehensive review of the algebraic approach to the genetic code originally proposed by two of the present authors, which aims at explaining the degeneracies encountered in the genetic code as the result of a sequence of symmetry breakings that have occurred during its evolution. We present the relevant background material from molecular biology and from mathematics, including the representation theory of (semi) simple Lie groups/algebras, together with considerations of general nature.

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Section: Discussionmentioning

confidence: 96%

Symmetry and Symmetry Breaking: An Algebraic Approach to the Genetic Code

Hornos

Forger

1999

Int. J. Mod. Phys. B

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“…Table 10. Branching of the codon representation of so (14) in the chain (11): third phase, first option. Table 6.7: Branching of the codon representation of so (14) in the chain (11): third phase, first option.…”

Section: Chain 233223 (With One Intermediate Step Omitted)mentioning

confidence: 99%

“…1, a similar analysis for Lie superalgebras has already been completed, 10,11 whereas the same investigation in the context of finite groups is under way and has so far given rise to partial results. 12 This article, which has grown out of previous work by Y.M.M.…”

Section: Introductionmentioning

confidence: 99%

Extending the Search for Symmetries in the Genetic Code

Antoneli

Braggion

Forger

et al. 2003

Int. J. Mod. Phys. B

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We report on the search for symmetries in the genetic code involving the medium rank simple Lie algebras B 6 = so(13) and D 7 = so (14), in the context of the algebraic approach originally proposed by one of the present authors, which aims at explaining the degeneracies encountered in the genetic code as the result of a sequence of symmetry breakings that have occurred during its evolution. Work supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and by FAPESP (Fundação de Amparoà Pesquisa do Estado de São Paulo), Brazil. 3135 Int. J. Mod. Phys. B 2003.17:3135-3204. Downloaded from www.worldscientific.com by ROYAL INSTITUTE OF TECHNOLOGY on 02/03/15. For personal use only. 3136 F. Antoneli et al.to allocate the aminoacids and the termination signal. This process of spontaneous symmetry breaking has been used before, in various fields of science. The crucial point in such a search for symmetries is to find the ancestor group G and the possible chains of subgroups. The first search for symmetries in the genetic code along this line of reasoning, as outlined in Ref. 1 (see also Ref. 2, and Refs. 3 and 4 for comments), has been carried out within the class of ordinary compact Lie groups G or, what amounts to the same thing, of semisimple Lie algebras g, based on Cartan's classification of semisimple Lie algebras, Dynkin's classification of their maximal semisimple subalgebras 5,6 and the tables of branching rules of McKay and Patera. 7 This search has shown that, on the one hand, there is no simple Lie algebra that can generate the standard genetic code directly in its present form. On the other hand, it turned out that the standard genetic code can be obtained from the symplectic algebra sp (6) if, in the last step, the process of symmetry breaking is allowed to be gently interrupted, or frozen. This means that a few of the multiplets (subspaces) that have resulted from previous steps of the process do not participate in the symmetry reduction implied by the last step.Two restrictions have been imposed on the search reported in Ref. 1. The first refers to the last phase of the process, after the ancestor algebra has already been broken to a direct sum of su(2)-subalgebras. In this last phase, further steps consist in breaking one or several of these su(2)-subalgebras completely, which in the language of atomic or nuclear physics can be implemented by introducing an appropriate generator L z into the Hamiltonian. Another possibility that was also considered is to introduce instead its square L 2 z , which leads to a softer form of symmetry breaking. However, the possibility of performing both of these breakings sequentially, with freezing applied to the second step only, was not contemplated. The second restriction was the exclusion of "diagonal breaking", which in the case of su(2)-subalgebras amounts to a Clebsch-Gordan summation law for quantum numbers.In Ref. 8, this search has been enlarged as a consequence of a new geometric interpretation of the operator L 2 z (rather than L z )...

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“…Дальнейшее развитие теоретико-групповых математических структур для исследования генети ческого кода привело к рассмотрению классических супергрупп и супералгебр Ли [32 ]. В этом подходе предполагается, что типичные представления кодо нов также описываются 64-мерными неприводимы ми представлениями основных классических супер алгебр Ли серии si (6/1) [10,[33][34][35] и других подходящих супералгебр, например, osp (512) [36 ]. Была проведена полная классификация всех ти пичных представлений кодонов -всего 18 пред ставлений для 12 различных супералгебр Ли -с целью нахождения наиболее подходящих для опи сания вырожденности генетического кода [10,36].…”

Section: Cg Cu саunclassified

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2000

Biopolym. Cell

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Харьковский национальный университет им. М. Н. Каразина Пл. Свободы, 4, Харьков, 61077, Украина Институт молекулярной биологии и генетики НАН Украины Ул. Академика Заболотного, !50> Киев, 03143, Украина В работе рассматриваются формальные аспекты проблемы генетического кода. Особое внимание уделено математическим подходам к описанию свойств кода. На основании неравномерной вырожденности генетического кода введена новая числовая характеристика нуклеотида -сте пень детерминации. Предложена модель дуплетного (ромбического) кода в виде матрицы, полученной внешним произведением, в которой наблюдаются симметрийные закономерности. Обсуждаются также различные алгебраические и суперсимметричные модели триплетного гене тического кода.

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Lie superalgebras and the multiplet structure of the genetic code. II. Branching schemes

Cited by 11 publications

References 15 publications

Symmetry and Symmetry Breaking: An Algebraic Approach to the Genetic Code

Symmetry and Symmetry Breaking: An Algebraic Approach to the Genetic Code

Extending the Search for Symmetries in the Genetic Code

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