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Indecomposable representations ofA 2 (su 3 ,su 2,1 ) and representations induced by them

Неприводимые представленияA 2 (su 3 ,su 2,1 ) и представления, индуцированные ими

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Il Nuovo Cimento A (1965-1970)

Summary

Indecomposable representations are investigated for the case of the simple complex Lie algebraA 2 (the complexification ofsu 3 andsu 2,1). The matrix elements are explicitly determined for the elementary representations, and the extremal vectors which characterize invariant subspaces are given in explicit form. Quotient spaces are used to derive other representations from the elementary representations, including the finite-dimensional irreducible representations, the infinite-dimensional irreducible representations which are bounded above, as well as new types of indecomposable representations. Again, the matrix elements for these representations are given in explicit form. In the appendix the same program is carried out for the simple complex lie algebraA 1 (the complexification ofsu 2 andsu 1,1) as an example. The branching of the elementary representations, as well as of the representations derived from the elementary representations, is analyzed with respect to two subalgebras of the typeA 1. Again, theA 1 extremal vectors are obtained in explicit form.

Riassunto

Si studiano le rappresentazioni non scomponibili per il caso della semplice algebra di Lie complessaA 2 (la complessificazione disu 3 esu 2,1). Si determinano esplicitamente gli elementi matriciali per le rappresentazioni elementari e si danno in forma esplicita i vettori estremali che caratterizzano sottospazi invarianti. Si usano gli spazi quoziente per derivare altre rappresentazioni da quelle elementari, comprese le rappresentazioni irriducibili a dimensione finita, le rappresentazioni irriducibili a dimensione infinita limitate superiorimente, ed anche nuovi tipi di rappresentazioni non scomponibili. Si danno inoltre in forma esplicita gli elementi matriciali per queste rappresentazioni. In appendice si esegue lo stesso programma per la semplice algebra di Lie complessaA 1 (la complessificazione disu 2 esu 1,1) come esempio. Si analizza la diramazione delle rappresentazioni elementari, ed anche delle rappresentazioni derivate da quelle elementari, rispetto a due subalgebre di tipoA 1. Si ottengono inoltre i vettori estremali diA 1 in forma esplicita.

Резюме

Исследуются неприводимые представления для случая простой комплексной алгебры ЛиA 2 (su 3 ,su 2,1 ). Матричные элементы выражаются в явном виде для элементарных представлений. В явном виде приводятся экстремальные векторы, которые характеризуют инвариантные подпространства. Используются частные пространства для вывода других представлений из элементармых представлений, включая конечномерные неприводимые представления, бесконечномерные неприводимые представления, которые являются ограниченными, а также новые типы неприводимых представлений. Матричные элементы для этих представлений записываются в явном виде. В приложении, как пример, реализуется та же программа для простой комплексной алгебры ЛиA 2(su 2,su 1,1). Ветвление элементарных представлений, а также представлений, выведенных из элементарных представлений, анализируется относительно двух субалгебр типаA 1. Экстремальные векторыA 1 получаются в явном виде.

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Authors and Affiliations

  1. Department of Physics and Astronomy, Southern Illinois University, 62901, Carbondale, Ill., USA

    B. Gruber

  2. Institute for Theoretical Physics, Academy of Sciences of the Ukrainian SSR, 130, Kiev, USSR

    A. U. Klimyk

  3. Institute for Nuclear Research, Moscow State University, Moscow, USSR

    Y. F. Smirnov

Authors
  1. B. Gruber
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  2. A. U. Klimyk
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  3. Y. F. Smirnov
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Traduzione a cura della Redazione.

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Gruber, B., Klimyk, A.U. & Smirnov, Y.F. Indecomposable representations ofA 2 (su 3 ,su 2,1 ) and representations induced by them. Nuov Cim A 69, 97–127 (1982). https://doi.org/10.1007/BF02902648

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  • DOI: https://doi.org/10.1007/BF02902648

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