Полугрупповые методы в суперсимметричных теориях элементарных частиц

UDC 539.12 POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS S.A. Duplij ∗ Center for Mathematics, Science and Education Rutgers University, 118 Frelinghuysen Rd., Piscataway, NJ 08854-8019 Е -mail: duplij@math.rutgers.edu, http://homepages.spa.umn.edu/˜duplij Received 20 May 2012 Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a ’heteromorphism’ which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented. KEY WORDS : n-ary group, Post theorem, commutativity, homomorphism, group action, Yang-Baxter equation ПОЛИАДИЧЕСКИЕ СИСТЕМЫ , ПРЕДСТАВЛЕНИЯ И КВАНТОВЫЕ ГРУППЫ С . А . Дуплий Центр математики , науки и образования , университет Ратгерса , Пискатавэй , 08854-8019, США Приведен обзор полиадических систем и их представлений , дана классификация общих полиадических систем . Построены многоместные обобщения гомоморфизмов , сохраняющие ассоциативность . Определены мультидействия и мультиместные пред - ставления . Приведены конкретные примеры матричных представлений для некоторых тернарных групп . Определены тернарные алгебры и Хопф алгебры , изучены их свойства . В заключение , предствлены некоторые тернарные обобщения квантовых групп и уравнения Янга - Бакстера . КЛЮЧЕВЫЕ СЛОВА : n - арная группа , теорема Поста , коммутативность , гомоморфизм , групповое действие , уравнение Янга - Бакстера ПОЛ I АДИЧН I СИСТЕМИ , ПРЕДСТАВЛЕННЯ I КВАНТОВ I ГРУПИ С . А . Дупл i й Центр математики , науки та осв i ти , ун i верситет Ратгерсу , П i скатавєй , 08854-8019, США Зроблено огляд пол i адичних систем та їх представлень , дана класиф i кац i я загальних пол i адичних систем . Побудован i багатом i с - н i узагальнення гомоморф i зми i в , що збер i гають асоц i ативн i сть . Визначен i мультид i ї i мультим i сн i представлення . Наведен i конкретн i приклади матричних представлень для деяких тернарних груп . Визначен i тернарна алгебра i алгебри Хопфа , вивчен i їх властивост i. На зак i нчення , предствлен i деяк i тернарн i узагальнення квантових груп та р i вняння Янга - Бакстера . КЛЮЧОВ I СЛОВА : n - арна група , теорема Поста , комутативн i сть , гомоморф i зм , групова д i я , р i вняння Янга - Бакстера One of the most promising directions in generalizing physical theories is the consideration of higher arity algebras [1], in other words ternary and n -ary algebras, in which the binary composition law is substituted by a ternary or n -ary one [2]. Firstly, ternary algebraic operations (with the arity n = 3 ) were introduced already in the XIX-th century by A. Cayley in 1845 and later by J. J. Silvester in 1883. The notion of an n -ary group was introduced in 1928 by [3] (inspired by E. N¨other) and is a natural generalization of the notion of a group. Even before this, in 1924, a particular case, that is, the ternary group of idempotents, was used in [4] to study infinite abelian groups. The important coset theorem of Post explained the connection between n -ary groups and their covering binary groups [5]. The next step in study of n -ary groups was the Gluskin-Hossz´u theorem [6,7]. Another definition of n -ary groups can be given as a universal algebra with additional laws [8] or identities containing special elements [9]. The representation theory of (binary) groups [10, 11] plays an important role in their physical applications [12]. It is initially based on a matrix realization of the group elements with the abstract group action realized as the usual matrix multiplication [13, 14]. The cubic and n -ary generalizations of matrices and determinants were made in [15, 16], and their physical application appeared in [17, 18]. In general, particular questions of n -ary group representations were considered, and matrix representations derived, by the author [19], and some general theorems connecting representations of binary and n -ary groups were presented in [20]. The intention here is to generalize the above constructions of n -ary group representations to more complicated and nontrivial cases. In physics, the most applicable structures are the nonassociative Grassmann, Clifford and Lie algebras [21–23], and so their higher arity generalizations play the key role in further applications. Indeed, the ternary analog of Clifford algebra was ∗ On leave of absence from V.N. Karazin Kharkov National University, Svoboda Sq. 4, Kharkov 61022, Ukraine. 28 «Journal of Kharkiv National University», №1017, 2012 S.A. Duplij physical series «Nuclei, Particles, Fields» , issue 3 /55/ Polyadic systems... © Duplij S.A., 2012

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