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. MSC classes: 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05

Published on: **Mar 4, 2016**

Published in:
Science

- 1. Polyadic Systems and their Multiplace Representations STEVEN DUPLIJ Institute of Mathematics, University of M¨unster http://wwwmath.uni-muenster.de/u/duplij 2014 1
- 2. 1 History 1 History Ternary algebraic operations (with the arity n = 3) were introduced by A. Cayley in 1845 and later by J. J. Sylvester in 1883. The notion of an n-ary group was introduced in 1928 by D¨ ORNTE [1929] (inspired by E. N¨ other). The coset theorem of Post explained the connection between n-ary groups and their covering binary groups POST [1940]. The next step in study of n-ary groups was the Gluskin-Hossz´u theorem HOSSZ´U [1963], GLUSKIN [1965]. The cubic and n-ary generalizations of matrices and determinants were made in KAPRANOV ET AL. [1994], SOKOLOV [1972], physical application in KAWAMURA [2003], RAUSCH DE TRAUBENBERG [2008]. 2
- 3. 1 History Particular questions of ternary group representations were considered, by the SD in BOROWIEC ET AL. [2006], DUPLIJ [2013a]. Some theorems connecting representations of binary and n-ary groups were presented in DUDEK AND SHAHRYARI [2012]. In physics, the most applicable structures are the nonassociative Grassmann, Clifford and Lie algebras L ˜OHMUS ET AL. [1994], GEORGI [1999]. The ternary analog of Clifford algebra was considered in ABRAMOV [1995], and the ternary analog of Grassmann algebra ABRAMOV [1996] was exploited to construct ternary extensions of supersymmetry ABRAMOV ET AL. [1997]. In the higher arity studies, the standard Lie bracket is replaced by a linear n-ary bracket, and the algebraic structure of the corresponding model is defined by the additional characteristic identity for this generalized bracket, corresponding to the Jacobi identity DE AZCARRAGA AND IZQUIERDO [2010]. 3
- 4. 1 History The infinite-dimensional version of n-Lie algebras are the Nambu algebras NAMBU [1973], TAKHTAJAN [1994], and their n-bracket is given by the Jacobian determinant of n functions, the Nambu bracket, which in fact satisfies the Filippov identity FILIPPOV [1985]. Ternary Filippov algebras were successfully applied to a three-dimensional superconformal gauge theory describing the effective worldvolume theory of coincident M2-branes of M-theory BAGGER AND LAMBERT [2008a,b], GUSTAVSSON [2009]. 4
- 5. 2 Plan 2 Plan 1. Classification of general polyadic systems and special elements. 2. Definition of n-ary semigroups and groups. 3. Homomorphisms of polyadic systems. 4. The Hossz´ u-Gluskin theorem and its “q-deformed” generalization. 5. Multiplace generalization of homorphisms - heteromorpisms. 6. Associativity quivers. 7. Multiplace representations and multiactions. 8. Examples of matrix multiplace representations for ternary groups. 5
- 6. 3 Notations 3 Notations Let G be a underlying set, universe, carrier, gi 2 G. The n-tuple (or polyad) g1, . . . , gn is denoted by (g1, . . . , gn). The Cartesian product G×n consists of all n-tuples (g1, . . . , gn). For equal elements g 2 G, we denote n-tuple (polyad) by (gn). If the number of elements in the n-tuple is clear from the context or is not important, we denote it with one bold letter (g), or