Polyadic systems and their multiplace representations
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
Transcripts - Polyadic systems and their multiplace representations
Polyadic Systems and their
Institute of Mathematics, University of M¨unster
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
 (inspired by E. N¨ other).
The coset theorem of Post explained the connection between n-ary
groups and their covering binary groups POST .
The next step in study of n-ary groups was the Gluskin-Hossz´u
, GLUSKIN .
The cubic and n-ary generalizations of matrices and determinants
were made in KAPRANOV ET AL. , SOKOLOV , physical
application in KAWAMURA , RAUSCH DE TRAUBENBERG .
Particular questions of ternary group representations were
considered, by the SD in BOROWIEC ET AL. , DUPLIJ [2013a].
Some theorems connecting representations of binary and n-ary
groups were presented in DUDEK AND SHAHRYARI .
In physics, the most applicable structures are the nonassociative
Grassmann, Clifford and Lie algebras L ˜OHMUS ET AL. ,
GEORGI . The ternary analog of Clifford algebra was
considered in ABRAMOV , and the ternary analog of
Grassmann algebra ABRAMOV  was exploited to construct
ternary extensions of supersymmetry ABRAMOV ET AL. .
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 .
The infinite-dimensional version of n-Lie algebras are the Nambu
algebras NAMBU , TAKHTAJAN , and their n-bracket is
given by the Jacobian determinant of n functions, the Nambu
bracket, which in fact satisfies the Filippov identity FILIPPOV .
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 .
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”
5. Multiplace generalization of homorphisms - heteromorpisms.
6. Associativity quivers.
7. Multiplace representations and multiactions.
8. Examples of matrix multiplace representations for ternary
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