Structure of unital 3-fields

Duplij, Steven; Werner, Wend

doi:10.1007/s00591-020-00285-1

“…The presented internalisation of the definition of a Lie affebra leading to systems of interacting ternary operations seems to mesh well with the recent resurgence of interest in such systems, both in pure mathematics (see e.g. [12] and references therein) and in mathematical physics (see e.g. [9], [18], [19]).…”

Section: Introductionmentioning

confidence: 57%

Lie trusses and heaps of Lie affebras

Brzeziński¹

2022

Preprint

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“…The presented internalisation of the definition of a Lie affebra leading to systems of interacting ternary operations seems to mesh well with the recent resurgence of interest in such systems, both in pure mathematics (see e.g. [12] and references therein) and in mathematical physics (see e.g. [9], [18], [19]).…”

Section: Introductionmentioning

confidence: 60%

Lie trusses and heaps of Lie affebras

Brzeziński¹

2022

Proceedings of Corfu Summer Institute 2021 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2021)

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“…This statement for the limiting case [[1]] 2 appeared in DUPLIJ AND WERNER [2015], while studying the ideal structure of the corresponding (3, 2)-ring. Proposition 6.20.…”

Section: Congruence Classes As Polyadic Ringsmentioning

confidence: 96%

“…Remark 2.12. The so-called 3-vector space introduced and studied in DUPLIJ AND WERNER [2015], corresponds to…”

Section: Two Set Polyadic Structuresmentioning

confidence: 99%

Arity Shape of Polyadic Algebraic Structures

Duplij¹

2019

Z. mat. fiz. anal. geom.

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Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to restrictions which determine their possible arity shapes and lead us to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic C * -algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators. Another application is connected with number theory, and it is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 6.16), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's last theorem are formulated. For nonderived polyadic ring operations (on polyadic numbers) neither of these statements holds, and counterexamples are given. Finally, a procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem to the Tarry-Escott problem is presented.Proof. Both statements follow from (5.21) and the definition of the polyadic operator norm (5.55). Therefore, we arrive toDefinition 5.28. The operator Banach algebra B T satisfying the multi-C * -relations is called a polyadic operator multi-C * -algebra.The first example of a multi-C * -algebra (as in the binary case) can be constructed from one isometry operator (see Definition 5.15).Definition 5.29. A polyadic algebra generated by one isometry operator T satisfying (5.46) on the inner pairing space H m K ,n K ,m V ,kρ=1,N represents a polyadic Toeplitz algebra T m T ,n T and has the arity shape m T = m V , n T = N.Example 5.30. The ternary Toeplitz algebra T 3,3 is represented by the operator T and relations(5.62)Example 5.31. If the inner pairing is semicommutative (5.31), then (⋆ 3 ) can be eliminated by (5.64) and the corresponding relations representing T 3,3 become(5.65)Let us consider M polyadic operators T 1 T 2 . . . T M ∈ B T and the related partial (in the usual sense) isometries (5.52) which are mutually orthogonal (5.50). In the binary case, the algebra generated by M operators, such that the sum of the related orthogonal partial projections is unity, represents the Cuntz algebra O M CUNTZ [1977].

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Structure of unital 3-fields

Cited by 4 publications

References 19 publications

Lie trusses and heaps of Lie affebras

Lie trusses and heaps of Lie affebras

Lie trusses and heaps of Lie affebras

Arity Shape of Polyadic Algebraic Structures

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