Steven Duplij
Theoretical and mathematical physicist, Doctor of Physical and Mathematical Sciences (Habilitation Doctorate), Humboldt Fellow and Fulbright Scholar, Editor-compiler of the "Concise Encyclopedia of Supersymmetry" (Springer), poet, musician.
Homepage: https://www.uni-muenster.de/IT.StepanDouplii
Arxiv: https://arxiv.org/search/?query=duplij&searchtype=all&abstracts=show&order=-announced_date_first&size=100
Orcid: https://orcid.org/0000-0003-1184-6952
LinkedIn: https://www.linkedin.com/in/duplij
Homepage: https://www.uni-muenster.de/IT.StepanDouplii
Arxiv: https://arxiv.org/search/?query=duplij&searchtype=all&abstracts=show&order=-announced_date_first&size=100
Orcid: https://orcid.org/0000-0003-1184-6952
LinkedIn: https://www.linkedin.com/in/duplij
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Papers by Steven Duplij
arxiv hep-th/2406 02188
https://arxiv.org/abs/2403.19361.
*) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures,
https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
https://arxiv.org/abs/2312.01366
https://arxiv.org/abs/2206.14840
arxiv hep-th/2406 02188
https://arxiv.org/abs/2403.19361.
*) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures,
https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
https://arxiv.org/abs/2312.01366
https://arxiv.org/abs/2206.14840
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead 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, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with number theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. 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 the derived polyadic ring operations neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem for the Tarry-Escott problem is presented.
The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to 'generalize' it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the 'concrete way' and provide consequent 'polyadization' of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang–Baxter equation, find its constant solutions, and introduce polyadic tensor categories. Suitable for university students of advanced level algebra courses and mathematical physics courses.
Key features
• Provides a general, unified approach • Widens readers perspective of the possibilities to develop standard algebraic structures
• Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
• Presents applications of 'polyadization' approach to concrete algebraic structures.
For the light paperback version of the book use MyPrint service here: https://iopscience.iop.org/book/978-0-7503-2648-3 also PDF, ePub and Kindle.
For libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Polyadic-Algebraic-Structures//?k=9780750326483
Amazon.com ordering: https://www.amazon.com/Polyadic-Algebraic-Structures-Steven-Duplij/dp/0750326468
LinkedIn (with free Front Matter and 1st Chaper): https://www.linkedin.com/posts/duplij_arity-algebraicstructure-field-activity-6940082232648720384-e2qk
A variety of topics has been covered: constraint-free description of singular theories in the partial Hamiltonian formalism, which is equivalent to many-time dynamics; generalized nonlinear (super) electrodynamics, its Lagrangian and non-Lagrangian description; nontrivial versions of the interaction in multigravity and the Pauli-Fierz model; noninvertibility properties on the quantum plane, universal enveloping algebras, and various superstructures: the hyperbolic superplane, supermanifolds and super matrix models; the von Neumann regular generalization of the antipode in Hopf algebras and of solutions to the Yang-Baxter equations and the R-matrix.
Amazon: http://www.amazon.com/Supersymmetry-Multigravity-Singular-Theories-Mathematics/dp/1925823091