## Dr. habil. Stepan DoupliiPen/scientific name Center for Information Technology University of Münster Röntgenstr. 7-13 (Rm. 115 in №9) 48149 Münster T: +49 251 83-31530 F: +49 251 83-31555 Previous homepage (with other activities): Steven Duplij's homepage
Ancient unupdated version : Steven Duplij's U.S. homepage |

## Research Areas

Quantum computing, mathematics, theoretical physics.

## On Web

## CV

PDF in GermanPDF in English, extended, clickable links

**Degrees**

1999 Habilitation Thesis: "Semigroup Methods in Supersymmetric Theories of Elementary Particles", Bogolyubov Institute of Theoretical Physics, Kiev, Ukraine, Doctor Habilitatus in Theoretical Physics (Doctor of Science in Physics and Mathematics)

1983 Ph.D. Thesis: "Theoretical Investigation of Hard Processes in QCD";

1978-1982 Post Graduate Course in Theoretical Physics, Kharkov State University; PhD (Candidate of Science in Physics and Mathematics); adviser M.P. Rekalo

**Education**

1974-1978 Department of Theoretical Physics, Kharkov State University; The Distinguished Diploma in Theoretical Nuclear Physics - M.Sc.

1973-1976 Special English Language Course, Kharkov State University; The Distinguished Diploma

1973-1974 Department of Theoretical Radiophysics, Kharkov State University

**Positions** (work experience 47 years)

2016-2017 Lecturer in Mathematics, University of Applied Sciences, Bochum, Germany

2014-2016 Scientific Researcher, Mathematisches Institut, Universität Münster, Münster, Germany

2000-2014 Lead Senior Staff Researcher, Nuclear Physics Laboratory, Kharkov National University, Kharkov, Ukraine

1983-1992 Staff Researcher, Nuclear Physics Laboratory, Kharkov State University, Kharkov, Ukraine

1978-1982 Half-time Researcher, Radiophysics Laboratory, Kharkov State University, Kharkov, Ukraine

1977-1978 Researcher, Kharkov Institute of Physics and Technology, Kharkov, Ukraine

**Grants**

2007 American Physical Society Travel Grant, Johns Hopkins University, USA (host: J. Bagger)

1994-1997 Alexander von Humboldt Fellowship, University of Kaiserslautern, Kaiserslautern, Germany (initial host: W. Rϋhl, follow up hosts: J. Cuntz, F. Müller-Hoissen, M. Zirnbauer) AvH S.D. Profile

**Scientific visits**

## Scientific results and innovative ideas

**HTML**(as separate page)

**supersymmetric**

**models**of elementary particles, based on the inclusion of semigroups is proposed (book, thesis). The concept of

*semisupermanifold*having noninvertible transition functions (satisfying higher von Neumann regularity) is introduced, and its deviation from being an ordinary manifold is given by a newly defined variable,

*obstructedness*. Based on this idea, the novel notions of

*category regularization, regular topos, regular functor, higher regular braiding, regular Yang-Baxter equation and regular module, regular algebra and coalgebra, regular graded algebras*are presented, and their role in topological quantum field theory is outlined.

Even- and odd-reduced supermatrices are introduced and considered on a par, being complementary in terms of the newely obtained

*Berezinian addition formula*, and are unified into a kind of

*"sandwich" semigroup*. A special subset of odd-reduced supermatrices represent

*higher order rectangular bands*for which new generalized

*"fine" Green's relations and egg-box diagrams*are constructed.

*One-parameter semigroups of idempotent odd-reduced supermatrices*and corresponding superoperator semigroups are introduced and studied by the new

*semigroup × semigroup method*. The linear idempotent superoperators and exponential superoperators are mutually dual in some sense, and the first gives rise to an additional noninvertible

*non-exponential solutions to the initial Cauchy problem*. A novel

*permanent-determinant symmetry*is found for even complex superplane. It is shown that the corresponding counterparts (

*per analogs*) of the cross ratio, distance and harmonic set are invariant under the introduced

*per mapping*, a special noninvertible subset of the fractional linear transformation. The per analogs of the

*Laguerre formula for distance and Schwarzian derivative*are presented.

An

*additional superextension of complex structure*is uncovered, which is noninvertible and can correspond to another (odd) superanalog of Riemann surfaces and to the counterpart of

*superconformal-like transformations which twist the parity of tangent space*and their

*nonlinear realization*, which together with the ordinary ones form the

*superconformal semigroup*having special unusual properties. A unique formula connecting

*berezinian, permanent and determinant*is obtained. From a physical viewpoint, the above conceptions can lead to

*semistatistics*, being

*von Neumann regular analog*of the ordinary statistics.

**Quantum**

**groups**: a generalization of the Hopf algebra is introduced by relaxing the requirement for inverses of the generators of the Cartan subalgebra, which leads to a

*regular quasi-R-matrix*structure. The classification of 6-vertex

*constant solutions to Yang-Baxter equation over Grassmann algebra*is presented, including noninvertible ones which correspond to

*von Neumann regular R-matrix*. The

*actions of universal enveloping quantum algebras*on quantum planes (also of

*arbitrary dimension*) are found. A novel

*double-graded quantum superplane*and corresponding

*double-graded Hopf algebra*are presented.

**Singular**

**theories**with degenerate Lagrangians are formulated

*without involving constraints*using Clairaut equation theory and the corresponding generalized

*Clairaut duality*. A new

*antisymmetric bracket*(an analogue of the Poisson bracket) describing the time evolution of singular systems is built. A novel

*partial Hamiltonian formalism*is constructed. It is shown that a

*singular theory can be interpreted as the multi-time dynamics*.

**Nonlinear**

**gauge**

**theories**: a generalized approach to nonlinear classical electrodynamics and supersymmetric electrodynamics is suggested, which takes into account all possible types of media and nonlocal effects, and is described in both Lagrangian and

*non-Lagrangian*theories. First steps in the formulation of a general

*nonlinear conformal-invariant electrodynamics*based on

*nonlinear constitutive equations and conformal compactification*were made.

**Gravity**: constitutive equations for

*nonlinear gravito-electromagnetism*and an exact form of the Maxwell gravitational field equations are obtained. A general approach to describing the

*interaction of multi-gravity models*in space-times of arbitrary dimension is formulated. The gauge gravity vacuum is investigated in the

*constraintless Clairaut-type formalism*(as in QCD). A special

*fermionic lineal gravity*model which differs from standard supersymmetry is presented.

**Quantum**

**computing**(book IOP, FrontMatter): a new conception of quantum computing which incorporates an additional kind of uncertainty, vagueness/fuzziness, by introducing a new

*"obscure" class of qudits/qubits*, is announced. A superqubit theory in super-Hilbert space is reconsidered, and a new kind of

*superqubit carrying odd parity*is introduced. A new kind of quantum gates, namely

*higher braiding gates*, is suggested, which lead to a special type of multiqubit entanglement that can speed up key distribution and accelerate various algorithms. A novel visualization of quantum walks in terms of newly defined objects,

*polyanders*, is also proposed.

**Polyadic**

**structures**(book IOP, FrontMatter):

*polyadization*, i.e. exchanging binary operations with higher arity ones, is proposed as a general new approach to the algebraic structures used in physics. A new form of the Hosszu-Gluskin theorem (giving the general shape of

*n*-ary multiplication by the chain formula) in terms of polyadic powers is given, and its

*“q-deformed” generalization*is found using the newly introduced

*quasi-endomorphism*. A polyadic analog of homomorphism, or

*heteromorphism*, a mapping between algebraic structures of different arities, is introduced, which leads to the definition of a new kind of

*n*-ary group representation,

*multiplace representations, as well as multiactions*and a

*polyadic direct product*.

The

*arity invariance principle*, a manifest expression of algebraic structure in terms of operations independent of their arities, is claimed. The relations of the von Neumann regular semigroups and the Artin braid group were found, and a higher arity generalization gave the

*polyadic-binary correspondence*, which allowed the definition of the following new structures:

*higher braid groups*,

*higher degree analogs of Coxeter group and Artin braid group*. The following were also uncovered: unusual

*polyadic rings and fields (which can, remarkably, be zeroless and nonunital)*having addition and multiplication of different arities,

*polyadic integer numbers*and

*p-adic integers*,

*polyadic convolution products*having multiplication and comultiplication of different arities and their corresponding

*polyadic Hopf algebra*and

*n-ary R-matrix*,

*polyadic multistar adjoints*and

*polyadic operator C*-algebras and Cuntz algebras*. The

*polyadic analogs of the Lander–Parkin–Selfridge conjecture and Fermat's Last Theorem*were formulated.

It is proposed that mediality as a principle is more natural, unique and universal than commutativity in generalizing the latter to

*n*-ary algebras (in the binary case commutativity directly follows from mediality). This is called the

*commutativity-to-mediality ansatz*, which is applied to obtain

*almost medial*, a new kind of tensor categories,

*n*-ary graded algebras*polyadic nonunital "groupal" categories*with

*"quertors"*(analogs of querelements in

*n*-ary groups),

*"medialed" tensor categories and querfunctors*. A principally new mechanism of additional

*"continuous noncommutativity"*, governed by a special

*"membership deformation" of commutativity*for algebras with the underlying set as obscure/fuzzy set, is introduced. Using the membership deformation factor together with the ordinary graded commutation factor, the almost commutative graded (

*n*-ary) algebras and Lie algebras with

*double commutativity*are obtained, and their projective representations are studied.

As a first step towards a the

*polyadic algebraic*, the

*K*-theory*Grothendieck construction*of the completion group for a monoid is generalized to the case, where both are of different, higher arities. As opposed to the binary case, an identity is not necessary for the initial

*m*-ary semigroup to obtain a class

*n*-ary group, which in turn need not contain an identity.

A new (infinite) class of division algebras, the

*hyperpolyadic algebras*, which correspond to the (only 4) binary division algebras R, C, H, O (reals, complex numbers, quaternions, octonions) are defined. A

*polyadic analog of the Cayley-Dickson construction*is proposed, and a novel iterative process gives

*"half-quaternions"*and

*"half-octonions"*.

*New polyadic algebras with higher brackets*which have (as opposed to

*n*-ary Lie algebras) different arity from the initial

*n*-ary algebra multiplication, are introduced. The

*sigma matrices and the Pauli group*are generalized to

*higher arities*. Using them, a toy model of one-dimensional supersymmetric quantum mechanics was constructed, as a first example of

*polyadic supersymmetry*, which is specially extended in a way different from the

*new multigraded SQM*previously proposed.

**DNA**

**theory**: a new characteristic of nucleotides, the

*determinative degree*, which is proportional to the dipole moment and the weight of hydration site, is unveiled. The physical characteristics of nucleotides such as

*dipole moment, heat of formation and energy of the most stable formation*are newly computed by advanced methods. The concept of a

*triander*is set up, which leads to a new method of visual sequence analysis and identification using

*DNA walk diagrams*.

## Scientific publications

PDF (1985-2024) mathematical physics, quantum computing and DNA structure (161 items)

PDF (1980-1985) radiophysics and nuclear physics, particle physics and quantum chromodynamics (31 items)

HTML (1985-2008) with links to articles (105 items)

In total: **192** publications, among them 9 books and 183 articles.

In addition: 130 entries (table in PDF) in Concise Encyclopedia of Supersymmetry.

**Coeditors**

J. Bagger, W. Siegel, M.L. Walker, J. Wess, V. Zima

**Coauthors**

V.P. Akulov, A.Yu. Berezhnoy, A. Borowiec, A.J. Bruce, N. Chashchin, V.V. Chitov, M. Chursin, A. Frydryszak, O.M. Getmanetz, E. Di Grezia, W. Dudek, D.R. Duplij, N.V. Duplii, V.P. Duplij, G. Esposito, G.A. Goldin, Y. Hong, V.V. Kalashnikov, O.I. Kotulska, A.T. Kotvytskiy, G.Ch. Kurinnoj, S.V. Landar, F. Li, Y.G. Mashkarov, W. Marcinek, E.A. Maslov, N.M. Pelykhaty, V.M. Puzh, M.P. Rekalo, M.A. Rukavitsyn, A. Sadovnikov, Z.S. Sagan, S. Sinel'shchikov, I.I. Shapoval, V.M. Shtelen, D.V. Soroka, V.A. Soroka, S.A. Steshenko, R. Vogl, M.L. Walker, W. Werner, I.I. Zalyubovsky

**Recent 2018-2024**

### 2024

S. Duplij, Polyadic sigma matrices, J. Math. Phys. 65 (2024), 083509 , listed in Editor's Picks, preprint arXiv:2403.19361 (PDF) [math.GT], 19 pages, 2024.S. Duplij, Hyperpolyadic structures, Mathematics 12 (2024), 2378, 29 pages.

S. Duplij, Polyadic supersymmerty, preprint arXiv:2406.02188 [hep-th], 14 pages, 2024.

V.P. Duplij, N.V. Duplii, S. Duplij, DNA walk diagram in Triander and jsTriander applications, Plant Phys. Genet. (Fiziol. rast. genet.) 56 (2024), no. 4, 353-361.

### 2023

**Books**

S. Duplij, R. Vogl, Innovative Quantum Computing, IOP Publishing (Bristol-London) 2023, 178 pages.

**Articles**

S. Duplij, Hyperpolyadic structures, preprint arXiv:2312.01366 [math.RA], 29 pages, 2023.

S. Duplij, R. Vogl, Polyander visualization of quantum walks, preprint arXiv:2311.00409 [quant-ph], 12 pages, 2023.

S. Duplij, R. Vogl, On superqubits, preprint arXiv:2310.09635 [quant-ph], 14 pages, 2023.

### 2022

**Books**

S. Duplij, Polyadic Algebraic Structures, IOP Publishing (Bristol-London) 2022, 461 pages.

Selected Topics in Gravity, Field Theory and Quantum Mechanics, S. Duplij and M.L. Walker, Editors, MDPI Books, Basel, 2022, 348 pages. https://www.mdpi.com/books/book/6455.

**Articles**

S. Duplij, Polyadic analogs of direct product, Universe 8 (2022), 230.

S. Duplij, Polyadic rings of

*p*-adic integers, Symmetry 14 (2022), 2591.

S. Duplij, Polyadization of algebraic structures, Symmetry 14 (2022), 1782.

M.L. Walker and S. Duplij, Gauge gravity vacuum in constraintless Clairaut-type formalism, Universe 8 (2022), 176.

S. Duplij and W. Werner, Extensions of special 3-fields, preprint arXiv:2212.08606 [math.RA], 22 pages, 2022.

### 2021

**Articles**

S. Duplij and W. Werner, Structure of unital 3-fields, Math. Semesterber. 68 (2021), 27–53.

S. Duplij, Polyadic Hopf algebras and quantum groups, East European J. Phys. 2 (2021), 5–50 (arXiv:1811.02712).

S. Duplij, Higher braid groups and regular semigroups from polyadic-binary correspondence, Mathematics 9 (2021), 972.

S. Duplij, Graded medial

*n*-ary algebras and polyadic tensor categories, Symmetry 13 (2021), 1038.

M.L. Walker and S. Duplij, Gauge gravity vacuum in constraintless Clairaut-type formalism, preprint arXiv: 2106.07723, 23 pages, 2021.

S. Duplij and R. Vogl, Polyadic braid operators and higher braiding gates, Universe 7 (2021), 301.

S. Duplij and R. Vogl, Obscure qubits and membership amplitudes, in Topics on Quantum Information Science, S. Curilef and A. R. Plastino, eds., IntechOpen, London, 2021, 20 pp.

S. Duplij, Membership deformation of commutativity and obscure

*n*-ary algebras, J. Math. Physics, Analysis, Geometry 17 (2021), 441–462 (arXiv:2006.07865).

A.J. Bruce and S. Duplij, Double-graded quantum superplane, Rep. Math. Phys. 86 (2020), 383–400 (arXiv:1910.12950).

A.J. Bruce and S. Duplij, Double-graded supersymmetric quantum mechanics, J. Math. Phys. 61 (2020), 063503, 13 ( arXiv:1904.06975).

### 2019

**Articles**

S. Duplij, Supergravity Was Discovered by D.V. Volkov and V.A. Soroka in 1973, Wasn’t it? East European J. Phys. (2019), 3, 81-82 (arXiv:1910.03259).

S. Duplij, Arity shape of polyadic algebraic structures, J. Math. Physics, Analysis, Geometry 15 (2019), 3–56 (arXiv:1703.10132).

A.J. Bruce and S. Duplij, Double-graded quantum superplane, Univ. Luxembourg, preprint arXiv:1910.12950, 2019.

### 2018

**Books**

S. Duplij, Exotic Algebraic and Geometric Structures in Theoretical Physics, Nova Publishers, New York, 2018, 410 pp.

**Articles**

G.A. Goldin, V.M. Shtelen, and S. Duplij, Conformal symmetry transformations and nonlinear Maxwell equations, in Mathematical Structures and Applications, T. Diagana and B. Toni, eds., Springer, Berlin, 2018, pp. 211–224.

S. Duplij, Polyadic Hopf algebras and quantum groups, preprint arXiv:math.RA/1811.02712, 57 pages, 2018.