Steven Duplij (Stepan Douplii) Homepage
with direct links to articles
A
new direction in 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 novel 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 arity shape, 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 n-ary graded algebras,
a new kind of tensor categories, 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 K-theory,
the 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 new iterative process gives "half-quaternions"
and
"half-octonions".
A novel polyadic
product of vectors
in
any vector space is defined, which is consistent with the
polyadization
procedure using vectorization.
Endowed with newly introduced product, the vector space becomes a
polyadic algebra which is a division algebra. 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.
The
fundamental
notion
of number theory, the positional
numeral system, is
generalized from
binary integer number ring Z=Z2,2
to
nonderived polyadic rings Zm,n
whose
addition takes m arguments and multiplication takes n. Our key
contributions include: 1) Demonstrating that every commutative
polyadic ring supports a place-value expansion that respects fixed
operation patterns. 2) Establishing a lower bound on digit counts
based on the arity of addition. 3) Identifying a representability
gap where only a subset of elements have finite expansions,
determined by structural invariants. These findings enable novel
approaches to arithmetic, data encoding, and hardware design that
extend beyond conventional binary logic.
We
present a novel
encryption and decryption method that combines polyadic
algebraic structures with signal processing techniques. The approach
begins by encoding information into signals with integer amplitudes
which
are then transformed into structured integer sequences using
polyadic operations. Decryption
involves applying tailored rules and solving dedicated systems
of equations to accurately reconstruct the original message.
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.
All subjects: 202 items
PDF
(1985-2025)
mathematical physics, quantum computing and DNA structure (171
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: 202
publications, among them 9 books and 191
articles.
In addition: 130
entries (table
in PDF) in Concise
Encyclopedia of Supersymmetry.
Listed in Universität Münster Highly Cited Researchers, n.25 from 37 (PDF, the final entry on the page).
Ukrainian National H-index Ranking
Coeditors
(5)
J. Bagger,
W. Siegel, M.L. Walker, J. Wess, V. Zima
Coauthors
(52)
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, Moukun
Fang, G.A. Goldin, Qiang Guo, Yanyong Hong, Shuai Huang, V.V.
Kalashnikov, M. Kaliuzhnyi, O.I. Kotulska, A.T. Kotvytskiy, G.Ch.
Kurinnoj, S.V. Landar, Fang Li, Li-Chao Liu, Y.G. Mashkarov, W.
Marcinek, E.A. Maslov, N.M. Pelykhaty, V.M. Puzh, Liangang Qi, 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, Yuhang Tian, R. Vogl, M.L. Walker, Yani Wang,
W. Werner, Tianfang Xu, I.I. Zalyubovsky, Wenjie Zhao