arXiv subjects
Publications and source records attributed to Steven Duplij.
A proposal of the concept of $n$-regular obstructed categories is given. The corresponding regularity conditions for mappings, morphisms and related structures in categories are considered. An n-regular TQFT is introduced. It is shown the connection of time reversibility with the regularity.
Regular and higher regular graded algebras (in simplest case satisfying Von Neumann regularity $Θ_{1}Θ_{2}Θ_{1}=Θ_{1}$ instead of anticommutativity) are introduced and their properties are studied. They are described in terms of obstructed categories with nonclosed invertible and noninvertible morphisms for which generalized (obstructed) functors and natural transformations are given. Regular algebras and bialgebras are considered with examples. Corresponding regularizations of the cross product and Wick algebras are made.
We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}(2)$ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}(2)$ and a $J$-weak Hopf algebra $v\mathfrak{sl}_{q}(2)$ which are studied in detail. It is shown that the monoids of group-like elements of $w\mathfrak{sl}_{q}(2)$ and $v\mathfrak{sl}_{q}(2)$ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w\mathfrak{sl}_{q}(2)$ a quasi-braided weak Hopf algebra $\overline{U}_{q}^{w}$ is constructed and it is shown that the corresponding quasi-$R$-matrix is regular $R^{w}\hat{R}^{w}R^{w}=R^{w}$.
The categories with noninvertible morphisms are studied analogously to the semisupermanifolds with noninvertible transition functions. The concepts of regular n-cycles, obstruction and the regularization procedure are introduced and investigated. It is shown that the regularization of a category with nonivertible morphisms and obstruction form a 2-category. The generalization of functors, Yang-Baxter equation, (co-) algebras, (co-) modules and some related structures to the regular case is given.
Various symmetries connected with purine-pyrimidine content of DNA sequences are studied in terms of the intruduced determinative degree, a new characteristics of nucleotide which is connected with codon usage. A numerological explanation of CG pressure is proposed. A classification of DNA sequences is given. Calculations with real sequences show that purine-pyrimidine symmetry increases with growing of organization. A new small parameter which characterizes the purine-pyrimidine symmetry breaking is proposed for the DNA theory.
A three-dimensional model of the genetic vocabulary is proposed in terms of determinative degree, an introduced characteristic of nucleotide, which reflects absolute difference between purin and pyrimidin bases in one DNA strand. In the fremework of this model manifest symmetry and group properties are described. It is proposed to use the determinative degree of codons to analyze genetic texts, and to explain the biological sense of various functional regions of DNA.
One-parameter semigroups of antitriangle idempotent supermatrices and corresponding superoperator semigroups are introduced and investigated. It is shown that $t$-linear idempotent superoperators and exponential superoperators are mutually dual in some sense, and the first gives additional to exponential solution to the initial Cauchy problem. The corresponding functional equation and analog of resolvent are found for them. Differential and functional equations for idempotent (super)operators are derived for their general $t$ power-type dependence.
We propose to extend ``invertibility'' to ``regularity'' for categories in general abstract algebraic manner. Higher regularity conditions and ``semicommutative'' diagrams are introduced. Distinction between commutative and ``semicommutative'' cases is measured by non-zero obstruction proportional to the difference of some self-mappings (obstructors) $e^{(n)}$ from the identity. This allows us to generalize the notion of functor and to ``regularize'' braidings and related structures in monoidal categories. A ``noninvertible'' analog of the Yang-Baxter equation is proposed.
The thesis is devoted to abstract, geometric and symmetric aspects of modern elementary particle theories. A new direction in constructing supersymmetric and superstring models based on consequent and strong consideration and inclusion of semigroups, ideals and noninvertible properties into their mathematical structure is proposed. A theory of semisupermanifolds (Chapter I) and noninvertible generalization of superconformal (Chapters II-III) and hyperbolic geometries (Chapter V) are introduced. New continuous supermatrix representations of semigroups (Chapter IV) are obtained. The carried out investigations will allow us to formulate a theoretical model of elementary particles based on supersymmetry in terms of more general categories and new structures - as a theory of abstract and supermatrix semigroups which includes previous theories as a particular invertible case in abstract sense.
We consider the nilpotent additions to classical trajectories in supersymmetric and nonsupersymmetric theories. The condition of anilpotence of action on some generalized solutions leads to the Witten supersymmetric Lagrangian. The condition of anilpotence of topological charge is the same as one of superpotential with spontaneous broken supersymmetry. We should vanish half of Grassmann constants of integration, because in this case only we obtain the same number of normalized bosonic and fermionic zero modes.
We propose the ``short'' version of q-deformed differential calculus on the light-cone using twistor representation. The commutation relations between coordinates and momenta are obtained. The quasi-classical limit introduced gives an exact shape of the off-shell shifting.
Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rectangular band analogs and the ``wreath'' version of them are studied in detail. We introduce the ``fine'' equivalence relations leading to ``multidimesional'' eggbox diagrams. They are full images of Green's relations on corresponding subsemigroups.
Supersymmetry contains initially noninvertible objects, but it is common to deal with the invertible ones only, factorizing former in some extent. We propose to reconsider this ansatz and try to redefine such fundamental notions as supermanifolds, fiber bundles and homotopies using some weakening invertibility conditions. The prefix semi- reflects the fact that the underlying morphisms form corresponding semigroups consisting of a known group part and a new ideal noninvertible part. We found that the absence of invertibility gives us the generalization of the cocycle conditions for transition functions of supermanifolds and fiber bundles in a natural way, which can lead to construction of noninvertible analogs of Čech cocycles. We define semi-homotopies, which can be noninvertible and describe mappings into the semi-supermanifolds introduced.
We found another N=1 odd superanalog of complex structure (the even one is widely used in the theory of super Riemann surfaces). New N=1 superconformal-like transformations are similar to anti-holomorphic ones of nonsupersymmetric complex function theory. They are dual to the ordinary superconformal transformations subject to the Berezinian addition formula presented, noninvertible, highly degenerated and twist parity of the tangent space in the standard basis. They also lead to the ''mixed cocycle condition'' which can be used in building noninvertible objects analogous to super Riemann surfaces. A new parametrization for the superconformal group is presented which allows us to extend it to a semigroup and to unify the description of old and new transformations.
We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the generalized noninvertible superconformal-like transformations. The features of even- and odd-reduced supermatrices are investigated on a par. They can be unified into some kind of "sandwich" semigroups. Also we define a special module over even- and odd-reduced supermatrix sets, and the generalized Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced supermatrices represent semigroup bands and Rees matrix semigroups over a unit group.
A new type of semigroups which appears while dealing with $N=1$ superconformal symmetry in superstring theories is considered. The ideal series having unusual abstract properties is constructed. Various idealisers are introduced and studied. The ideal quasicharacter is defined. Green's relations are found and their connection with the ideal quasicharacter is established.