| RUS ENG |
| Full version |
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| PEOPLE |
| Duplij Steven |
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| Head Scientist Researcher |
| Doctor of physico-mathematical sciences (1999) |
In the theory of polyadic systems a concept of heteromorphism is proposed – a multiplace generalization of the homomorphism, allowing to obtain another arity of multiplication. Multiactions and changing arity representations of $n$-ary groups and semigroups are introduced. Ternary Hopf algebras and ternary generalization of quantum groups and Yang–Baxter equation are studied. It is shown that the Hosszu–Gluskin fundamental formula representing $n$-ary product in the theory of polyadic groups is not unique, and can be generalized ("deformed") using an integer parameter, also a "deformed" analogue of the Hosszu-Gluskin theorem is formulated.
Singular theories (with degenerate Lagrangian) are formulated without involving the Dirac constraints. A partial Hamiltonian formalism in the reduced phase space (with a pre-fixed number of generalized momenta) is built. The equivalence of singular theories and multi-time dynamics in the framework of the proposed partial Hamiltonian formalism is shown. A new anti-symmetric bracket (similar to the Poisson bracket), which describes the time evolution of singular systems, could lead to a new quantization scheme without using constraints. Also for singular theories a new version of the Hamiltonian formalism is proposed by generalizing the Legendre transform to the case, when the Hessian is zero, using mixed (envelope/general) solutions of the multidimensional Clairaut differential equation.
A direction in the construction of supersymmetric and superstring models, based on a consistent inclusion in the study semigroups and their mathematical structure is proposed. Noninvertible generalization of superconformal and hyperbolic geometries are found, noninvertible analogs of superconformal transformations and super Riemann surfaces are given. A concept the semisupermanifold generalizing the notion of supermanifold to the noninvertible transition functions case is proposed. This approach is applied to the generalization of categories, for which an analogue of "obstruction morphisms" are cyclic $n$-morphisms. A corresponding generalization of functors and other objects in monoidal categories, noninvertible analog of the Yang-Baxter equation and noninvertible generalization of (co-) algebras, the left and right (co-) modules, the tensor product of the action modules, also a double regular $R$-matrix, are definned.
A noninvertible pemutation statistics – semistatistics is proposed – by changing the conditions under which the double exchange leads to the identity transformation to the von Neumann regularity condition.
Generalization of Hopf algebras introduced by weakening the conditions of invertibility for the generators of the Cartan subalgebra is presented. A quasi-braided weak Hopf algebra is constructed and it is shown that the corresponding quasi-$R$-matrix is von Neumann regular.
The Peirce decomposition is applied to the generalization of quantum algebras, and their finite-dimensional representations are proposed. The actions of the quantum universal enveloping algebras on the quantum plane (of arbitrary dimensions) are obtained and classified. The classical limits of the obtained actions are found.
Semigroups of supermatrices are considered, and their various reductions are determined. The semigroups of antitriangle supermatrices (which are bands) are introduced, and the properties of the corresponding one-parameter semigroups of superoperators are studied. It is shown that the $t$-linear idempotent superoperators and usual exponential superoperators are dual in some sense, and the first ones give an extra (odd and nonexponential) solution of the Cauchy problem. Antitriangle supermatrices are used to construct representations of semigroups bands, also new generalized (fine) Green's relations are found.
The classical mechanics with nilpotent even directions is built, and their role in supersymmetric quantum mechanics is considered.
A generalized approach to nonlinear classical electrodynamics and supersymmetric electrodynamics, taking into account all the possible types of media (anisotropic, pyroelectric, chiral and ferromagnetic), ans non-local effects that can be described as Lagrangian and non-Lagrangian theories, is presented. Generalized constitutive equations and constitutive tensors of the general form are introduced. The nonlinear equations for Gravitoelectromagnetism are studied, and the problem of finding a particular solution of equations for the Maxwell-gravitational field exact shape corresponding to nonlinear constitutive equation is solved.
A general approach to the description of the interaction in multigravity models in spacetime of arbitrary dimension is proposed, as well as various possibilities of generalization of the invariant volume and the most general form of the interaction potential is given.
