Polymorphism of Representations of Universal Algebra
In this paper I consider the polymorpism of representations of universal algebra and tensor product of representations of universal algebra.
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arXiv subjects
Aleks Kleyn
Publications and source records attributed to Aleks Kleyn.
Correspondence between Row-Column Determinants and Quasideterminants of Matrices over Quaternion Algebra
In this paper, we considered the theory of quasideterminants and row and column determinants. We considered the application of this theory to the solving of a system of linear equations in quaternion algebra. We established correspondence between row and column determinants and quasideterminants of matrix over quaternion algebra.
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Polylinear Mapping of Free Algebra
In this paper I consider the structure of the polylinear mapping of the free algebra over the commutative ring.
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Linear Mappings of Free Algebra
For arbitrary F-algebra, in which the operation of addition is defined, I explore biring of matrices of mappings. The sum of matrices is determined by the sum in F-algebra, and the product of matrices is determined by the product of mappings. The system of equations, whose matrix is a matrix of mappings, is called a system of additive equations. I considered the methods of solving system of additive equations. As an example, I consider the solution of a system of linear equations over the complex field provided that the equations contain unknown quantities and their conjugates. Linear mappings of algebra over a commutative ring preserve the operation of addition in algebra and the product of elements of the algebra by elements of the ring. The representation of tensor product A\otimes A in algebra A generates the set of linear transformations of algebra A. The results of this research will be useful for mathematicians and physicists who deal with different algebras.
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Vector Space Over Division Ring
A system of linear equations over a skew field has properties similar to properties of a system of linear equations over a field. Even noncommutativity of a product creates a new picture the properties of system of linear equations and of vector space over skew-field have a close relationship.
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Biring of Matrices
Matrices allow two products linked by transpose. Biring is algebra which defines on the set two correlated structures of the ring. According to each product we can extend the definition of a quasideterminant given in [1, 2] and introduce two different types of a quasideterminant.
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The Matrix of Linear Mappings
On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we define the operation of product. The operation is based on the product of mappings. If the matrix a of linear mappings has an inverse matrix, then the quasideterminant of the matrix a and the inverse matrix are matrices of linear mappings. In the paper, I consider conditions when a matrix of linear mappings has inverse matrix, as well methods of solving a system of linear equations in an associative algebra.
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The Gateaux Derivative of Map over Division Ring
I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent. I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function of original number \bar a=a+iai+jaj+kak . In quaternion algebra this difference leads to the absence of analogue of the Cauchy Riemann equations that are well known in the theory of complex function.
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Morphism of T*-Representations
Importance of theorem dedicated to isomorphisms consist in statement that they allow to identify different mathematical objects which have something common from the point of view of certain model. This paper considers morphisms of \Ts representation of $\mathfrak{F}$\Hyph algebra and morphisms of \Ts representation of fibered $\mathfrak{F}$\Hyph algebra.
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Tidal Force in Metric-Affine Gravity
Generalization of an idea may lead to very interesting result. Learning how torsion influences on tidal force reveals similarity between tidal equation for geodesic and the Killing equation of second type. The relationship between tidal acceleration, curvature and torsion gives an opportunity to measure torsion.
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Metric-Affine Manifold
We call a manifold with torsion and nonmetricity the metric-affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport and moving basis. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection. The analysis of the Frenet transport leads to the concept of the Cartan transport and an introduction of the connection compatible with the metric tensor. The dynamics of a particle follows to the Cartan transport. We need additional physical constraints to make a nonmetricity observable.
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Reference Frame in General Relativity
A reference frame in event space is a smooth field of orthonormal bases. Every reference frame is equipped by anholonomic coordinates. Using anholonomic coordinates allows to find out relative speed of two observers and appropriate Lorentz transformation. Synchronization of a reference frame is an anholonomic time coordinate. Simple calculations show how synchronization influences time measurement in the vicinity of the Earth. Measurement of Doppler shift from the star orbiting the black hole helps to determine mass of the black hole. According observations of Sgr A, if non orbiting observer estimates age of S2 about 10 Myr, this star is 0.297 Myr younger.
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Fibered Correspondence
Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. Reduced fibered correspondence in bundle is called 2-ary fibered relation. We considered fibered equivalence and isomorphism theorem in case of fibered morphisms.
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Fibered F-Algebra
The concept of F-algebra and its representation can be extended to an arbitrary bundle. We define operations of fibered F-algebra in fiber. The paper presents the representation theory of of fibered F-algebra as well as a comparison of representation of F-algebra and of representation of fibered F-algebra.
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Basis Manifold
In this paper we study a symmetry group of vector space. Basis manifold is a homogeneous space of a symmetry group. This concept leads us to the definition of active and passive transformations on basis manifold. Active transformation can be expressed as a transformation of vector space. Passive transformation gives ability to define concepts of invariance and of geometrical object.
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