arXiv subjects
Publications and source records attributed to Aleks Kleyn.
From the symmetry between definitions of left and right divisors in associative $D$-algebra $A$, the possibility to define quotient as $A\otimes A$-number follows. In the paper, I considered division and division with remainder. I considered also definition of prime $A$-number.
In this book i treat linear algebra over division ring. A system of linear equations over a division ring has properties similar to properties of a system of linear equations over a field. However, noncommutativity of a product creates a new picture. Matrices allow two products linked by transpose. Biring is algebra which defines on the set two correlated structures of the ring. As in the commutative case, solutions of a system of linear equations build up right or left vector space depending on type of system. We study vector spaces together with the system of linear equations because their properties have a close relationship. As in a commutative case, the group of automorphisms of a vector space has a single transitive representation on a frame manifold. This gives us an opportunity to introduce passive and active representations. Studying a vector space over a division ring uncovers new details in the relationship between passive and active transformations, makes this picture clearer. Considering of twin representations of division ring in Abelian group leads to the concept of D vector space and their linear map. Based on polyadditive map I considered definition of tensor product of rings and tensor product of D vector spaces.
The common in ring, module and algebra is that they are Abelian group with respect to addition. This property is enough to study integration. I treat integral of measurable map into normed Abelian $Ω$-group. Theory of integration of maps into $Ω$-group has a lot of common with theory of integration of functions of real variable. However I had to change some statements, since they implicitly assume either compactness of range or total order in $Ω$-group.
Since sum which is not necessarily commutative is defined in Ω-algebra A, then Ω-algebra A is called Ω-group. I also considered representation of Ω-group. Norm defined in Ω-group allows us to consider continuity of operations and continuity of representation.
In this book, I explored differential equations for operation in Lie group and for representations of group Lie in a vector space.
Based on twin representations of division ring in an Abelian group I consider $D$\Hyph vector spaces over division ring. Morphism of $D$\Hyph vector spaces is linear map of $D$\Hyph vector spaces. I consider derivative of function $f$ of continuous division ring as linear map the most close to function $f$. I explore expansion of map into Taylor series and method to find solution of differential equation. The norm in $D$\Hyph vector space allows considering of continuous mapping of $D$\Hyph vector spaces. Differential of mapping $f$ of $D$\Hyph vector spaces is defined as linear mapping the most close to map $f$.
In this book I treat the structure of D-module which has countable basis. If we do not care for topology of D-module, then we consider Hamel basis. If norm is defined in D-module, then we consider Schauder basis. In case of Schauder basis, we consider vectors whose expansion in the basis converges normally.
In the paper I considered mappings of conjugation of quaternion algebra. I proved the theorem that there is unique expansion of R-linear mapping of quaternion algebra relative to the given set of mappings of conjugation.
In the paper, I considered construction of algebra of fractions of algebra with conjugation. I also considered algebra of polynomials and algebra of rational mappings over algebra with conjugation.
In the paper I considered methods for solving equations of the form axb+cxd=e in the algebra which is finite dimensional over the field.
In the paper, I consider properties and mappings of free algebra with unit. I consider also conjugation of free algebra with unit.
In the paper I considered definition and structure of linear mapping of Banach algebra over commutative ring. Based on this definition I explore derivative of continuous mapping.
Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, we considered the definition of orthonormal basis in Minkowski space, the structure of metric tensor relative to orthonormal basis, procedure of orthogonalization. Linear transformation of Minkowski space mapping at least one orthonormal basis into orthonormal basis is called motion. The set of motions of Minkowski space V generates not complete group SO(V) which acts single transitive on the basis manifold. Passive transformation of Minkowski space mapping at least one orthonormal basis into orthonormal basis is called quasimotion of Minkowski space. The set of passive transformations of Minkowski space generates passive representation of not complete group SO(V) on basis manifold. Since twin representations (active and passive) of not complete group SO(V) on basis manifold are single transitive, then we may consider definition of geometric object.
We say that there is a representation of the universal algebra B in the universal algebra A if the set of endomorphisms of the universal algebra A has the structure of universal algebra B. Therefore, the role of representation of the universal algebra is similar to the role of symmetry in geometry and physics. Morphism of the representation is the mapping that conserves the structure of the representation. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. The set of automorphisms of the representation of the universal algebra forms the group. Twin representations of this group in basis manifold of the representation are called active and passive representations. Passive representation in basis manifold is underlying of concept of geometric object and the theory of invariants of the representation of the universal algebra.
Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, I considered the definition of orthonormal basis in Minkowski space, the structure of metric tensor relative to orthonormal basis, procedure of orthogonalization. Linear transformation of Minkowski space which preserves the scalar product is called a motion. Linear transformation which maps the orthonormal basis into an infinitely close orthonormal basis is infinitesimal motion. An infinitesimal motion maps orthonormal basis into orthonormal basis. The set of infinitesimal motions generates Lie algebra, which acts single transitive on basis manifold of Minkowski space. Element of twin representation is called quasimotion of Minkowski space. Quasimotion of event space is called Lorentz transformation.
In the paper I considered linear and antilinear automorphisms of quaternion algebra. I proved the theorem that there is unique expansion of R-linear mapping of quaternion algebra relative to the given set of linear and antilinear automorphisms.
In the paper I considered linear and antilinear mappings of finite dimensional algebra over complex field, as well I considered involution. I considered also an example of $C^*$\Hyph algebra.
In this paper I explore the set of quaternion algebras over field. Quaternion algebra E(C,-1,-1) is isomorphic to tensor product of complex field C and quaternion algebra H=E(R,-1,-1). Considered two sets of quaternion functions, which satisfy to equation similar to Cauchy-Riemann equation for a complex function.