The aim of the paper is to give a full classification of factorizations of groups in terms of descent cohomology (pointed) sets introduced in Mesablishvili (Trans. A. Razmadze Math. Inst.
173
(2), ...137–155,
2019
). We show that descent cohomology includes Serre’s non-abelian group cohomology as a special case. This enables us to generalize Serre’s theory further to include monoids.
Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids ...and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described.
The connections between the properties of associative rings that are Lie-solvable (Engel, n-Engel, locally finite, respectively) and the properties of their adjoin subgroups are investigated.
A ring R is said to be of stable range 1.5 if for each a,b∈R and 0≠c∈R satisfying aR+bR+cR=R there exists r∈R such that (a+br)R+cR=R. Let R be a commutative domain in which all finitely generated ...ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R.
Abstract
There are analyzed two physically reasonable generalizations of the Kardar-Parisi-Zhang equation describing the spin glasses growth models and possessing important from physical point of ...view properties. The first one proved to be a completely integrable Hamiltonian dynamical system with an infinite hierarchy of commuting to each other conservation laws, and the second one proved to be linearized modulo some nonlinear constraints, imposed on its solutions.
Some ranks of modules over group rings Bovdi, Victor A.; Kurdachenko, Leonid A.
Communications in algebra,
20/3/4/, Volume:
49, Issue:
3
Journal Article
Peer reviewed
A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be ...generated by r elements. Such rings are closely related to Prüfer domains. In the present paper, we investigate some analogs of these concepts for modules over group rings.
We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially ...constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows.
Let U be a semiunitary space; i.e., a complex vector space with scalar product given by a positive semidefinite Hermitian form 〈⋅,⋅〉. If a linear operator A:U→U is bounded (i.e., ‖Au‖⩽c‖u‖ for some ...c∈R and all u∈U), then the subspace U0:={u∈U|〈u,u〉=0} is invariant, and so A defines the linear operators A0:U0→U0 and A1:U/U0→U/U0.
Let A be an indecomposable bounded operator on U such that 0≠U0≠U. Let λ be an eigenvalue of A0. We prove that the algebraic multiplicity of λ in A1 is not less than the geometric multiplicity of λ in A0, and the geometric multiplicity of λ in A1 is not less than the number of Jordan blocks Jt(λ) of each fixed size t×t in the Jordan canonical form of A0.
We give canonical forms of selfadjoint and isometric operators on U, and of Hermitian forms on U.
For an arbitrary system of semiunitary spaces and linear mappings on/between them, we give an algorithm that reduces their matrices to canonical form. Its special cases are Belitskii's and Littlewood's algorithms for systems of linear operators on vector spaces and unitary spaces, respectively.
V.I. Arnold (1971) constructed a simple normal form to which all complex matrices B in a neighborhood U of a given square matrix A can be reduced by similarity transformations that smoothly depend on ...the entries of B. We calculate the radius of the neighborhood U. A.A. Mailybaev (1999, 2001) constructed a reducing similarity transformation in the form of Taylor series; we construct this transformation by another method. We extend Arnold's normal form to matrices over the field Qp of p-adic numbers and the field F((T)) of Laurent series over a field F.
Let (A,B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum(A__,B__)⊕(A1,B1)⊕…⊕(At,Bt) that is congruent to (A,B), in which ...(A__,B__) is a pair of nonsingular matrices and (A1,B1),…,(At,Bt) are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of (A,B) under congruence over an algebraically closed field of characteristic not 2.
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