Related papers: Linear vector fields and exponential law
In engineering practice one often encounters planar problems, where the corresponding vector space of forces, velocities or (infinitesimal) displacements is three dimensional. This paper shows how these spaces can be factorized, such that…
We compute equivariant fundamental classes of orbits in GL(2)-representations. As applications, we find degrees of the orbit closures corresponding to elliptic fibrations and self-maps of the projective line.
These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.
In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space…
The principal series of unitary representations of the Lorentz group has been considered in the helicity basis. Decompositions of tensor products of the spinspaces are studied in the framework of projective representations of the symmetric…
We comment on the article by M. Ozdemir and M. Erdogdu. We indicate that the exponential map onto the Lorentz group can be obtained in two elementary ways. The first way utilizes a commutative algebra involving a conjugate of a…
In this work we solve the problem of providing a Morita invariant definition of Lie and Courant algebroids over Lie groupoids. By relying on supergeometry, we view these structures as instances of vector fields on graded groupoids which are…
The main purpose of this paper is to study the vector groupoids. This is an algebraic structure which combines the concepts of Brandt groupoid and vector space such that these are compatible.
We give a very concise review of the group field theory formalism for non-perturbative quantum gravity, a higher dimensional generalisation of matrix models. We motivate it as a simplicial and local realisation of the idea of 3rd…
We consider the groups of regular circulant matrices over finite fields and integer residue class rings. In both cases we present a formula for the order of these groups. We also make a first step towards finding the algebraic structure of…
We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and…
We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.
We use representation theory to construct spaces of matrices of constant rank. These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more…
An enlarged group G of nonlinear transformations, modeled on the general linear group GL(2,R), leads to a beautiful, apparently unremarked symmetry between the wave function's phase and the logarithm of its amplitude. Equations Doebner and…
In this book, I explored differential equations for operation in Lie group and for representations of group Lie in a vector space.
We study generic representations of general linear groups over a finite ring R with coefficients in a field k in which the cardinality of R is invertible, that is functors from finitely-generated projective R-modules to k-vector spaces. We…
This article provides a geometric representation for the well-known isomorphism between the special orthogonal group of an isotropic quadratic space of dimension 3 and the group of projective transformations of a projective line. This…
We study a vector field of R^3 equivariant under the D_2 symmetry group, called "the D_2 field" in the literature. We construct the complete list of Darboux polynomials for it, solving the partial differential equation defining them. We…
On the affine space containing the space $\mathcal{S}$ of quantum states of finite-dimensional systems there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding relevant…
We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define…