Related papers: Commutator-free Cayley methods
In this paper, we extend the popular integral control technique to systems evolving on Lie groups. More explicitly, we provide an alternative definition of "integral action" for proportional(-derivative)-controlled systems whose…
We study the connection between Lagrangian and Hamiltonian descriptions of closed/open dynamics, for a collection of particles with quadratic interaction (closed system) and a sub-collection of particles with linear damping (open system).…
By means of contractions of Lie algebras, we obtain new classes of indecomposable quasi-classical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise…
Matrix Lie groups provide a language for describing motion in such fields as robotics, computer vision, and graphics. When using these tools, we are often faced with turning infinite-series expressions into more compact finite series (e.g.,…
The aim of the present article is to construct quadratically integrable three dimensional systems in non-vanishing magnetic fields which possess so-called non-subgroup type integrals. The presence of such integrals means that the system…
We implement two algorithms in MATHEMATICA for classifying automorphisms of lower-dimensional non-commutative Lie algebras. The first algorithm is a brute-force approach whereas the second is an evolutionary strategy. These algorithms are…
The non-commutative differential calculus on the quantum groups $SL_q(N)$ is constructed. The quantum external algebra proposed contains the same number of generators as in the classical case. The exterior derivative defined in the…
New classes of Lie-Hamilton systems are obtained from the six-dimensional fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(6,\mathbb{R})$. The ansatz is based on a recently proposed procedure for constructing…
The theory of Poisson-Lie groups and Lie bialgebras plays a major role in the study of one dimensional integrable systems; many families of integrable systems can be recovered from a Lax pair which is constructed from a Lie bialgebra…
There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only…
By combining the ideas of Cartan's equivalence method and the method of the equivariant moving frame for pseudo-groups, we develop an efficient method for solving equivalence problems arising from horizontal Lie pseudo-group actions. The…
In this paper we present a new procedure to obtain unitary and irreducible representations of Lie groups starting from the cotangent bundle of the group (the cotangent group). We discuss some applications of the construction in…
Theory of differential operators on associative algebras is not extended to the non-associative ones in a straightforward way. We consider differential operators on Lie algebras. A key point is that multiplication in a Lie algebra is its…
Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor-coefficients of the…
We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One…
The purpose of this paper is to connect two subjects: the theory of quantum integrable systems (complete commutative rings of differential operators), and differential Galois theory. We define quantum completely integrable systems (QCIS),…
High dimensional integrals are abundant in many fields of research including quantum physics. The aim of this paper is to develop efficient recursive strategies to tackle a class of high dimensional integrals having a special product…
We introduce new families of pure quantum states that are constructed on top of the well-known Gilmore-Perelomov group-theoretic coherent states. We do this by constructing unitaries as the exponential of operators quadratic in Cartan…
The Gauss decompositions of the quantum groups, related to classical Lie groups and supergroups are considered by the elementary algebraic and $R$-matrix methods. The commutation relations between new basis generators (which are introduced…
In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on…