A minimal-variable symplectic method for isospectral flows

Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie--Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the \textit{spherical midpoint method} on \SO(3). In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie--Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature.Comment: 17 pages, 9 figure

Post-Lie Algebras, Factorization Theorems and Isospectral-Flows

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl

Post-Lie Algebras and Isospectral Flows

In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical $R$-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation

Stratified Manifold of Quantum States, actions of the complex special linear group

We review the geometry of the space of quantum states $\mathscr{S}(\mathcal{H})$ of a finite-level quantum system with Hilbert space $\mathcal{H}$ from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups: the special unitary group $\mathcal{SU}(\mathcal{H})$ and its complexification $\mathcal{SL}(\mathcal{H})$, the complex special linear group. A stratum of the stratification generated by $\mathcal{SU}(\mathcal{H})$ is composed of isospectral states, that is, density operators with the same spectrum, A stratum of the stratification generated by $\mathcal{SL}(\mathcal{H})$ is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of $\mathcal{SL}(\mathcal{H})$ which is related with the canonical K\"{a}hler structure on isospectral quantum states. The fundamental vector fields of this $\mathcal{SL}(\mathcal{H})$-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve the von Neumann entropy and the convex structure of $\mathscr{S}(\mathcal{H})$, while the latter give rise to invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve the von Neumann entropy but not the convex structure of $\mathscr{S}(\mathcal{H})$. A similar decomposition is given for the $\mathcal{SL}(\mathcal{H})$-action generating the stratification of $\mathscr{S}(\mathcal{H})$ into manifolds of quantum states with the same rank, where gradient vector fields preserve the rank but do not preserve entropy. Some comments on multipartite quantum systems are made. It is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the action of the complex special linear group associated with the partition