1. Fundamentos, Estructuras y Generalizaciones de Lie.
La investigación actual se centra en la profundización de los aspectos teóricos de las álgebras y grupos de Lie, buscando clasificaciones, extensiones y generalizaciones a estructuras como superálgebras, pseudogrupos y sistemas de Lie. Existe un interés particular en trascender las representaciones puramente matriciales para explorar la geometría subyacente.
- I need to finish reading my book about lie algebras. The one i have now is annoyingly concrete rn tho all the theory only applies to matrix groups and doesnt really take advantage of the geometrical aspect which is what i actually care about.
- Hello and welcome back to our series of seminars introducing mathematical concepts. This week what in the everloving fuck is. A lie algebra?
- Mixed superposition rules for lie systems and compatible geometric structures rutwig campoamor-stursberg, oscar carballal, francisco j. Herranz, javier de lucas.
- Classification of nilpotent lie algebras of nilpotency class 3 having a derived subalgebra of dimension three.
- Foundations of noncommutative carrollian geometry via lie-rinehart pairs andrew james bruce.
2. Aplicaciones en Control, Dinámica Cuántica y Sistemas Físicos.
La teoría de Lie se está aplicando activamente para modelar y controlar sistemas complejos, desde la dinámica cuántica y la optimización de circuitos, hasta el control de vehículos autónomos y la mejora de métodos numéricos como las Redes Neuronales Informadas por la Física (PINNs).
- Enhancing pinn performance through lie symmetry group.
- A lie theoretic framework for controlling open quantum systems corey omeara.
- Finite-dimensional lie algebras in bosonic quantum dynamics the single-mode case tim heib, andreea silvia goia, sona baghiyan, robert zeier, david edward bruschi.
- Online learning-enhanced lie algebraic mpc for robust trajectory tracking of autonomous surface vehicles.
- Analytical solution and lie algebra of the relativistic boltzmann equation yi wang, xuan zhao, zhe xu, jin hu.
- Runge-kutta approximations for direct coning compensation applying lie theory.
3. Geometría, Topología y Formas Reales.
Una línea de investigación crucial aborda la conexión entre las álgebras de Lie y sus grupos correspondientes, incluyendo la complexificación de grupos y el estudio de formas reales. También se exploran las propiedades geométricas de los grupos de Lie en el contexto de variedades y espacios homogéneos.
- Can you complexify a group? i suppose if it is a lie group you can complexify its lie algebra and then lie-integrate that?
- Mathematicians working on lie theory often go the other way! they start with complex lie algebras and their corresponding complex lie groups, and then look at real forms of those. For example, gln,c has gln,r and un as real forms.
- The most beautiful spaces are the irreducible compact riemannian symmetric spaces! there are the infinite series the lie group son, sun, spn and the 6 series listed below.
- Naive lie theory by john stillwell is a beginners guide to lie theory through matrix lie theories, i.e. Subgroups of the general linear groups over the reals.
- Normal curves in sub-finsler lie groups branching for strongly convex norms and face stability for polyhedral norms.
4. Computación Simbólica y Aprendizaje Automático Equivariante.
La computación se utiliza para clasificar y manipular estructuras de Lie, mientras que el aprendizaje automático está adoptando la teoría de Lie para construir redes neuronales equivariantes que respetan las simetrías físicas, especialmente en el contexto de las álgebras excepcionales.
- Symbolic computation of optimal systems of subalgebras of three- and four-dimensional real lie algebras.
- Equivariant neural networks for general linear symmetries on lie algebras.
- Hyperdimensional enumeration of exceptional lie algebras for accelerated quantum circuit optimization abstract this paper introduces a novel methodology for efficiently enumerating exceptional lie algebras.
- I can only really describe it as harmonic decomposition with sigmoidal ridge functions but the practical upshot is a system that learns exceptional lie algebra and unlocks a completely new mode of harmonic analysis that can be applied to classification and synthesis of sequence data.
- Lie algebra-derived fermionic unitaries enable compact quantum circuits with reduced optimization parameters.