Deformations of adjoint orbits for semisimple Lie algebras and Lagrangian submanifolds
نویسندگان
چکیده
We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and semi-direct product given by Cartan decomposition. The two structures admit Hermitian symplectic form defined in complex Lie algebra. provide some applications such as constructions of Lagrangian submanifolds.
منابع مشابه
Star-shapedness and K-orbits in Complex Semisimple Lie Algebras
Given a complex semisimple Lie algebra g = k + ik (k is a compact real form of g), let π : g → h be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra h := t + it, where t is a maximal abelian subalgebra of k. Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to k, and show that it is star-shaped. The result extends a resul...
متن کاملCounting Semisimple Orbits of Finite Lie Algebras by Genus
The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of semisimple orbits of a given split genus. This conjecture is proved for type A, and partial results are obtained for other types. For type A a probabilistic interpretation is given in terms of card shuffling.
متن کاملCounting Split Semisimple Orbits of Finite Lie Algebras by Genus
The adjoint action of a nite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of split semisimple orbits of a given genus. This conjecture is proved for type A, and partial results are obtained for other types. For type A a probabilistic interpretation is given in terms of Solomon's descent algebra and card shu ing. 3
متن کاملSmall Semisimple Subalgebras of Semisimple Lie Algebras
The goal of Section 2 is to provide a proof of Theorem 2.0.1. Section 3 introduces the necessary facts about Lie algebras and representation theory, with the goal being the proof of Proposition 3.5.7 (ultimately as an application of Theorem 2.0.1), and Proposition 3.3.1. In Section 4 we prove the main theorem, using Propositions 3.3.1 and 3.5.7. In Section 5, we apply the theorem to the special...
متن کاملRepresentations of Semisimple Lie Algebras
Let L be a finite-dimensional, semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let H be a fixed Cartan subalgebra of L, and Φ be the root system. Fix a base ∆ = {α1, · · · , αl} of Φ. Let Λ denote the set of dominant, integral linear functions on H. Theorem 0.1. There is a one-to-one correspondence Λ ∼ −→ {isomorphism classes of finite-dimensional irreducible L-...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Differential Geometry and Its Applications
سال: 2021
ISSN: ['1872-6984', '0926-2245']
DOI: https://doi.org/10.1016/j.difgeo.2021.101719