74,089 research outputs found
Classical dynamical r-matrices and homogeneous Poisson structures on and
Let G be a finite dimensional simple complex group equipped with the standard
Poisson Lie group structure. We show that all G-homogeneous (holomorphic)
Poisson structures on , where is a Cartan subgroup, come
from solutions to the Classical Dynamical Yang-Baxter equations which are
classified by Etingof and Varchenko. A similar result holds for the maximal
compact subgroup K, and we get a family of K-homogeneous Poisson structures on
, where is a maximal torus of K. This family exhausts all
K-homogeneous Poisson structures on up to isomorphisms. We study some
Poisson geometrical properties of members of this family such as their
symplectic leaves, their modular classes, and the moment maps for the T-action
Hopf algebroids and quantum groupoids
We introduce the notion of Hopf algebroids, in which neither the total
algebras nor the base algebras are required to be commutative. We give a class
of Hopf algebroids associated to module algebras of the Drinfeld doubles of
Hopf algebras when the -matrices act properly. When this construction is
applied to quantum groups, we get examples of quantum groupoids, which are
semi-classical limits of Poisson groupoids. The example of quantum is
worked out in details.Comment: 30 pages, in Late
On a Dimension Formula for Twisted Spherical Conjugacy Classes in Semisimple Algebraic Groups
Let be a connected semisimple algebraic group over an algebraically
closed field of characteristic zero, and let be an automorphism of .
We give a characterization of -twisted spherical conjugacy classes in
by a formula for their dimensions in terms of certain elements in the Weyl
group of , generalizing a result of N. Cantarini, G. Carnovale, and M.
Costantini when is the identity automorphism. For simple and an
outer automorphism of , we also classify the Weyl group elements that appear
in the dimension formula.Comment: 8 page
Generative Exploration and Exploitation
Sparse reward is one of the biggest challenges in reinforcement learning
(RL). In this paper, we propose a novel method called Generative Exploration
and Exploitation (GENE) to overcome sparse reward. GENE automatically generates
start states to encourage the agent to explore the environment and to exploit
received reward signals. GENE can adaptively tradeoff between exploration and
exploitation according to the varying distributions of states experienced by
the agent as the learning progresses. GENE relies on no prior knowledge about
the environment and can be combined with any RL algorithm, no matter on-policy
or off-policy, single-agent or multi-agent. Empirically, we demonstrate that
GENE significantly outperforms existing methods in three tasks with only binary
rewards, including Maze, Maze Ant, and Cooperative Navigation. Ablation studies
verify the emergence of progressive exploration and automatic reversing.Comment: AAAI'2
Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras
We introduce and study some mixed product Poisson structures on product
manifolds associated to Poisson Lie groups and Lie bialgebras. For
quasitriangular Lie bialgebras, our construction is equivalent to that of
fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann-
Schwarzbach, and Meinrenken. Our primary examples include four series of
holomorphic Poisson structures on products of flag varieties and related spaces
of complex semi-simple Lie groups.Comment: 37 pages, submitted to IMR
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