Classical dynamical r-matrices and homogeneous Poisson structures on G/HG/H and K/TK/T

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 $G/H$, where $H \subset G$ 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 $K/T$, where $T = K \cap H$ is a maximal torus of K. This family exhausts all K-homogeneous Poisson structures on $K/T$ 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 $R$-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 $sl(2)$ is worked out in details.Comment: 30 pages, in Late

On a Dimension Formula for Twisted Spherical Conjugacy Classes in Semisimple Algebraic Groups

Let $G$ be a connected semisimple algebraic group over an algebraically closed field of characteristic zero, and let $\th$ be an automorphism of $G$. We give a characterization of $\th$-twisted spherical conjugacy classes in $G$ by a formula for their dimensions in terms of certain elements in the Weyl group of $G$, generalizing a result of N. Cantarini, G. Carnovale, and M. Costantini when $\th$ is the identity automorphism. For $G$ simple and $\th$ an outer automorphism of $G$, 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

A Poisson structure on compact symmetric spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.Comment: 11 pages, exposition streamline