Polyubles, Poisson homogeneous spaces and multi-flag varieties

TL;DR

This paper explores the structure of polyubles of Manin triples, establishing isomorphisms and constructing Poisson homogeneous spaces and multi-flag varieties, advancing understanding in Poisson geometry and Lie theory.

Contribution

It introduces a unique isomorphism between polyubles of different orders and constructs classes of Poisson homogeneous spaces and global isomorphisms for multi-flag varieties.

Findings

Established a unique isomorphism between polyubles of different orders.

Constructed classes of Poisson homogeneous spaces and homeomorphisms.

Applied results to multi-flag and multi-double flag varieties, creating global Poisson isomorphisms.

Abstract

A polyuble of a Manin triple can be regarded as the ``$n$-th power'' of it, which plays an important rule in the study of Poisson geometry, mathematical physics and Lie theory. In this paper, we first construct an isomorphism between the $mn$-ble and the $n$-ubles of $m$-uble by colored graph and point out it is unique. Then, we construct a class of Poisson homogeneous spaces and obtain a class of Poisson homeomorphisms between them based on the first main result. Last, we apply first two main results to multi-flag varieties as well as multi-double flag varieties and construct a class of global Poisson isomorphisms between them as well as their $T$-leaves.