Dirac Lie Groups

TL;DR

This paper extends the classical theory of Poisson Lie groups to Dirac Lie groups, classifying simply connected cases via Dirac Manin triples and providing explicit constructions and properties.

Contribution

It introduces Dirac Lie groups, classifies them using Dirac Manin triples, and constructs their structures explicitly, advancing the understanding of multiplicative Courant algebroids.

Findings

Classification of simply connected Dirac Lie groups via Dirac Manin triples

Explicit construction of Dirac Lie group structures

Development of basic properties of Dirac Lie groups

Abstract

A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups H is isomorphic to the category of Manin triples (d, g, h), where h is the Lie algebra of H. In this paper, we consider Dirac Lie groups, that is, Lie groups H endowed with a multiplicative Courant algebroid A and a Dirac structure E /subset A for which the multiplication is a Dirac morphism. It turns out that the simply connected Dirac Lie groups are classified by so-called Dirac Manin triples. We give an explicit construction of the Dirac Lie group structure defined by a Dirac Manin triple, and develop its basic properties.