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Publication# Dirac Group(oid)s and Their Homogeneous Spaces

Abstract

A theorem of Drinfel'd (Drinfel'd (1993)) classifies the Poisson homogeneous spaces of a Poisson Lie group (G,πG) via a special class of Lagrangian subalgebras of the Drinfel'd double of its Lie bialgebra. This result is extended in Liu et al. (1998) to a classification of the Poisson homogeneous spaces of a Poisson groupoid (G⇉P, πG) via a special class of Dirac structures in the Courant algebroid defined by the corresponding Lie bialgebroid. It is hence natural to ask what corresponds via these classifications, or extensions of them, to arbitrary Dirac structures in the Drinfel'd double or the Courant algebroid associated to a Poisson group(oid). We show in this thesis that there is a bigger class of Dirac structures that fits in this correspondence. They correspond naturally to Dirac homogeneous spaces of the Poisson group(oid). Dirac Lie groups and Dirac groupoids have been defined by Ortiz (2009) as a generalization of Poisson Lie groups, Poisson groupoids and presymplectic groupoids. We prove in an alternative manner the existence of a natural Lie bialgebra associated to a Dirac Lie group, and we find good candidates for the infinitesimal data of a Dirac groupoid; a square of morphisms of Lie algebroids associated to the multiplicative Dirac structure. These objects generalize simultaneously the Lie bialgebroid of Poisson groupoids and the IM-2-forms associated to presymplectic groupoids (Bursztyn et al. (2004)). We show also the existence of a Courant algebroid associated to these algebroids. This Courant algebroid structure is induced in a natural way by the ambient standard Courant algebroid TG ×G T*G and is isomorphic to the Courant algebroid AG ×P A*G in the Poisson case. We show that Dirac homogeneous spaces of Dirac Lie groups (respectively Dirac groupoids) correspond to certain classes of Dirac structures in the Drinfel'd double of the Lie bialgebra (respectively in the Courant algebroid). These results generalize the classification theorems in Drinfel'd (1993) and Liu et al. (1998). Along the way, we study involutive multiplicative foliations on Lie groupoids and show when and how there is a natural groupoid structure defined on the leaf spaces in the set of objects and the set of units of the Lie groupoid. This is a fact that is always true in the Lie group case, where a multiplicative foliation is automatically the left and right invariant image of an ideal in the Lie algebra, i.e., the leaves are the cosets of a normal subgroup of the Lie group.

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Lie group

In mathematics, a Lie group (pronounced liː ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction).

Lie algebra

In mathematics, a Lie algebra (pronounced liː ) is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative.

Dirac delta function

In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., ) to its value at zero of its domain (), or as the weak limit of a sequence of bump functions (e.g.