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slides
The Classification of Dirac Homogeneous Spaces
Authors
Patrick James Robinson
Publication date
1 January 2014
Publisher
View
on
arXiv
Abstract
A well known result of Drinfeld classifies Poisson Lie groups
(
H
,
Î
)
(H,\Pi)
(
H
,
Î
)
in terms of Lie algebraic data in the form of Manin triples
(
d
,
g
,
h
)
(\mathfrak{d},\mathfrak{g},\mathfrak{h})
(
d
,
g
,
h
)
; he also classified compatible Poisson structures on
H
H
H
-homogeneous spaces
H
/
K
H/K
H
/
K
in terms of Lagrangian subalgebras
l
⊂
d
\mathfrak{l}\subset\mathfrak{d}
l
⊂
d
with
l
∩
h
=
k
=
L
i
e
(
K
)
\mathfrak{l}\cap\mathfrak{h}=\mathfrak{k}=\mathrm{Lie}(K)
l
∩
h
=
k
=
Lie
(
K
)
. Using the language of Courant algebroids and groupoids, Li-Bland and Meinrenken formalized the notion of \emph{Dirac Lie groups} and classified them in terms of so-called "
H
H
H
-equivariant Dirac Manin triples"
(
d
,
g
,
h
)
β
(\mathfrak{d}, \mathfrak{g}, \mathfrak{h})_\beta
(
d
,
g
,
h
)
β
​
; this generalizes the first result of Drinfeld, as each Poisson Lie group gives a unique Dirac Lie group structure. In this thesis, we consider a notion of homogeneous space for Dirac Lie groups, and classify them in terms of
K
K
K
-invariant coisotropic subalgebras
c
⊂
d
\mathfrak{c}\subset\mathfrak{d}
c
⊂
d
, with
c
∩
h
=
k
\mathfrak{c}\cap\mathfrak{h} = \mathfrak{k}
c
∩
h
=
k
. The relation between Poisson and Dirac morphisms makes Drinfeld's second result a special case of this classification.Comment: 110 pages, PhD Thesi
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