Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classical Yang-Baxter Equation and Left Invariant Affine Geometry on Lie
  Groups   Let G be a Lie group with Lie algebra $ \Cal G: = T_\epsilon G$ and $T^*G =
\Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G
acts on $\Cal G^*$ via the coadjoint action.
  We show that there is a 1-1 correspondance between the skew-symmetric
solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter Equation in G,
and the set of connected Lie subgroups of $T^*G$ which carry a left invariant
affine structure and whose Lie algebras are lagrangian graphs in $ \Cal G
\oplus \Cal G^*$.
  An invertible solution r endows G with a left invariant symplectic structure
and hence a left invariant affine structure. In this case we prove that the
Poisson Lie tensor $\pi := r^+ - r^-$ is polynomial of degree at most 2 and the
double Lie groups of $(G,\pi)$ also carry a canonical left invariant affine
structure.
  In the general case of (non necessarly invertible) solutions r, we supply a
necessary and suffisant condition to the geodesic completness of the associated
affine structure

37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg