Degree of Parabolic Quantum Groups - Dipartimento di Matematica ...

1. Poisson algebraic Groups 5
1.2 Lie bialgebras and Manin triples
1.2.1 Lie bialgebras
Definition 1.2.1. A Lie bialgebra is a pair (g, φ) where g is a finite dimensional
Lie algebra over k, and φ : g → g ∧ g, called cobracket, satisfies the
following conditions:
1. the dual map φ ∗ : g ∗ ∧ g ∗ → g ∗ makes g ∗ into a Lie algebra;
2. the cobracket φ : g → g ∧ g is a 1-cocycle on g, i.e. for any a, b ∈ g,
φ([a, b]) = a.φ(b) − bφ(a)
where a.(b ⊗ c) = [a ⊗ 1 + 1 ⊗ a, b ⊗ c] = [a, b] ⊗ c + b ⊗ [a, c].
Lie bialgebras form a category, with morphisms being Lie algebra homomorphism
which commute with the cobracket.
Property. Given a Lie bialgebra g, the vector space g ∗ carries a canonical
structure of Lie bialgebra, the Lie bracket being the map dual to the cobracket
in g and the cobracket being the map dual to the bracket in g.
Definition 1.2.2. The Lie bialgebra g ∗ is called the dual Lie bialgebra of g.
Example 1.2.3. Any Lie algebra g with trivial cobracket (i.e., zero) is a Lie
bialgebra.
Example 1.2.4. Let g be a Lie algebra. Consider the dual vector space g ∗
as a commutative Lie algebra. Then the map φ : g ∗ → g ∗ ∧ g ∗ dual to the
Lie bracket on g, define a Lie bialgebra structure on g ∗ , it is the dual Lie
bialgebra to the one in example 1.2.3.
Example 1.2.5. Let g be a complex simple Lie algebra with a fixed non
degenerate invariant symmetric bilinear form, which is necessarily a scalar
multiple of the Killig form. Choose a Cartan subalgebra h ⊂ g, with n =
dimh, the rank of g. Choose a set of simple roots α1, . . .,αn ∈ h∗ . This
gives a decomposition
g = n+ ⊕ h ⊕ n−
(1.2)
where n+ (resp. n−) is the nilpotent subalgebra spanned by the positive
(resp. negative) root subspaces.
Let X ± i , Hi, i = 1, . . .,n, be the Chevalley generators corresponding
to the simple root αi, and A = (ai,j) the Cartan matrix whose entries are
ai,j = 2 (αi,αj)
(αi,αi) , where (, ) is the symmetric bilinear form on h∗ induced by
the bilinear form on g.
Recall that g is generated by X ± i , Hi and the Serre relations