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

[Hi, Hj] = 0,
X + i , X− j
1. Poisson algebraic Groups 6
Hi, X ± j
= δi,jHi, ad 1−ai,j
= ±ai,jX ± j ,
± ±
X i X j = 0,
where ad(a)(b) = [a, b] is the adjoint action of g on itself.
Then the following cobracket φ defines a Lie bialgebra structure on g:
φ(Hi) = 0,
φ(X ± i ) = diX ± i
∧ Hi,
where di, i = 1, . . .,n, are positive rational numbers that satisfy diai,j =
djaj,i. For more details see [KS98].
Definition 1.2.6. The Lie bialgebra structure described in example 1.2.5 is
called the standard Lie bialgebra structure on g
Example 1.2.7. Consider the simple complex Lie algebra g = sl2(C), then
the Chevalley generator are
X +
0 1 1 0
= , H = , X
0 0 0 −1
−
0 0
= ,
1 0
then the cobracket is
We verify only that φ is a cocycle.
φ(H) = 0,
φ(X ± ) = X ± ∧ H.
φ([H, X ± ]) = φ(±2X ± ) = ±2X ± ∧ H
H · φ(X ± ) − X ± · φ(H) = H · (X ± ∧ H)
φ([X + , X − ]) = φ(H) = 0
= [H, X ± ] ∧ H = ±2X ± ∧ H
X + · φ(X − ) − X − φ(X + ) = X + · X − ∧ H − X − · X + ∧ H
= [X + , X − ] ∧ H + X − ∧ [X + , H]
−[X − , X + ] ∧ H − X + ∧ [X − , H]
= H ∧ H + 2X − ∧ X +
= 0
+H ∧ H + 2X + ∧ 2X −
The fact that φ ∗ defines a Lie bracket is an easy exercise.