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

1. Poisson algebraic Groups 3
Definition 1.1.5. 1. Let M and N be smooth Poisson manifolds. A
smooth map f : M → N is called a Poisson map if the induced map
f ∗ : C ∞ (N) → C ∞ (M) is a Poisson homomorphism.
2. Let M and N be algebraic Poisson k-variety. An algebraic map f :
M → N is called a Poisson map if the induced map f ∗ : k[N] → k[M]
is a Poisson homomorphism.
It is clear that smooth, and algebraic, Poisson manifolds form a category,
with morphisms being Poisson map.
Definition 1.1.6. Suppose that M is a smooth Poisson manifold, let A =
C ∞ (M).
1. Given φ ∈ A, the vector field ξφ associated to the Hamiltonian derivation
{φ, } of A is called Hamiltonian vector field.
2. A submanifold (not necessarily closed) N ⊂ M is called Poisson submanifold
if the vector ψφ(n) is tangent to N for any n ∈ N and φ ∈ A.
Let M a smooth Poisson manifold, then there is an equivalent definition
of Poisson manifold in term of bivector fields. Recall that an n-vector field
is a section of the bundle n TM where TM is the tangent vector bundle of
M. In particular, we call 2-vector fields bivector fields.
Recall also the definition of the Schouten bracket of n-vector fields which
generalize the usual Lie bracket on vector fields. The Schouten bracket of
an m-vector field with an n vector field is an (m + n − 1)-vector field which
is locally defined by
[u1 ∧ . . . ∧ um, v1 ∧ . . . ∧ vm]
=
(−1) i+j [ui, vj] ∧ u1 ∧ . . . ∧ ûi ∧ um ∧ v1 ∧ . . . ∧ ˆvj ∧ . . . ∧ vn
i,j
where u1, . . .,um, v1, . . .,vn ∈ TmM, m ∈ M, and [, ] denote the Lie bracket
of vector fields.
Denote by T ∗ M the cotangent bundle to M. Given a bivector field π on
M, we define a bilinear operation { , } on C ∞ (M) by
{φ, ψ} = 〈π, dφ ∧ dψ〉 (1.1)
where 〈, 〉 is the natural paring between the sections of the bundle T ∗ M ∧
T ∗ M and TM ∧ TM.
Proposition 1.1.7. The bracket 1.1 defines a Poisson manifold structure
on M if and only if
[π, π] = 0