8 NOTATIONS AND CONVENTIONS
More generally, we use the notation
(ΔU Id) ΔU (X) =
(X) (X(1))
(X(1))(1) (X(1))(2) X(2) (3)
=:
(X)
X(11) X(12) X(2) ,
and obvious generalization of it.
To a fixed ordered basis {X1,...,Xm} of the Lie algebra g, we associate a PBW
basis of U(g):
(4)
{Xβ,
β
Nm}
,

:= X1
β1
X2
β2
. . . Xmm
β
.
This induces a filtration
U(g) =
k∈N
Uk(g) , Uk(g) Ul(g) , k l ,
in terms of the subsets
(5) Uk(g) :=
|β|≤k


, R , k N ,
where |β| := β1 + · · · + βm. For β, β1,β2
Nm,
we define the ‘structure constants’
ωβ1,β2 β
R of U(g), by
(6)
Xβ1 Xβ2
=
|β|≤|β1|+|β2|
ωβ1,β2 β
U|β1|+|β2|(g) .
We endow the finite dimensional vector space Uk(g), with the
1-norm
|.|k within
the basis
{Xβ,
|β| k}:
(7) |X|k :=
|β|≤k
|Cβ| if X =
|β|≤k


Uk(g) .
We observe that the family of norms {|.|k}k∈N is compatible with the filtered struc-
ture of U(g), in the sense that if X Uk(g), then |X|k = |X|l whenever l k.
Considering a subspace V g, we also denote by U(V ) the unital subalgebra of
U(g) generated by V :
U(V ) = span X1X2 . . . Xn : Xj V , n N , (8)
that we may filtrate using the induced filtration of U(g). We also observe that)
the co-product preserves the latter subalgebras, in the sense that ΔU
(
U(V )
U(V ) U(V ).
Regarding the uniform structures on a locally compact group G, we say that a
function f : G C is right (respectively left) uniformly continuous if for all ε 0,
there exists U, an open neighborhood of the neutral element e, such that for all
(g, h) G × G we have
|f(g) f(h)| ε , whenever
g−1h
U
(
respectively
hg−1
U
)
.
We call a Lie group G (with Lie algebra g) exponential, if the exponential map
exp : g G is a global diffeomorphism.
Let f1,f2 be two real valued functions on G. We say that f1 and f2 have the
same behavior, that we write
(9) f1 f2,
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