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

4. General Theory 37
Now we use the braid group to construct analogues of the root vectors
associated to non simple roots.
Let us take a reduced expression ω0 = si1 . . .siN for the longest element
in the Weyl group W. Setting βj = si1 · · ·sij−1 (αj), we get a total order on
the set of positive root. We define the elements Eβj = Ti1 . . . Tij−1 (Eij ) and
Fβj = Ti1 . . .Tij−1 (Fij ). Note that this elements depend on the choice of the
reduced expression.
Lemma 4.1.3. (i) Eβj ∈ U+ , ∀ i = 1...N
(ii) Fβj ∈ U − , ∀ i = 1...N
Lemma 4.1.4. (i) The monomials E k1 · · ·EkN β1 βN
(ii) The monomials F k1 kN · · ·F β1 βN
are a C(q) basis of U −
Poincaré Birkhoff Witt Theorem. The monomials
E k1
β1
kN k1
· · ·EkN KαF · · ·F βN βN β1
are a C(q) basis of U+
are a C(q) basis of U. In fact as vector spaces, we have the tensor product
decomposition,
U = U + ⊗ U 0 ⊗ U −
Proof. See [Lus93].
Levendorskii Soibelman relation. For i < j one has
(i)
(ii)
EβjEβi − q(βi|βj)
EβiEβj =
k∈Z N +
ckE k
(4.6)
where ck ∈ C[q, q −1 ] and ck = 0 only when k = (k1, . . .,kN) is such
that ks = 0 for s ≤ i and s ≥ j, and E k = E k1
β1
FβjFβi − q−(βi|βj)
FβiFβj =
k∈Z N +
· · ·EkN
βN .
ckF k
(4.7)
where ck ∈ C[q, q −1 ] and ck = 0 only when k = (k1, · · · , kN) is such
that ks = 0 for s ≤ i and s ≥ j, and F k = F kN
βN
Proof. See [LS91b].
· · ·F k1
β1 .
An immediate corollary is the following: Let ω ∈ W. Choose a reduce
expression for it, ω = si1 . . .sik , which we complete to a reduced expression
ω0 = si1 . . .siN of the longest element of W. Consider the elements Eβj ,
j = 1, . . .,k. Then we have: