14 2. THE SERRE RELATIONS
Relations:
tiejti
−1
=
vαj (hi)ej,
tifj ti
−1
=
v−αj (hi)fj,
[ei, fj ] = δij
ti ti
−1
v v−1
,
[ti, tj] = 0, titi
−1
= ti
−1ti
= 1,
ei
2ej
(v +
v−1)eiej
ei + ejei
2
= 0 (i j = ±1),
eiej = ejei (otherwise),
fi2fj
(v +
v−1)fifj
fi +
fjfi2
= 0 (i j = ±1),
fifj = fjfi (otherwise).
These relations are called the (deformed) Serre relations.
Definition 2.6. Let [k] =
vk−v−k
v−v−1
, for k N, and let [n]! =
n
k=1
[k]. Then
fi(n)
is defined by
fi(n)
=
fin
[n]!
.
Roughly speaking, the quantum algebra is the algebra which is obtained by
“integrating”the Cartan subalgebra and deforming the other relations “nicely”.
We may obtain representations of Uv(slr) by deforming the representations of
g. We can also define tensor product representations by deforming the coproduct
of the enveloping algebra as follows.
Δ(ti) = ti ti, Δ(ei) = 1 ei + ei ti
−1,
Δ(fi) = fi 1 + ti fi.
Exercise 2.7. Verify that Δ defines an algebra homomorphism from Uv(slr)
to Uv(slr) Uv(slr).
Exercise 2.8. Let V = Kr and define ρ : Uv(slr) End(V ) by
ρ(ti) = I + (v 1)Ei,i +
(v−1
1)Ei+1,i+1,
ρ(ei) = Ei,i+1, ρ(fi) = Ei+1,i.
Show that (ρ, V ) is a representation of Uv(slr). This is called the natural (or vector)
representation of Uv(slr).
Exercise 2.9. Let V be the natural representation of Uv(sl2). Decompose
V V into a sum of irreducible Uv(sl2)-submodules.
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