8 DANIEL K. NAKANO
In this section we will prove that the projective covers and irreducibles in the graded category are
also sent under the forgetful functor to the projective covers and irreducibles of /7(g). Consequently,
each projective and irreducible module for U(g) is endowed with the structure of a graded module.
Shen [Sh-2] has proved that a finite-dimensional irreducible module V over U(Q) is isomorphic to a
graded module if and only if
JV+
and N~ act nilpotently on V. Using different techniques we will
recover this result and extend the result to the projective modules.
The theorems and proofs in this section will hold for both categories C and Cgr. We will use V
to mean either C and Cgr. The first one of these statements is a general "Frobenius reciprocity"
law for graded restricted Lie algebras.
Theore m 1.2.1. Let [) be a, graded restricted subalgebra, of$\ with M, M' G ob Vg and N G ob V\).
Then for each n = 0,1,2,.... we have
(1) Extnu(i){M,M') = Extl(i){M'*,M')
(2) Ext^B)(N ®u{h) U(S),M) = Exrm)(N,M \m))
(3) Extl(t)(M, (JV ®m)
U(S)m))
= Ext^M \m), N).
Proof. The standard proofs [Cu-R-1] work for (1) and (2) if we look at graded morphisms. For (3),
use (1) and (2)
Ext^9)(M,(N* ®u(t)) 17(g)*) 3 Ext%lt)(N* ®
U ( H
I%),M* )
*Extlw{N*,M\*m))
*Extnm){M\m),N).Q
L e m m a 1.2.2. Let M 6 ob
VB±.
Then
M ®U(B±) Ufa) \u(B*)= M ®U(G0) U{B*).
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