Theorem 4.7 in [Reference Shu1] should be corrected as follows. All notations and assumptions are the same as in [Reference Shu1]. Especially, all Lie superalgebras are assumed to be finite-dimensional.
Theorem 0.1 Let
${\mathfrak {g}}$
be a solvable Lie superalgebra. Any irreducible module V of
${\mathfrak {g}}$
is associated with some
$\chi \in {\mathfrak {g}}_{\bar {0}}^*$
, which has dimension 
where
${\mathfrak {h}}$
is a subalgebra with
$\chi ({\mathfrak {h}}^{(1)})=0$
for its derived subalgebra
${\mathfrak {h}}^{(1)}:=[{\mathfrak {h}},{\mathfrak {h}}]$
, and V contains a one-dimensional
${\mathfrak {h}}$
-module, and n depends on
${\mathfrak {h}}$
.
Proof For any given irreducible
${\mathfrak {g}}$
-module
$(V,\rho )$
, V becomes an irreducible
${\mathfrak {g}}_p$
-modules associated with
$\Upsilon \in ({\mathfrak {g}}_{\bar {0}})_p^*$
where
${\mathfrak {g}}_p$
is a minimal finite-dimensional p-envelope of
${\mathfrak {g}}$
with
${\mathfrak {g}}_p=({\mathfrak {g}}_{\bar {0}})_p+{\mathfrak {g}}_{\bar {1}}$
such that
$({\mathfrak {g}}_{\bar {0}})_p$
is a p-envelope of
${\mathfrak {g}}_{\bar {0}}$
(see [Reference Shu1, Lemma A.3]). Moreover, V is associated with
$\chi :=\Upsilon |_{{\mathfrak {g}}_{\bar {0}}}\in {\mathfrak {g}}^*_{\bar {0}}$
. By [Reference Shu1, Corollary 4.6], there exists a restricted subalgebra
$\frak {H}$
of
${\mathfrak {g}}_p$
with
$\Upsilon (\frak {H}^{(1)})=0$
such that
${V\cong U_\chi ({\mathfrak {g}}_p)\otimes _{U_\chi (\frak {H})}S}$
where
$S\subset V$
is a one-dimensional
$\frak {H}$
-module. Correspondingly, 
.
Take
${\mathfrak {h}}=\frak {H}\cap {\mathfrak {g}}$
. By definition,
${\mathfrak {h}}_{\bar {0}}=\frak {H}_{\bar {0}}\cap {\mathfrak {g}}_{\bar {0}}$
, and
${\mathfrak {h}}_{\bar {1}}=\frak {H}_{\bar {1}}\cap {\mathfrak {g}}_{\bar {1}}=\frak {H}_{\bar {1}}$
. Then
${\chi ({\mathfrak {h}}^{(1)})=0}$
, and
${\mathfrak {h}}$
has one-dimensional module S. Set 
. Then we have
The proof is completed.
Remark 0.2 When
${\mathfrak {g}}$
is a restricted solvable Lie superalgebra, n coincides with 
. In this case, the above theorem is actually a strengthened version of [Reference Shu1, Corollary 4.6] on the irreducible modules and their dimensions. So in this special case, the theorem implies [Reference Shu1, Proposition 5.4(1)] which states:
-
• Each irreducible
$U_\chi ({\mathfrak {g}})$
-module V for a restricted solvable Lie superalgebra
${\mathfrak {g}}$
is associated with certain restricted subalgebra
${\mathfrak {h}}$
with
$\chi ({\mathfrak {h}}^{(1)})=0$
, such that V has dimension and there is a one-dimensional
${\mathfrak {h}}$
-submodule in V.
