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Generalized restricted representations of the Zassenhaus superalgebras
Accepted Manuscript Generalized restricted representations of the Zassenhaus superalgebras
Yu-Feng Yao, Bin Shu, Yi-Yang Li
PII: DOI: Reference:
S0021-8693(16)30237-X http://dx.doi.org/10.1016/j.jalgebra.2016.08.007 YJABR 15849
To appear in:
Journal of Algebra
Received date:
10 January 2016
Please cite this article in press as: Y.-F. Yao et al., Generalized restricted representations of the Zassenhaus superalgebras, J. Algebra (2016), http://dx.doi.org/10.1016/j.jalgebra.2016.08.007
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GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS YU-FENG YAO, BIN SHU AND YI-YANG LI Abstract. Let F be an algebraically closed field of prime characteristic p > 2, and n ∈ N+ . Let Z(n) be the Zassenhaus superalgebra defined over F, which, as the simplest nonrestricted simple Lie superalgebra, is the superversion of the Zassenhaus algebra. More precisely, Z(n) is the Lie superalgebra of the special super derivations of the superalgebra Π(n). Here Π(n) is the tensor product of the divided power algebra of one variable and the Grassmann superalgebra of one variable. In this paper we study generalized restricted simple modules over the Zassenhaus superalgebra Z(n). Classification of isomorphism classes of generalized restricted simple modules and their dimensions are precisely determined. A sufficient and necessary condition for irreducibility of generalized restricted Kac modules is provided.
1. Introduction The finite-dimensional simple Lie superalgebras over the field of complex numbers were classified by Kac in the 1970s (cf. [2]). Over an algebraically closed field of prime characteristic, the classification of finite-dimensional simple Lie superalgebras remains unknown up to the present time. However, a natural class is known, namely the modular version of the complex simple Lie superalgebras. These Lie superalgebras are restricted ones. In [18], more simple Lie superalgebras of Cartan type in prime characteristic were constructed. They are not restricted Lie superalgebras in general. Representation theory of Lie superalgebras in characteristic 0 has been systematically developed over the past decades. However, there seems to lack rich theory on the modular representations of Lie superalgebras over an algebraically closed field of prime characteristic. In recent years, there has been an increasing interest in modular representation theory of 2010 Mathematics Subject Classification. 17B10, 17B50, 17B66, 17B70. Key words and phrases. the Zassenhaus superalgebra, irreducible module, (generalized) restricted module, (generalized) restricted Kac module, (atypical) typical weight. This work is supported by Natural Science Foundation of Shanghai (Grant No. 16ZR1415000) and National Natural Science Foundation of China (Grant Nos. 11271130, 11571008 and 11526139). 1
2
YU-FENG YAO, BIN SHU AND YI-YANG LI
restricted Lie superalgebras (cf. [11, 17, 7, 8, 1, 6, 12, 13, 14, 10, 15]). In this paper, we initiate the study of representations of non-restricted Lie superalgebras in prime characteristic. We concern the representation theory of the Zassenhaus superalgebra, which is the simplest non-restricted simple Lie superalgebra. We classify the isomorphism classes of generalized restricted simple modules over the Zassenhaus superalgebra. Moreover, we precisely determine the dimensions of the generalized restricted simple modules. We also give a sufficient and necessary condition for generalized restricted Kac modules to be irreducible. 2. Preliminaries In this paper, we always assume that the base field F is algebraically closed and of characteristic p > 2. All vector spaces (modules) V are over F, and Z2 -graded, i.e., V = V¯0 ⊕V¯1 . For each homogeneous element v ∈ V , we denote by v¯ the parity of v. Let A = A¯0 ⊕ A¯1 be a superalgebra. A homogeneous super derivation D of A is a homogeneous linear transformation of A such that D(ab) = D(a)b + (−1)D a¯ aD(b), ∀ a, b ∈ A¯0 ∪ A¯1 . n
2.1. The Zassenhaus superalgebra. Let n ∈ N+ . Set A(n) = F[x]/(xp ), the divided power algebra of one variable x, and (1) the Grassmann superalgebra of one variable ξ. For convenience, we also denote by xi the image of xi under the quotient map from F[x] to A(n) for 0 ≤ i ≤ pn − 1. Set Π(n) = A(n) ⊗F (1). Then Π(n) has an F-basis {xi ξ j | 0 ≤ i ≤ pn − 1, j = 0, 1}, and is Z2 -graded with Π(n)¯0 = spanF {xi | 0 ≤ i ≤ pn − 1} and Π(n)¯1 = spanF {xi ξ | 0 ≤ i ≤ pn − 1}. Moreover, Π(n) is an associative superalgebra in which the multiplication is given by i + s i+s j+t i j s t x ξ , xξ ·x ξ = i
0 ≤ i, s ≤ pn − 1, j = 0, 1, t = 0, 1
with the convention that xk ξ l = 0,
for k ≥ pn or l > 1.
A super derivation D of Π(n) is called special if for 0 ≤ i ≤ pn − 1 and j = 0, 1, D(xi ξ j ) = xi−1 D(x)ξ j + jxi D(ξ). Define two linear maps D and ∂ on Π(n) by D(xi ξ j ) = xi−1 ξ j , ∂(xi ξ j ) = jxi , for 0 ≤ i ≤ pn − 1, j = 0, 1.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
3
Then both D and ∂ are special derivations of Π(n). Let Z(n) be the set of all special super derivations of Π(n). A straightforward calculation implies that Z(n) = spanF {xi ξ j D, xi ξ j ∂ | 0 ≤ i ≤ pn − 1, j = 0, 1} = Z(n)¯0 ⊕ Z(n)¯1 , where Z(n)¯0 = spanF {xi D, xi ξ∂ | 0 ≤ i ≤ pn − 1} and Z(n)¯1 = spanF {xi ∂, xi ξD | 0 ≤ i ≤ pn − 1}. It is a routine to check that Z(n) is a Lie subsuperalgebra of the derivation superalgebra of Π(n) under the usual Lie bracket (cf. [18]). Moreover, it is a simple Lie superalgebra and referred as the Zassenhaus superalgebra, the even part of which contains a subalgebra isomorphic to the usual Zassenhaus algebra (cf. [9, §7.6]). Furthermore, the Zassenhaus superalgebra Z(n) is a restricted Lie superalgebra if and only if n = 1. Although in general it is not restricted, it is a generalized restricted Lie superalgebra in the following sense (see Definition 2.1). 2.2. Generalized restricted Lie superalgebras and their generalized χ-reduced representations. The following notion of generalized restricted Lie superalgebra generalizes the notion of a restricted Lie superalgebra (cf. [3]). Definition 2.1. A Lie superalgebra g = g¯0 ⊕ g¯1 is said to be a generalized restricted one if the even part g¯0 is a generalized restricted Lie algebra and the odd part g¯1 as an adjoint g¯0 -module is a generalized restricted module (cf. [4, 5]). More precisely speaking , there exist an ordered basis E = {ei | i ∈ I} of g¯0 , an |I|-tuple s = (si )i∈I of positive integers and a
generalized restricted map ϕs : E → g¯0 sending ei to eϕi s such that (ad(ei ))p i (y) = ad(eϕi s )(y) s
for any i ∈ I, y ∈ g. Remarks 2.2.
(i) One usually denotes the generalized restricted Lie superalgebra men-
tioned in Definition 2.1 by (g, E, ϕs ) to indicate the ordered basis E and the generalized restricted map ϕs . (ii) Any restricted Lie superalgebra (L, [p]) is a generalized restricted Lie superalgebra where ϕs = [p]|E and s = (si )i∈I with si = 1, ∀ i ∈ I. Different from the case of characteristic zero, any irreducible representation of a generalized restricted Lie superalgebra (g, E, ϕs ) is attached with a unique χ ∈ g∗¯0 . Let Jχ be the ideal si
of the universal enveloping superalgebra U (g) generated by epi − eϕi s − χ(ei )p i , i ∈ I. Set s
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YU-FENG YAO, BIN SHU AND YI-YANG LI
Uχ (g, E, ϕs ) = U (g)/Jχ which is called the generalized χ-reduced enveloping superalgebra of g. One usually denotes the generalized χ-reduced enveloping superalgebra Uχ (g, E, ϕs ) si
by Uχ (g) for brevity. A g-module M is said to be a generalized χ-reduced one if epi ·v−eϕi s ·v = s
χ(ei )p i v, ∀ v ∈ M, i ∈ I. Then a generalized χ-reduced g-module is a Uχ (g)-module. All generalized χ-reduced g-modules constitute a full subcategory of the category of g-modules, denoted by Uχ (g)-mod. In the case χ = 0, a generalized χ-reduced module is said to be a generalized restricted module, and we call U0 (g) the generalized restricted enveloping superalgebra of g, denoted by u(g) for brevity. It should be remarked that the notion of generalized χ-reduced enveloping superalgebras (resp. generalized χ-reduced representations, generalized restricted representations etc.) for a restricted Lie superalgebra coincides with that of the ordinary χ-reduced enveloping superalgebras (resp. χ-reduced representations, restricted representations etc.). 2.3. Basic structure of the Zassenhaus superalgebra. In this paper we always assume g = Z(n) is the Zassenhaus superalgebra which is a generalized restricted Lie superalgebra with an ordered basis {xi D, xi ξ∂ | 0 ≤ i ≤ pn − 1} of g¯0 , and the generalized restricted map ϕs defined as t
ϕs (xp D) = −xp
t+1 −p+1
D for 1 ≤ t ≤ n − 1,
ϕs (xD) = xD, ϕs (ξ∂) = ξ∂, and ϕs (xj D) = ϕs (xk ξ∂) = 0 for other j and 1 ≤ k ≤ pn − 1, where s = (n, 1, 1, · · · , 1). There is a natural Z-grading structure on g, i.e., g = where
n −1 p
i=−1
g[i] ,
g[i] = spanF {xk ξ l D, xk ξ l ∂ | k + l = i + 1, 0 ≤ k ≤ pn − 1, l = 0, 1} for − 1 ≤ i ≤ pn − 1. Associated with this grading, one has the following filtration: (2.2.1)
g = g−1 ⊃ g0 ⊃ · · · ⊃ gpn −1 ,
where gi =
j≥i
g[j] for − 1 ≤ i ≤ pn − 1.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
5
The zero graded component g[0] ∼ = gl(1|1) under the map φ : g[0] −→ gl(1|1), xD → E11 , ξ∂ → E22 , x∂ → E12 , ξD → E21 . Furthermore, we have the triangular decomposition g[0] = n− ⊕ h ⊕ n+ , where n− = FξD, n+ = Fx∂, and h = FxD + Fξ∂. Set b = h ⊕ n+ , B = b ⊕ g1 = b + i≥1 g[i] , and g+ = g[0] ⊕ g1 = g0 . It is easy to check that the subalgebras b, B and g+ are (generalized) restricted subalgebras of g.
3. Generalized restricted representations of the Zassenhaus superalgebra Keep the same notations and convention as in §2. In particular, g = Z(n) is the Zassenhaus superalgebra over an algebraically closed field F of prime characteristic p > 2. Recall that g = g[−1] ⊕ g+ with g+ = g[0] ⊕ g1 and g[0] ∼ = gl(1|1). 3.1. Restricted irreducible representations of g[0] . Recall that g[0] = n− ⊕ h ⊕ n+ with n− = FξD, h = FxD + Fξ∂, and n+ = Fx∂. The isomorphism classes of restricted simple modules of g[0] are parameterized by Λ := {λ = (λ1 , λ2 ) | λi ∈ Fp , i = 1, 2}, the set of restricted weights, which coincides with F2p (cf. [11, 16]). More precisely, for each λ = (λ1 , λ2 ) ∈ Λ, one can define the one-dimensional h-module Fvλ with xD · vλ = λ1 vλ and ξ∂·vλ = λ2 vλ . This one-dimensional h-module Fvλ can be regarded as a b-module with trivial n+ -action. One then get the so-called baby Verma module V (λ) as the induced g[0] -module
u(g[0] ) u(b) Fvλ , where u(g[0] ) and u(b) are the restricted enveloping superalgebras of g[0] and b, respectively. As a vector space, V (λ) = spanF {1 ⊗ vλ , ξD ⊗ vλ }. Each baby Verma module V (λ) has a unique simple quotient, denoted by L0 (λ). We have the following easy observation which describes simple restricted g[0] -modules. The proof is straightforward. Lemma 3.1. Let λ = (λ1 , λ2 ) ∈ Λ. Then the following statements hold. (i) If λ1 = −λ2 , then L0 (λ) = V (λ) = spanF {1 ⊗ vλ , ξD ⊗ vλ } is two-dimensional. (ii) If λ1 = −λ2 , then W = FξD ⊗ vλ is the unique maximal submodule of V (λ) and L0 (λ) = V (λ)/W . In this case, we also denote by 1 ⊗ vλ the image of 1 ⊗ vλ in the quotient V (λ)/W for brevity. Then L0 (λ) = spanF {1 ⊗ vλ } is one-dimensional. 3.2. Generalized restricted irreducible representations of g. In this subsection, we exploit the structure of generalized restricted simple g-modules based on the known information on restricted simple g[0] -modules presented in § 3.1. Recall that g+ = g[0] + g1 , and g1 = spanF {xi ξ j D, xi ξ j ∂ | i + j ≥ 2, 1 ≤ i ≤ pn − 1, j = 0, 1}.
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YU-FENG YAO, BIN SHU AND YI-YANG LI
We first need the following result which implies that simple restricted modules for g[0] and g+ coincide. Lemma 3.2. Each simple restricted g+ -module is simple as a g[0] -module with trivial g1 action. Conversely, each simple restricted g[0] -module can be extended to a simple restricted g+ -module with trivial g1 -action. Proof. Since g1 is an ideal of g+ , the second statement is obvious. We only need to show the first statement. Let M be a simple restricted g+ -module. Note that g1 ∩g¯0 has an ordered basis {X1 , · · · , Xs } [p]2
such that Xi
= 0, ∀ 1 ≤ i ≤ s. Since M is a restricted g+ -module, 2
[p]2
Xip · m = Xi
· m = 0, ∀ 1 ≤ i ≤ s, m ∈ M.
On the other hand, 1 Y 2 = [Y, Y ] ∈ g1 ∩ g¯0 , ∀ Y ∈ g1 ∩ g¯1 . 2 This implies that each element of a basis of g1 acts on M nilpotently. It follows from the superversion of Jacobson’s weakly nilpotent theorem (see [18]) that M g1 := {m ∈ M | z · m = 0, ∀ z ∈ g1 } = 0. It is easy to check that M g1 is a g+ -submodule, since g1 is an ideal of g+ . The simplicity of M as a g+ -module implies that M g1 = M . Consequently, g1 acts on M trivially.
By Lemma 3.2, each simple restricted g[0] -module L0 (λ) can be regarded as a simple restricted g+ -module, where λ = (λ1 , λ2 ) ∈ Λ. Then one can define the generalized restricted Kac g-module K(λ) as the induced g-module u(g) ⊗u(g+ ) L0 (λ). We have the following description for generalized restricted Kac g-modules and simple generalized restricted gmodules. Proposition 3.3. The following statements hold. (i) Each generalized restricted Kac g-module K(λ) has a unique simple submodule, i.e., K(λ) has a simple socle. (ii) Each generalized restricted Kac g-module K(λ) has a unique maximal submodule J(λ). Hence, K(λ) has a unique simple quotient, denoted by L(λ). (iii) Each simple generalized restricted g-module is of the form L(λ) for some λ ∈ Λ. (iv) Let λ, μ ∈ Λ, then K(λ) ∼ = K(μ) if and only if λ = μ.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
7
(v) Let λ, μ ∈ Λ, then L(λ) ∼ = L(μ) if and only if λ = μ. Proof. (i) Let S be an arbitrary simple g-submodule of K(λ). Take a nonzero element pn −1
v=
1
Di ∂ j ⊗ vij ∈ S,
i=0 j=0
where vij ∈ L0 (λ), ∀ 0 ≤ i ≤ pn − 1, j = 0, 1. Let i0 = min{i | vij = 0 for some j} and j0 = min{j | vi0 j = 0}. Then Dp ξD · Dp
n −1
∂ ⊗ vi0 j0 = −Dp
n −1
n −1−i
0
∂ 1−j0 v = Dp
n −1
∂ ⊗ vi0 j0 ∈ S. Note that
∂ ⊗ ξD · vi0 j0 , x∂ · Dp
n −1
∂ · ⊗vi0 j0 = −Dp
n −1
∂ ⊗ x∂ · vi0 j0
and h · Dp
n −1
∂ ⊗ vi0 j0 ∈ FDp
n −1
∂ ⊗ vi0 j0 + FDp
n −1
∂ ⊗ h · vi0 j0 , ∀ h ∈ h.
Since L0 (λ) is a simple g[0] -module, L0 (λ) = u(g[0] )vi0 j0 . Hence, u(g[0] )Dp
n −1
∂ ⊗ vi0 j 0 = D p
Consequently, S = u(g)Dp
n −1
n −1
∂ ⊗ L0 (λ) ∼ = L0 (λ + (1, −1)) as u(g[0] )-modules.
∂ ⊗ L0 (λ).
(ii) Let N − = FξD + g[−1] which is a generalized restricted subalgebra of g. Note that n
n
Dp = 0 and (ξD)2 = ∂ 2 = 0 in u(N − ). Then for each f ∈ u(N − )N − , we have f 2p = 0 in u(N − ). We claim that each nonzero proper submodule U of K(λ) is included in u(N − )N − ⊗ vλ . Indeed, suppose U u(N − )N − ⊗ vλ , then there exists a nonzero element ν = (1 + f ) ⊗ vλ ∈ U for some f ∈ u(N − )N − . Consequently, (1−f +f 2 −f 3 +· · ·−f 2p
n −1
)ν = 1⊗vλ ∈ U .
Hence, U = K(λ), a contradiction. Now let J(λ) be the sum of all proper submodules of K(λ). Then J(λ) ⊂ u(N − )N − ⊗ vλ K(λ). Therefore, J(λ) is the unique maximal submodule, and K(λ) has a unique simple quotient L(λ) = K(λ)/J(λ). (iii) Let M be a simple generalized restricted g-module. Then M contains a simple restricted g[0] -submodule L0 (λ) for some λ ∈ Λ. We then have the following natural g-module homomorphism ϕ : K(λ) = u(g) ⊗u(g+ ) L0 (λ) −→ M u ⊗ v −→ u · v. The simplicity of M as a g-module implies that ϕ is surjective. Hence M is isomorphic to a quotient of K(λ). Now it follows from the statement (ii) that M ∼ = L(λ). (iv) It suffices to show that λ = μ if K(λ) ∼ = K(μ). For that, let
K(λ)D,∂ := {w ∈ K(λ) | D · w = ∂ · w = 0} and K(μ)D,∂ := {w ∈ K(μ) | D · w = ∂ · w = 0}.
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YU-FENG YAO, BIN SHU AND YI-YANG LI
Then K(λ)D,∂ and K(μ)D,∂ are g[0] -submodules of K(λ) and K(μ), respectively. Moreover, it is easy to check that K(λ)D,∂ = Dp
n −1
∂ ⊗ L0 (λ) ∼ = L0 (λ + (1, −1))
and K(μ)D,∂ = Dp
n −1
∂ ⊗ L0 (μ) ∼ = L0 (μ + (1, −1)).
Since K(λ) ∼ = K(μ) as g-modules, it follows that K(λ)D,∂ ∼ = K(μ)D,∂ as g[0] -modules, i.e., we get an isomorphism of g[0] -modules: L0 (λ + (1, −1)) ∼ = L0 (μ + (1, −1)). It follows from the discussion in §3.1 that λ + (1, −1) = μ + (1, −1). Consequently, λ = μ, as desired. (v) It suffices to show that λ = μ if L(λ) = L(μ). Indeed, suppose ψ : L(λ) = K(λ)/J(λ) −→ L(μ) = K(μ)/J(μ) is an isomorphism between L(λ) and L(μ). Assume ψ(v) = 1 ⊗ vμ for some v ∈ K(λ) \ J(λ). Then y · v ∈ J(λ), ∀ y ∈ n+ + g1 , and h · v − μ(h)v ∈ J(λ), ∀ h ∈ h.
(3.3.1) We can write v = a ⊗ vλ +
ui ⊗ fi vλ where a ∈ F \ {0} and
ui ⊗ fi ∈ u(g[−1] )g[−1] ⊗ u(n− ) + u(g[−1] ) ⊗ u(n− )n− , ∀ i. Hence, (3.3.2)
h · v = aλ(h) ⊗ vλ +
hui ⊗ fi vλ , ∀ h ∈ h.
It follows from (3.3.1) and (3.3.2) that a(λ(h) − μ(h)) ⊗ vλ ∈ u(g[−1] )g[−1] ⊗ vλ + u(g[−1] ) ⊗ u(n− )n− vλ , ∀ h ∈ h. This implies that λ(h) = μ(h), ∀ h ∈ h, i.e., λ = μ, as desired.
Definition 3.4. Let M be a g-module and λ ∈ h∗ . A nonzero vector v ∈ M is said to be a singular vector of weight λ if h · v = λ(h)v, ∀ h ∈ h and y · v = 0, ∀ y ∈ n+ + g1 .
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
9
In the following, we will precisely determine the structure of the simple generalized restricted g-modules L(λ) for λ = (λ1 , λ2 ) ∈ Λ. Type I: λ1 = −λ2 . Let λ = (λ1 , −λ1 ) ∈ Λ. Then the simple restricted g[0] -module L0 (λ) is one-dimensional with a basis vλ such that x∂ · vλ = ξD · vλ = 0, and xD · vλ = λ1 vλ , ξ∂ · vλ = −λ1 vλ .
(3.4.1)
The generalized restricted Kac g-module K(λ) has a basis {Di ∂ j ⊗ vλ | 0 ≤ i ≤ pn − 1, j = 0, 1}. Let M be a nonzero submodule of K(λ). It follows from Proposition 3.3 (i) that M contains the vector Dp (3.4.2)
x
pn −1
D·D
n −1
∂ ⊗ vλ . Hence,
pn −1
pn −1
∂ ⊗ vλ =
(−1)
i
i=0
pn − 1 n n Dp −1−i xp −1−i D∂ ⊗ vλ i
= D∂ ⊗ vλ + DxD∂ ⊗ vλ = D∂ ⊗ vλ + D∂ ⊗ xD · vλ = (1 + λ1 )D∂ ⊗ vλ ∈ M. Case (i): λ1 = −1, 0. In this case, it follows from (3.4.2) that D∂ ⊗ vλ ∈ M . Moreover, we have x2 D · D∂ ⊗ vλ = [x2 D, D]∂ ⊗ vλ = −xD∂ ⊗ vλ = −∂ ⊗ xD · vλ = −λ1 ∂ ⊗ vλ ∈ M. Since λ1 = 0, we have ∂ ⊗ vλ ∈ M . Furthermore, xξD · ∂ ⊗ vλ = [xξD, ∂] ⊗ vλ = xD ⊗ vλ = 1 ⊗ xD · vλ = λ1 ⊗ vλ ∈ M. Hence 1 ⊗ vλ ∈ M due to the assumption that λ1 = 0. Consequently, M = K(λ). We get the following result. Proposition 3.5. Let λ = (λ1 , −λ1 ) ∈ Λ with λ1 = −1, 0. Then the generalized restricted Kac g-module K(λ) is irreducible. Case (ii): λ1 = λ2 = 0. Let λ = (0, 0). It follows from (3.4.2) that D∂ ⊗ vλ ∈ M . Since ξD · D∂ ⊗ vλ = DξD∂ ⊗ vλ = D2 ⊗ vλ ∈ M,
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YU-FENG YAO, BIN SHU AND YI-YANG LI
it follows that x2 D · D2 ⊗ vλ = D ⊗ vλ ∈ M . Furthermore, (−x∂) · D ⊗ vλ = ∂ ⊗ vλ ∈ M . Consequently, I := spanF {Di ∂ j ⊗ vλ | 0 ≤ i ≤ pn − 1, j = 0, 1, and i + j ≥ 1} ⊆ M. It is easy to check that I is a submodule. Hence, M = I or K(λ). In conclusion, I is the unique maximal submodule of K(λ) and L(λ) = K(λ)/I which is a one-dimensional trivial g-module. We get the following result. Proposition 3.6. Let λ = (0, 0). Then the irreducible g-module L(λ) is a one-dimensional trivial g-module. Case (iii): λ1 = −1, λ2 = 1. Let λ = (−1, 1). Note that Dp
n −1
∂ ⊗ vλ ∈ M . Set M1 := spanF {Dp
n −1
∂ ⊗ vλ }. We claim
that M1 is a g-submodule. Indeed, D · (Dp x∂ · (Dp xD · (D and for i ≥ 1, we have i
xD·D
pn −1
∂ ⊗ vλ
n −1
n −1
∂ ⊗ vλ ) = ∂ · (Dp
n −1
n −1
∂ ⊗ vλ ) = 0,
pn −1
∂ ⊗ vλ ) = 0,
∂ ⊗ vλ ) = ξD · (Dp
pn −1
∂ ⊗ vλ ) = ξ∂ · (D
∂ ⊗ vλ ) = 0,
pn − 1 n Dp −1−j xi−j D∂ ⊗ vλ = (−1) j j=0 n n n i p −1 pn −i i−1 p − 1 = (−1) ∂ ⊗ vλ + (−1) D Dp −i xD∂ ⊗ vλ i i−1 i
j
= Dp
n −i
∂ ⊗ vλ + D p
n −i
∂ ⊗ xD · vλ
= Dp
n −i
∂ ⊗ vλ − D p
n −i
∂ ⊗ vλ
= 0, i
x∂·D
pn −1
∂ ⊗ vλ = = = =
pn − 1 n Dp −1−j xi−j ∂ ∂ ⊗ vλ (−1) j j=0 n n i−1 p − 1 (−1) Dp −i x∂ ∂ ⊗ vλ i−1 n n i−1 p − 1 −(−1) Dp −i ∂ ⊗ x∂ · vλ i−1 0, i
j
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS i
x ξD · D
pn −1
11
pn − 1 n = (−1) Dp −1−j xi−j ξD∂ ⊗ vλ j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) ξD∂ ⊗ vλ + (−1) D Dp −i xξD∂ ⊗ vλ i i−1 n n p −1 p −1 n n = (−1)i Dp −i ⊗ vλ + (−1)i−1 Dp −i xD ⊗ vλ i i−1 i
∂ ⊗ vλ
= Dp
j
n −i
⊗ vλ − D p
n −i
⊗ vλ
= 0, and i
x ξ∂ · D
pn −1
∂ ⊗ vλ
pn − 1 n Dp −1−j xi−j ξ∂ ∂ ⊗ vλ = (−1) j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) ξ∂ ∂ ⊗ vλ + (−1) D Dp −i xξ∂ ∂ ⊗ vλ i i−1 n n n i p −1 pn −1−i i p −1 D Dp −1−i ∂ ⊗ ξ∂ · vλ = −(−1) ∂ ⊗ vλ + (−1) i i n p −1 n −(−1)i−1 Dp −i x∂ ⊗ vλ i−1 i
= −Dp
j
n −1−i
∂ ⊗ vλ + D p
n −1−i
∂ ⊗ vλ − D p
n −i
∂ ⊗ x∂ · vλ
= 0. Hence, M1 is a g-submodule. Furthermore, we claim that M = M1 or K(λ). In fact, suppose M1 M .Then there exists v=
aij Di ∂ j ⊗ vλ ∈ M \ M1 .
Let i0 = min{i | aij = 0 for some j}. If i0 = pn − 1, then v = aDp
n −1
⊗ vλ + bDp
with a = 0. We have pn −1
x
pn − 1 n n ∂·v = (−1) a Dp −1−i xp −1−i ∂ ⊗ vλ i i=0 n pn −1 p −1 n n i + (−1) b Dp −1−i xp −1−i ∂ ∂ ⊗ vλ i i=0 pn −1
i
= a∂ ⊗ vλ ∈ M.
n −1
∂ ⊗ vλ
12
YU-FENG YAO, BIN SHU AND YI-YANG LI
Hence ∂ ⊗ vλ ∈ M . Moreover, xξD · ∂ ⊗ vλ = xD ⊗ vλ = 1 ⊗ xD · vλ = −1 ⊗ vλ ∈ M, so that M = K(λ). If i0 < pn − 1. Let
⎧ ⎨1, if a = 0, i0 0 j0 = ⎩0, if a = 0. i0 0
A direct computation implies that Dp
n −2−i
0
∂ j0 · v = ai0 ,1−j0 Dp
where ai0 ,1−j0 = 0. Hence Dp
n −2
∂ ⊗ vλ + ai0 +1,0 Dp
n −1
∂ ⊗ vλ ∈ M
n −2
∂ ⊗ vλ ∈ M . We then have n pn −2 n n pn −1 pn −2 i p −2 D·D ∂ ⊗ vλ = (−1) Dp −2−i xp −1−i D∂ ⊗ vλ x i i=0 = −xD∂ ⊗ vλ = −∂ ⊗ xD · vλ = ∂ ⊗ vλ ∈ M.
Moreover, xξD · ∂ ⊗ vλ = xD ⊗ vλ = 1 ⊗ xD · vλ = −1 ⊗ vλ ∈ M. Therefore, M = K(λ). In conclusion, M1 is the unique maximal submodule of K(λ). Consequently, L(λ) = K(λ)/M1 . We get the following result. Proposition 3.7. Let λ = (−1, 1). Then the irreducible generalized restricted g-module L(λ) = K(λ)/M1 , where M1 = spanF {Dp
n −1
∂ ⊗ vλ }. In particular, L(λ) has dimension
2pn − 1. Type II: λ1 = −λ2 . Let λ = (λ1 , λ2 ) ∈ Λ with λ1 = −λ2 . Then by Lemma 3.1, the simple restricted g[0] module L0 (λ) coincides with the baby Verma module V (λ) which has a basis {v1 , v2 } with the action of g[0] defined as follows: (3.7.1)
xD · v1 = λ1 v1 , ξ∂ · v1 = λ2 v1 , xD · v2 = (λ1 − 1)v2 , ξ∂ · v2 = (λ2 + 1)v2 ,
and (3.7.2)
x∂ · v1 = 0, ξD · v1 = v2 , x∂ · v2 = (λ1 + λ2 )v1 , ξD · v2 = 0.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
13
The generalized restricted Kac g-module K(λ) has a basis {Di ∂ j ⊗ v1 , Di ∂ j ⊗ v2 | 0 ≤ i ≤ pn − 1, j = 0, 1}. Let M be a nonzero submodule of K(λ). It follows from Proposition 3.3 (i) that M contains Dp
n −1
∂ ⊗ L0 (λ).
Case (i): λ1 = −1, 0. In this case, since Dp
n −1
∂ ⊗ v1 ∈ M , we can replace Dp
n −1
∂ ⊗ vλ by Dp
n −1
∂ ⊗ v1 in (3.4.2).
A similar argument as in the situation of Type I yields that 1 ⊗ v1 ∈ M . Consequently, M = K(λ). We get the following result. Proposition 3.8. Let λ = (λ1 , λ2 ) ∈ Λ with λ1 = −λ2 and λ1 = −1, 0. Then the generalzied restricted Kac g-module K(λ) is irreducible. Case (ii): λ1 = 0 and λ2 = ±1. In this case, λ2 = 0. Set W = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}. We have the following result on the simple generalized restricted g-module L (0, λ2 ) . Proposition 3.9. Let λ = (0, λ2 ) ∈ Λ with λ2 = 0, ±1. Then W is the unique maximal submodule of K(λ), and L(λ) = K(λ)/W . In particular, dim L(λ) = 2pn . Proof. Let M be a nonzero submodule of K(λ). It follows from Proposition 3.3 (i) that Dp D
n −1
pn −1
∂ ⊗ v1 ∈ M . A similar computation as in (3.4.2) with Dp
n −1
∂ ⊗ vλ replaced by
k
∂ ⊗ v1 yields that D∂ ⊗ v1 ∈ M . Hence D ∂ ⊗ v1 ∈ M for 1 ≤ k ≤ pn − 1. Moreover,
since xξ∂ · D∂ ⊗ v1 = [xξ∂, D]∂ ⊗ v1 + D[xξ∂, ∂] ⊗ v1 = −ξ∂ ∂ ⊗ v1 − D ⊗ x∂ · v1 = −∂ ⊗ ξ∂ · v1 − [ξ∂, ∂] ⊗ v1 = −λ2 ∂ ⊗ v1 + ∂ ⊗ v1 = (1 − λ2 )∂ ⊗ v1 ∈ M and λ2 = 1, it follows that ∂ ⊗ v1 ∈ M . Furthermore, ξD · ∂ ⊗ v1 = [ξD, ∂] ⊗ v1 − ∂ ⊗ ξD · v1 = D ⊗ v1 − ∂ ⊗ v2 ∈ M. Hence, Dl · (D ⊗ v1 − ∂ ⊗ v2 ) = Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ∈ M for 0 ≤ l ≤ pn − 1.
14
YU-FENG YAO, BIN SHU AND YI-YANG LI
In particular, Dp
n −1
∂ ⊗ v2 ∈ M . Consequently, W ⊆ M . What remain to show are the
following two statements. (i) W is a g-submodule. (ii) W is maximal. To show (i), we need to check that W is invariant under the action of g. Indeed, it is obvious that W is invariant under the action of g[−1] . Moreover, for i ≥ 0, we have k (−1) Dk−j (xi−j D)∂ ⊗ v1 ∈ W for 0 ≤ k ≤ pn − 1, j
min{k,i} i
k
x D · D ∂ ⊗ v1 =
j
j=0
min{k,i} i
k
x ξD · D ∂ ⊗ v1 =
j=0
=
k Dk−j (xi−j ξD)∂ ⊗ v1 (−1) j j
⎧ ⎨(−1)i−1 ⎩0,
k
x ∂ · D ∂ ⊗ v1 =
j=0
(Dk+2−i ⊗ v1 − Dk+1−i ∂ ⊗ v2 ) ∈ W, if k ≥ i − 1, if k < i − 1,
min{k,i} i
k i−1
k Dk−j (xi−j ∂)∂ ⊗ v1 = 0 for 0 ≤ k ≤ pn − 1, (−1) j j
and k (−1) Dk−j (xi−j ξ∂)∂ ⊗ v1 j
min{k,i} i
k
x ξ∂ · D ∂ ⊗ v1 =
j=0
j
⎧ ⎨(−1)i (λ − 1)kDk−i ∂ ⊗ v ∈ W, if k ≥ i, 2 1 i = ⎩0, if k < i.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
For 0 ≤ l ≤ pn − 2, we have
= =
=
= =
= =
=
= =
xi D · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l D Dl−j (xi−j D)∂ ⊗ v2 (−1) x D ⊗ v1 − (−1) j j j=0 j=0 l+1 l (−1)i Dl+2−i ⊗ v1 − (−1)i Dl+1−i ∂ ⊗ v2 i i l Dl+1−i (xD)∂ ⊗ v2 −(−1)i−1 i−1 i l+1 l+2−i i l D Dl+1−i ∂ ⊗ v2 (−1) ⊗ v1 − (−1) i i l Dl+1−i ∂ ⊗ v2 +(−1)i−1 i−1 l l i l+1 l+2−i i D + Dl+1−i ∂ ⊗ v2 (−1) ⊗ v1 − (−1) i i i−1 i l+1 Dl+2−i ⊗ v1 − Dl+1−i ∂ ⊗ v2 ∈ W, (−1) i
xi ξD · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l (−1) x ξD ⊗ v1 − (−1) D Dl−j (xi−j ξD)∂ ⊗ v2 j j j=0 j=0 l+1 l Dl+1−i ⊗ ξD · v1 − (−1)i Dl−i (ξD)∂ ⊗ v2 (−1)i i i l i−1 Dl+1−i (xξD)∂ ⊗ v2 −(−1) i−1 i l+1 l+1−i i l D Dl+1−i ⊗ v2 (−1) ⊗ v2 − (−1) i i l i−1 Dl+1−i ⊗ xD · v2 −(−1) i−1 l i l+1 l+1−i i l l+1−i i−1 (−1) ⊗ v2 − (−1) ⊗ v2 + (−1) D D Dl+1−i ⊗ v2 i i i−1 0,
15
16
YU-FENG YAO, BIN SHU AND YI-YANG LI
xi ∂ · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l D Dl−j (xi−j ∂)∂ ⊗ v2 (−1) x ∂ ⊗ v1 − (−1) = j j j=0 j=0 l+1 l = (−1)i Dl+1−i ∂ ⊗ v1 + (−1)i−1 Dl+1−i ∂ ⊗ x∂ · v2 i i−1 l+1 l = (−1)i + (−1)i−1 λ2 Dl+1−i ∂ ⊗ v1 ∈ W, i i−1
and
= =
=
=
= =
xi ξ∂ · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l (−1) x ξ∂ ⊗ v1 − (−1) D Dl−j (xi−j ξ∂)∂ ⊗ v2 j j j=0 j=0 l+1 l Dl+1−i ⊗ ξ∂ · v1 − (−1)i Dl−i (ξ∂)∂ ⊗ v2 (−1)i i i l i−1 Dl+1−i (xξ∂)∂ ⊗ v2 −(−1) i−1 l+1 i l+1−i i l D Dl−i ∂ ⊗ v2 (−1) λ2 ⊗ v1 + (−1) i i l i l l−i i−1 D ∂ ⊗ ξ∂ · v2 + (−1) Dl+1−i ⊗ x∂ · v2 −(−1) i i−1 l+1 i l+1−i i l (−1) λ2 ⊗ v1 + (−1) D Dl−i ∂ ⊗ v2 i i l l i l−i i−1 −(−1) (λ2 + 1) D ∂ ⊗ v2 + (−1) λ2 Dl+1−i ⊗ v1 i i−1 l+1 l l i l+1−i i (−1) λ2 ⊗ v1 − (−1) λ2 − D Dl−i ∂ ⊗ v2 i i−1 i l l+1−i (−1)i λ2 ⊗ v1 − Dl−i ∂ ⊗ v2 ∈ W. D i
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
Moreover, we have
n
xi D · (Dp −1 ∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j D)∂ ⊗ v2 j j=0 n n n i p −1 pn −i i−1 p − 1 D Dp −i ∂ ⊗ xD · v2 = (−1) ∂ ⊗ v2 + (−1) i i−1 = Dp
n −i
∂ ⊗ v2 − D p
n −i
∂ ⊗ v2
= 0,
n
xi ξD · (Dp −1 ∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j ξD)∂ ⊗ v2 j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) (ξD)∂ ⊗ v2 + (−1) D Dp −i (xξD)∂ ⊗ v2 i i−1 = Dp
n −i
⊗ v2 + D p
n −i
⊗ xD · v2
= Dp
n −i
⊗ v2 − D p
n −i
⊗ v2
= 0,
xi ∂ · (Dp
n −1
∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j ∂)∂ ⊗ v2 j j=0 n n i−1 p − 1 = (−1) Dp −i (x∂)∂ ⊗ v2 i−1 = −Dp
n −i
= −λ2 Dp
∂ ⊗ x∂ · v2
n −i
∂ ⊗ v1 ∈ W
17
18
YU-FENG YAO, BIN SHU AND YI-YANG LI
and n
xi ξ∂ · (Dp −1 ∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j ξ∂)∂ ⊗ v2 j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) (ξ∂)∂ ⊗ v2 + (−1) D Dp −i (xξ∂)∂ ⊗ v2 i i−1 = −Dp
n −1−i
∂ ⊗ v2 + D p
= −Dp
n −1−i
∂ ⊗ v2 + (λ2 + 1)Dp
= −λ2 (Dp
n −i
⊗ v1 − D p
n −1−i
n −1−i
∂ ⊗ ξ∂ · v2 − Dp n −1−i
n −i
⊗ x∂ · v2
∂ ⊗ v2 − λ2 Dp
n −i
⊗ v1
∂ ⊗ v2 ) ∈ W.
Therefore, W is a g-submodule. To show the statement (ii), it suffices to prove that K(λ)/W is irreducible. Indeed, we can take {Di ⊗ v1 , Di ⊗ v2 | 0 ≤ i ≤ pn − 1} as a basis of K(λ)/W . Let M be a nonzero n −1 p 2 submodule of K(λ)/W . Let 0 = v = aij Di ⊗ vj ∈ M . Set i0 = min{i | aij = 0 for some j}. Then Dp
n −1−i
i=0 j=1
0
· v = aDpn −1 ⊗ v1 + bDpn −1 ⊗ v2 ∈ M for some a, b ∈ F with
a = 0 or b = 0. If a = 0, then xp
n −1
D · (aDpn −1 ⊗ v1 + bDpn −1 ⊗ v2 ) n n pn −1 pn −1 p −1 p − 1 n n i i (−1) a Dp −1−i xp −1−i D ⊗ v1 + (−1) b Dpn −1−i xpn −1−i D ⊗ v2 = i i i=0 i=0 = aD ⊗ v1 + bD ⊗ v2 + bD ⊗ xD · v2 = aD ⊗ v1 ∈ M . Hence D ⊗ v1 ∈ M . Furthermore, xξ∂ · D ⊗ v1 = [xξ∂, D] ⊗ v1 = −ξ∂ ⊗ v1 = −λ2 ⊗ v1 ∈ M . Since λ2 = 0, it follows that 1 ⊗ v1 ∈ M . Therefore, M = K(λ)/W . If a = 0, then b = 0 and Dpn −1 ⊗ v2 ∈ M . Consequently, x
pn −1
∂·D
⊗ v2 =
i=0
pn − 1 Dpn −1−i xpn −1−i ∂ ⊗ v2 i
pn −1 pn −1
(−1)
i
= ∂ ⊗ v2 + D ⊗ x∂ · v2 = ∂ ⊗ v2 + λ2 D ⊗ v1 ∈ M
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
19
and xξD · (∂ ⊗ v2 + λ2 D ⊗ v1 ) = [xξD, ∂] ⊗ v2 + λ2 [xξD, D] ⊗ v1 = xD ⊗ v2 − λ2 ξD ⊗ v1 = −(λ2 + 1) ⊗ v2 ∈ M . Since λ2 = −1, it follows that 1 ⊗ v2 ∈ M . Consequently, we also get M = K(λ)/W . Therefore, K(λ)/W is simple, so that W is maximal. We complete the proof.
Case (iii): λ1 = 0 and λ2 = 1. Let W1 = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 1 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}
and W = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}
We then have the following result. Proposition 3.10. Let λ = (0, 1) ∈ Λ. Then W1 and W are the unique minimal and maximal submodules of K(λ), respectively. Consequently, L(λ) = K(λ)/W . In particular, dim L(λ) = 2pn . Proof. Let M be a nonzero submodule of K(λ). A similar argument as in the proof of Proposition 3.9 implies that Dk ∂ ⊗ v1 ∈ M for 1 ≤ k ≤ pn − 1. Moreover, since xξD · D∂ ⊗ v1 = [xξD, D]∂ ⊗ v1 + D(xξD)∂ ⊗ v1 = −ξD∂ ⊗ v1 + D(xD) ⊗ v1 = −[ξD, ∂] ⊗ v1 + ∂ ⊗ ξD · v1 + D ⊗ xD · v1 = −D ⊗ v1 + ∂ ⊗ v2 ∈ M, it follows that Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ∈ M for 0 ≤ l ≤ pn − 1. In particular Dp
n −1
∂ ⊗ v2 ∈ M .
Consequently, W1 ⊆ M . It is a routine to check that W1 and W are g-submodules by a similar computation as in the proof of Proposition 3.9. Hence, W1 is the unique minimal submodule of K(λ). To show that W is a maximal submodule, it suffices to prove that K(λ)/W is irreducible. We can take {Di ⊗ v1 , Di ⊗ v2 | 0 ≤ i ≤ pn − 1} as a basis of K(λ)/W . Let M n −1 p 2 aij Di ⊗ vj in M . be a nonzero submodule of K(λ)/W . Take a nonzero element v = i=0 j=1
20
YU-FENG YAO, BIN SHU AND YI-YANG LI
Set i0 = min{i | aij = 0 for some j}. Then Dp
n −1−i
0
· v = aDpn −1 ⊗ v1 + bDpn −1 ⊗ v2 ∈ M
with a = 0 or b = 0. If a = 0, then M = K(λ)/W by a similar computation as in the proof of Proposition 3.9. If a = 0, then Dpn −1 ⊗ v2 ∈ M . Since n pn −1 p − 1 n pn −1 j ξ∂·Dp −1 ⊗ v2 = (−1) Dpn −1−j (xpn −1−j ξ∂) ⊗ v2 = ξ∂ ⊗ v2 = 2 ⊗ v2 ∈ M x j j=0 and char F = p > 2, it follows that 1 ⊗ v2 ∈ M . Hence, we also get M = K(λ)/W . Consequently, K(λ)/W is irreducible, i.e., W is maximal. We complete the proof.
Case (iv): λ1 = 0 and λ2 = −1. Let W2 = spanF {Dk ∂⊗v1 , Dl+1 ⊗v1 −Dl ∂⊗v2 , Dp
n −1
∂⊗v2 , Dp
n −1
⊗v2 | 0 ≤ k ≤ pn −1, 0 ≤ l ≤ pn −2}
and W = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}.
We then have the following result. Proposition 3.11. Let λ = (0, −1) ∈ Λ. Then W and W2 are the unique minimal and maximal submodules of K(λ), respectively. Consequently, L(λ) = K(λ)/W2 . In particular, dim L(λ) = 2pn − 1. Proof. Let M be a nonzero submodule of K(λ). A similar argument as in the proof of Proposition 3.9 implies that W ⊆ M , and W is a g-submodule. Hence, W is the unique minimal submodule of K(λ). Moreover, for i ≥ 0, since n i n n n i pn −1 j p −1 ⊗ v2 = (−1) x D·D Dp −1−j xi−j D ⊗ v2 = Dp −i ⊗ v2 + Dp −i ⊗ xD · v2 = 0, j j=0 i
x ξD · D
pn −1
⊗ v2 =
i j=0
i
x∂·D
pn −1
(−1)
j
pn − 1 n n Dp −1−j xi−j ξD ⊗ v2 = Dp −1−i ⊗ ξD · v2 = 0, j
⊗ v2 =
i
(−1)
j=0
j
pn − 1 n Dp −1−j xi−j ∂ ⊗ v2 j
= Dp
n −1−i
∂ ⊗ v2 + D p
n −i
⊗ x∂ · v2
= Dp
n −1−i
∂ ⊗ v2 − D p
n −i
⊗ v1 ∈ W2 ,
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
and i
x ξ∂ · D
pn −1
⊗ v2 =
i
21
(−1)
j
j=0
pn − 1 n n Dp −1−j xi−j ξ∂ ⊗ v2 = Dp −1−i ⊗ ξ∂ · v2 = 0, j
it follows that W2 is also a g-submodule. To show that W2 is maximal, it suffices to prove that K(λ)/W2 is irreducible. Indeed, we can take {Di ⊗ v1 , Dj ⊗ v2 | 0 ≤ i ≤ pn − 1, 0 ≤ j ≤ pn − 2} as a basis of K(λ)/W2 . Let M be a nonzero submodule of K(λ)/W2 and let pn −1
0 = v =
pn −2
ai Di ⊗ v1 +
i=0
bj D j ⊗ v2 ∈ M .
j=0
Set i0 = min{i | ai = 0 or bi = 0} and
⎧ ⎨0, if a = 0, i0 l= ⎩1, if a = 0. i0
Then Dp
n −1−l−i
0
· v = aDpn −1 ⊗ v1 + bDpn −2 ⊗ v2 + cDpn −1 ⊗ v2 ∈ M
where a, b, c ∈ F, a = 0 or b = 0. Hence aDpn −1 ⊗ v1 + bDpn −2 ⊗ v2 ∈ M . If a = 0, then xp
n −1
D · (aDpn −1 ⊗ v1 + bDpn −2 ⊗ v2 ) n pn −1 p −1 j (−1) a Dpn −1−j (xpn −1−j D) ⊗ v1 = j j=0
pn − 2 + (−1) b Dpn −2−j (xpn −1−j D) ⊗ v2 j j=0
pn −2
j
= aD ⊗ v1 + aD ⊗ xD · v1 − b ⊗ xD · v2 = aD ⊗ v1 + b ⊗ v2 ∈ M , and xξ∂ · (aD ⊗ v1 + b ⊗ v2 ) = a[xξ∂, D] ⊗ v1 = −aξ∂ ⊗ v1 = a ⊗ v1 ∈ M . It follows that M = K(λ)/W2 . If a = 0, then Dpn −2 ⊗ v2 ∈ M . Since n pn −2 n n −2 p −1 j p −2 p D·D ⊗ v2 = (−1) xpn −1−j D ⊗ v2 = −xD ⊗ v2 = 1 ⊗ v2 ∈ M , x j j=0
22
YU-FENG YAO, BIN SHU AND YI-YANG LI
we also get M = K(λ)/W2 . Consequently, K(λ)/W2 is irreducible. Therefore, W2 is the unique maximal submodule
of K(λ), and L(λ) = K(λ)/W2 . Case (v): λ1 = −1. In this case, λ2 = 1. Let M be a nonzero submodule of K(λ). Since Dp Proposition 3.3 (i), we can replace D
pn −1
∂ ⊗ vλ by D
pn −1
n −1
∂ ⊗ v2 ∈ M by
∂ ⊗ v2 in (3.4.2). It follows from a
similar computation as in (3.4.2) that D∂ ⊗ v2 ∈ M . Furthermore, we have x∂ · D∂ ⊗ v2 = D(x∂)∂ ⊗ v2 − ∂ ∂ ⊗ v2 = −D∂(x∂) ⊗ v2 = (1 − λ2 )D∂ ⊗ v1 ∈ M. Since λ2 = 1, it follows that D∂ ⊗v1 ∈ M . Moreover, we have x2 D·D∂ ⊗v1 = −(xD)∂ ⊗v1 = ∂ ⊗ v1 ∈ M , so that xξD · ∂ ⊗ v1 = xD ⊗ v1 = −1 ⊗ v1 ∈ M . Consequently, M = K(λ). We get the following result. Proposition 3.12. Let λ = (−1, λ2 ) ∈ Λ with λ2 = 1. Then the generalized restricted Kac g-module K(λ) is irreducible. We are now in the position to present the following main result on irreducible generalized restricted representations of the Zassenhaus superalgebra. Theorem 3.13. Let g = Z(n) be the Zassenhaus superalgebra over an algebraically closed field F of characteristic p > 2. Let λ = (λ1 , λ2 ) ∈ Λ, and J(λ), R(λ) the maximal and minimal submodules of the generalized restricted Kac g-module K(λ), respectively. Set L(λ) = K(λ)/J(λ) be the irreducible generalized restricted g-module with a singular vector of weight λ. Then the following statements hold. Case I: λ1 = −λ2 . (I-1) λ1 ∈ {−1, 0}. J(λ) = R(λ) = 0 and L(λ) = K(λ). Consequently, dim L(λ) = 2pn . (I-2) λ1 = 0.
J(λ) = R(λ) = spanF {Di ∂ j ⊗ v0 | 0 ≤ i ≤ pn − 1, 0 ≤ j ≤ 1, i + j ≥ 1} ∼ = L (0, −1) , where v0 is a basis of the one-dimensional trivial g[0] -module L0 (0, 0) . Hence, L(λ) = K(λ)/J(λ) = spanF {1 ⊗ v0 } is one-dimensional.
(I-3) λ1 = −1. J(λ) = R(λ) = FDp
n −1
∂ ⊗ vλ ∼ = L (0, 0) , where vλ is a basis of the one-dimensional
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
23
n restricted g[0] -module L0 (λ) = L0 (−1, 1) . Hence, L(λ) = K(λ)/FDp −1 ∂ ⊗ vλ and dim L (−1, 1) = 2pn − 1.
Case II: λ1 = −λ2 . (II-1) λ1 = 0. J(λ) = R(λ) = 0 and L(λ) = K(λ). Consequently, dim L(λ) = 4pn . (II-2) λ1 = 0. (i) λ2 ∈ {±1}. n
J(λ) = R(λ) = spanF {Dk ∂ ⊗v1 , Dl+1 ⊗v1 −Dl ∂ ⊗v2 , Dp −1 ∂ ⊗v2 | 0 ≤ k ≤ pn −1, 0 ≤ l ≤ pn − 2} ∼ = L (0, λ2 − 1) , where {v1 , v2 } is a basis of the two-dimensional simple restricted g[0] -module L0 (λ) with xD · v1 = 0, ξ∂ · v1 = λ2 v1 , xD · v2 = −v2 , ξ∂ · v2 = (λ2 + 1)v2 , and x∂ · v1 = ξD · v2 = 0, ξD · v1 = v2 , x∂ · v2 = λ2 v1 . Hence, L(λ) = K(λ)/J(λ) = spanF {Di ⊗ vj | 0 ≤ i ≤ pn − 1, j = 1, 2} has dimension 2pn . (ii) λ2 = 1. J(λ) = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ n
pn − 2} and R(λ) = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp −1 ∂ ⊗ v2 | 1 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2} ∼ = L (−1, 1) where v1 and v2 are defined as in the case (i). Hence, L(λ) = K(λ)/J(λ) = spanF {Di ⊗ vj | 0 ≤ i ≤ pn − 1, j = 1, 2} has dimension 2pn . (iii) λ2 = −1. J(λ) = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 , D p
n −1
⊗ v2 | 0 ≤ k ≤ pn − n
1, 0 ≤ l ≤ pn −2} and R(λ) = spanF {Dk ∂ ⊗v1 , Dl+1 ⊗v1 −Dl ∂ ⊗v2 , Dp −1 ∂ ⊗v2 | 0 ≤ k ≤ pn −1, 0 ≤ l ≤ pn −2} ∼ = L (0, −2) where v1 and v2 are defined as in the case (i). Hence, L(λ) = K(λ)/J(λ) = spanF {Di ⊗ v1 , Dj ⊗ v2 | 0 ≤ i ≤ pn −1, 0 ≤ j ≤ pn −2} has dimension 2pn − 1. Moreover, 0, J(λ) and R(λ) are all the proper submodules of K(λ) in any case mentioned above. Proof. It suffices to show that R (0, λ2 ) ∼ = L (0, λ2 − 1) for λ2 = 1, R (0, 1) ∼ = L (−1, 1) and R (−1, 1) ∼ = L (0, 0) . Other statements follow from Propositions 3.5–3.12 directly.
24
YU-FENG YAO, BIN SHU AND YI-YANG LI
Define the following linear maps: Φ1 : R (0, 0)
−→
L (0, −1) = K (0, −1) /J (0, −1) = spanF {Di ⊗ v1 , Dj ⊗ v2 | 0 ≤ i ≤ pn − 1, 0 ≤ j ≤ pn − 2}
D i ⊗ v0
−→
Di−1 ⊗ v2
D i ∂ ⊗ v0
−→
D i ⊗ v1 ,
Φ2 : R (0, λ2 ) −→
L (0, λ2 − 1) = K (0, λ2 − 1) /J (0, λ2 − 1) = spanF {Di ⊗ vj | 0 ≤ i ≤ pn − 1, j = 1, 2}
D k ∂ ⊗ v1
−→
D k ⊗ v1
(0 ≤ k ≤ pn − 1)
Dl+1 ⊗ v1 − Dl ∂ ⊗ v2
−→
D l ⊗ v2
(0 ≤ l ≤ pn − 2)
∂ ⊗ v2
−→
Dpn −1 ⊗ v2 ,
Φ3 : R (0, 1)
−→
L (−1, 1) = K (−1, 1) /J (−1, 1)
Dp
n
−1
= spanF {Di ∂ j ⊗ v((−1,1)) | 0 ≤ i ≤ pn − 1, j = 0, 1, i + j < pn } D k ∂ ⊗ v1
−→
Dk−1 ∂ ⊗ v((−1,1))
Dl+1 ⊗ v1 − Dl ∂ ⊗ v2
−→
Dl ⊗ v((−1,1))
∂ ⊗ v2
−→
Dpn −1 ⊗ v((−1,1)) ,
Φ4 : R (−1, 1)
−→
L (0, 0) = K (0, 0) /J (0, 0) = spanF {1 ⊗ v0 }
−→
1 ⊗ v0 ,
Dp
Dp
n
−1
n
−1
∂ ⊗ v((−1,1))
(1 ≤ k ≤ pn − 1) (0 ≤ l ≤ pn − 2)
where λ2 = 0, 1. It follows from a direct computation that these Φi ’s are isomorphisms for 1 ≤ i ≤ 4, as desired.
Definition 3.14. Let λ = (λ1 , λ2 ) ∈ Λ. The weight λ is called typical (resp. atypical) if L(λ) = K(λ) (resp. L(λ) = K(λ)). As a direct consequence of Theorem 3.13, we have Corollary 3.15. The following statements hold. (i) The weight λ = (λ1 , λ2 ) is atypical if and only if λ1 = 0, λ2 ∈ Fp or λ1 = −1, λ2 = 1. (ii) We have the following non-split short exact sequences: 0 −→ L (0, −1) −→ K (0, 0) −→ L (0, 0) −→ 0;
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
0 −→ J (0, 1) −→ K (0, 1) −→ L (0, 1) −→ 0; 0 −→ L (−1, 1) −→ J (0, 1) −→ L (0, 0) −→ 0; 0 −→ J (0, −1) −→ K (0, −1) −→ L (0, −1) −→ 0; 0 −→ L (0, −2) −→ J (0, −1) −→ L (0, 0) −→ 0; 0 −→ L (0, i − 1) −→ K (0, i) −→ L (0, i) −→ 0 (2 ≤ i ≤ p − 2); 0 −→ L (0, 0) −→ K (−1, 1) −→ L (−1, 1) −→ 0.
25
References [1] F. F. Duan and B. Shu, Representations of the Witt superalgebra W (2), J. Algebra 396, 272-286, 2013. [2] V. Kac, Lie superalgebras, Adv. Math. 26, 8-96, 1977. [3] V. Petrogradski, Identities in the enveloping algebras for modular Lie superalgebras, J. Algebra 145, 1-21, 1992. [4] B. Shu, The generalized restricted representations of graded Lie algebras of Cartan type, J. Algebra 194, 157-177, 1997. [5] B. Shu, Generalized restricted Lie algebras and representations of the Zassenhaus algebra, J. Algebra 204, 549-572, 1998. [6] B. Shu and Y. F. Yao, Character formulas for restricted simple modules of the special superalgebras, Math. Nachr. 285(8-9), 1107-1116, 2012. [7] B. Shu and C. W. Zhang, Restricted representations of the Witt superalgebras, J. Algebra 324, 652-672, 2010. [8] B. Shu and C. W. Zhang, Representations of the restricted Cartan type Lie superalgebra W (m, n, 1), Algebr. Represent. Theory 14, 463-481, 2011. [9] H. Strade, Simple Lie algebras over fields of positive characteristic, I. Structure Theory, De Gruyter Expositions in Mathematics 38, Walter de Gruyter, Berlin, 2004. [10] S. J. Wang and W. D. Liu, On restricted representations of restricted contact Lie superalgebras of odd type, J. Algebra Appl. 15 (4), 1650075 (14 pages), 2016. [11] W. Q. Wang and L. Zhao, Representations of Lie superalgebras in prime characteristic I, Proc. Lond. Math. Soc. 99 (1), 145-167, 2009. ¯ [12] Y. F. Yao, On restricted representations of the extended special type Lie superalgebra S(m, n, 1), Monatsh. Math. 170, 239-255, 2013. [13] Y. F. Yao, Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type, Sci. China Ser. A 56(2), 239-252, 2013. [14] Y. F. Yao and B. Shu, Restricted representations of Lie superalgebras of Hamiltonian type, Algebr. Represent. Theory 16(3), 615-632, 2013. [15] J. X. Yuan and W. D. Liu, Restricted Kac module of Hamiltonian Lie superalgebras of odd type, Monatsh. Math. 178 (3), 473-488, 2015.
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YU-FENG YAO, BIN SHU AND YI-YANG LI
[16] L. S. Zheng and B. Shu, On representations of gl(m|n) and infinitesimal subgroups of GL(m|n), Chinese J. Contemp. Math. 31 (2), 107-122, 2010. [17] C. W. Zhang, On the simple modules for the restricted Lie superalgebra sl(n|1), J. Pure Appl. Algebra 213 (5), 756-765, 2009. [18] Y. Z. Zhang and W. D. Liu, Modular Lie superalgebras (in Chinese), Scientific Press, Beijing, 2001. Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China. E-mail address: [email protected] Department of Mathematics, East China Normal University, Shanghai, 200241, China. E-mail address: [email protected] School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China. E-mail address: yiyang [email protected]
Yu-Feng Yao, Bin Shu, Yi-Yang Li
PII: DOI: Reference:
S0021-8693(16)30237-X http://dx.doi.org/10.1016/j.jalgebra.2016.08.007 YJABR 15849
To appear in:
Journal of Algebra
Received date:
10 January 2016
Please cite this article in press as: Y.-F. Yao et al., Generalized restricted representations of the Zassenhaus superalgebras, J. Algebra (2016), http://dx.doi.org/10.1016/j.jalgebra.2016.08.007
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GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS YU-FENG YAO, BIN SHU AND YI-YANG LI Abstract. Let F be an algebraically closed field of prime characteristic p > 2, and n ∈ N+ . Let Z(n) be the Zassenhaus superalgebra defined over F, which, as the simplest nonrestricted simple Lie superalgebra, is the superversion of the Zassenhaus algebra. More precisely, Z(n) is the Lie superalgebra of the special super derivations of the superalgebra Π(n). Here Π(n) is the tensor product of the divided power algebra of one variable and the Grassmann superalgebra of one variable. In this paper we study generalized restricted simple modules over the Zassenhaus superalgebra Z(n). Classification of isomorphism classes of generalized restricted simple modules and their dimensions are precisely determined. A sufficient and necessary condition for irreducibility of generalized restricted Kac modules is provided.
1. Introduction The finite-dimensional simple Lie superalgebras over the field of complex numbers were classified by Kac in the 1970s (cf. [2]). Over an algebraically closed field of prime characteristic, the classification of finite-dimensional simple Lie superalgebras remains unknown up to the present time. However, a natural class is known, namely the modular version of the complex simple Lie superalgebras. These Lie superalgebras are restricted ones. In [18], more simple Lie superalgebras of Cartan type in prime characteristic were constructed. They are not restricted Lie superalgebras in general. Representation theory of Lie superalgebras in characteristic 0 has been systematically developed over the past decades. However, there seems to lack rich theory on the modular representations of Lie superalgebras over an algebraically closed field of prime characteristic. In recent years, there has been an increasing interest in modular representation theory of 2010 Mathematics Subject Classification. 17B10, 17B50, 17B66, 17B70. Key words and phrases. the Zassenhaus superalgebra, irreducible module, (generalized) restricted module, (generalized) restricted Kac module, (atypical) typical weight. This work is supported by Natural Science Foundation of Shanghai (Grant No. 16ZR1415000) and National Natural Science Foundation of China (Grant Nos. 11271130, 11571008 and 11526139). 1
2
YU-FENG YAO, BIN SHU AND YI-YANG LI
restricted Lie superalgebras (cf. [11, 17, 7, 8, 1, 6, 12, 13, 14, 10, 15]). In this paper, we initiate the study of representations of non-restricted Lie superalgebras in prime characteristic. We concern the representation theory of the Zassenhaus superalgebra, which is the simplest non-restricted simple Lie superalgebra. We classify the isomorphism classes of generalized restricted simple modules over the Zassenhaus superalgebra. Moreover, we precisely determine the dimensions of the generalized restricted simple modules. We also give a sufficient and necessary condition for generalized restricted Kac modules to be irreducible. 2. Preliminaries In this paper, we always assume that the base field F is algebraically closed and of characteristic p > 2. All vector spaces (modules) V are over F, and Z2 -graded, i.e., V = V¯0 ⊕V¯1 . For each homogeneous element v ∈ V , we denote by v¯ the parity of v. Let A = A¯0 ⊕ A¯1 be a superalgebra. A homogeneous super derivation D of A is a homogeneous linear transformation of A such that D(ab) = D(a)b + (−1)D a¯ aD(b), ∀ a, b ∈ A¯0 ∪ A¯1 . n
2.1. The Zassenhaus superalgebra. Let n ∈ N+ . Set A(n) = F[x]/(xp ), the divided power algebra of one variable x, and (1) the Grassmann superalgebra of one variable ξ. For convenience, we also denote by xi the image of xi under the quotient map from F[x] to A(n) for 0 ≤ i ≤ pn − 1. Set Π(n) = A(n) ⊗F (1). Then Π(n) has an F-basis {xi ξ j | 0 ≤ i ≤ pn − 1, j = 0, 1}, and is Z2 -graded with Π(n)¯0 = spanF {xi | 0 ≤ i ≤ pn − 1} and Π(n)¯1 = spanF {xi ξ | 0 ≤ i ≤ pn − 1}. Moreover, Π(n) is an associative superalgebra in which the multiplication is given by i + s i+s j+t i j s t x ξ , xξ ·x ξ = i
0 ≤ i, s ≤ pn − 1, j = 0, 1, t = 0, 1
with the convention that xk ξ l = 0,
for k ≥ pn or l > 1.
A super derivation D of Π(n) is called special if for 0 ≤ i ≤ pn − 1 and j = 0, 1, D(xi ξ j ) = xi−1 D(x)ξ j + jxi D(ξ). Define two linear maps D and ∂ on Π(n) by D(xi ξ j ) = xi−1 ξ j , ∂(xi ξ j ) = jxi , for 0 ≤ i ≤ pn − 1, j = 0, 1.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
3
Then both D and ∂ are special derivations of Π(n). Let Z(n) be the set of all special super derivations of Π(n). A straightforward calculation implies that Z(n) = spanF {xi ξ j D, xi ξ j ∂ | 0 ≤ i ≤ pn − 1, j = 0, 1} = Z(n)¯0 ⊕ Z(n)¯1 , where Z(n)¯0 = spanF {xi D, xi ξ∂ | 0 ≤ i ≤ pn − 1} and Z(n)¯1 = spanF {xi ∂, xi ξD | 0 ≤ i ≤ pn − 1}. It is a routine to check that Z(n) is a Lie subsuperalgebra of the derivation superalgebra of Π(n) under the usual Lie bracket (cf. [18]). Moreover, it is a simple Lie superalgebra and referred as the Zassenhaus superalgebra, the even part of which contains a subalgebra isomorphic to the usual Zassenhaus algebra (cf. [9, §7.6]). Furthermore, the Zassenhaus superalgebra Z(n) is a restricted Lie superalgebra if and only if n = 1. Although in general it is not restricted, it is a generalized restricted Lie superalgebra in the following sense (see Definition 2.1). 2.2. Generalized restricted Lie superalgebras and their generalized χ-reduced representations. The following notion of generalized restricted Lie superalgebra generalizes the notion of a restricted Lie superalgebra (cf. [3]). Definition 2.1. A Lie superalgebra g = g¯0 ⊕ g¯1 is said to be a generalized restricted one if the even part g¯0 is a generalized restricted Lie algebra and the odd part g¯1 as an adjoint g¯0 -module is a generalized restricted module (cf. [4, 5]). More precisely speaking , there exist an ordered basis E = {ei | i ∈ I} of g¯0 , an |I|-tuple s = (si )i∈I of positive integers and a
generalized restricted map ϕs : E → g¯0 sending ei to eϕi s such that (ad(ei ))p i (y) = ad(eϕi s )(y) s
for any i ∈ I, y ∈ g. Remarks 2.2.
(i) One usually denotes the generalized restricted Lie superalgebra men-
tioned in Definition 2.1 by (g, E, ϕs ) to indicate the ordered basis E and the generalized restricted map ϕs . (ii) Any restricted Lie superalgebra (L, [p]) is a generalized restricted Lie superalgebra where ϕs = [p]|E and s = (si )i∈I with si = 1, ∀ i ∈ I. Different from the case of characteristic zero, any irreducible representation of a generalized restricted Lie superalgebra (g, E, ϕs ) is attached with a unique χ ∈ g∗¯0 . Let Jχ be the ideal si
of the universal enveloping superalgebra U (g) generated by epi − eϕi s − χ(ei )p i , i ∈ I. Set s
4
YU-FENG YAO, BIN SHU AND YI-YANG LI
Uχ (g, E, ϕs ) = U (g)/Jχ which is called the generalized χ-reduced enveloping superalgebra of g. One usually denotes the generalized χ-reduced enveloping superalgebra Uχ (g, E, ϕs ) si
by Uχ (g) for brevity. A g-module M is said to be a generalized χ-reduced one if epi ·v−eϕi s ·v = s
χ(ei )p i v, ∀ v ∈ M, i ∈ I. Then a generalized χ-reduced g-module is a Uχ (g)-module. All generalized χ-reduced g-modules constitute a full subcategory of the category of g-modules, denoted by Uχ (g)-mod. In the case χ = 0, a generalized χ-reduced module is said to be a generalized restricted module, and we call U0 (g) the generalized restricted enveloping superalgebra of g, denoted by u(g) for brevity. It should be remarked that the notion of generalized χ-reduced enveloping superalgebras (resp. generalized χ-reduced representations, generalized restricted representations etc.) for a restricted Lie superalgebra coincides with that of the ordinary χ-reduced enveloping superalgebras (resp. χ-reduced representations, restricted representations etc.). 2.3. Basic structure of the Zassenhaus superalgebra. In this paper we always assume g = Z(n) is the Zassenhaus superalgebra which is a generalized restricted Lie superalgebra with an ordered basis {xi D, xi ξ∂ | 0 ≤ i ≤ pn − 1} of g¯0 , and the generalized restricted map ϕs defined as t
ϕs (xp D) = −xp
t+1 −p+1
D for 1 ≤ t ≤ n − 1,
ϕs (xD) = xD, ϕs (ξ∂) = ξ∂, and ϕs (xj D) = ϕs (xk ξ∂) = 0 for other j and 1 ≤ k ≤ pn − 1, where s = (n, 1, 1, · · · , 1). There is a natural Z-grading structure on g, i.e., g = where
n −1 p
i=−1
g[i] ,
g[i] = spanF {xk ξ l D, xk ξ l ∂ | k + l = i + 1, 0 ≤ k ≤ pn − 1, l = 0, 1} for − 1 ≤ i ≤ pn − 1. Associated with this grading, one has the following filtration: (2.2.1)
g = g−1 ⊃ g0 ⊃ · · · ⊃ gpn −1 ,
where gi =
j≥i
g[j] for − 1 ≤ i ≤ pn − 1.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
5
The zero graded component g[0] ∼ = gl(1|1) under the map φ : g[0] −→ gl(1|1), xD → E11 , ξ∂ → E22 , x∂ → E12 , ξD → E21 . Furthermore, we have the triangular decomposition g[0] = n− ⊕ h ⊕ n+ , where n− = FξD, n+ = Fx∂, and h = FxD + Fξ∂. Set b = h ⊕ n+ , B = b ⊕ g1 = b + i≥1 g[i] , and g+ = g[0] ⊕ g1 = g0 . It is easy to check that the subalgebras b, B and g+ are (generalized) restricted subalgebras of g.
3. Generalized restricted representations of the Zassenhaus superalgebra Keep the same notations and convention as in §2. In particular, g = Z(n) is the Zassenhaus superalgebra over an algebraically closed field F of prime characteristic p > 2. Recall that g = g[−1] ⊕ g+ with g+ = g[0] ⊕ g1 and g[0] ∼ = gl(1|1). 3.1. Restricted irreducible representations of g[0] . Recall that g[0] = n− ⊕ h ⊕ n+ with n− = FξD, h = FxD + Fξ∂, and n+ = Fx∂. The isomorphism classes of restricted simple modules of g[0] are parameterized by Λ := {λ = (λ1 , λ2 ) | λi ∈ Fp , i = 1, 2}, the set of restricted weights, which coincides with F2p (cf. [11, 16]). More precisely, for each λ = (λ1 , λ2 ) ∈ Λ, one can define the one-dimensional h-module Fvλ with xD · vλ = λ1 vλ and ξ∂·vλ = λ2 vλ . This one-dimensional h-module Fvλ can be regarded as a b-module with trivial n+ -action. One then get the so-called baby Verma module V (λ) as the induced g[0] -module
u(g[0] ) u(b) Fvλ , where u(g[0] ) and u(b) are the restricted enveloping superalgebras of g[0] and b, respectively. As a vector space, V (λ) = spanF {1 ⊗ vλ , ξD ⊗ vλ }. Each baby Verma module V (λ) has a unique simple quotient, denoted by L0 (λ). We have the following easy observation which describes simple restricted g[0] -modules. The proof is straightforward. Lemma 3.1. Let λ = (λ1 , λ2 ) ∈ Λ. Then the following statements hold. (i) If λ1 = −λ2 , then L0 (λ) = V (λ) = spanF {1 ⊗ vλ , ξD ⊗ vλ } is two-dimensional. (ii) If λ1 = −λ2 , then W = FξD ⊗ vλ is the unique maximal submodule of V (λ) and L0 (λ) = V (λ)/W . In this case, we also denote by 1 ⊗ vλ the image of 1 ⊗ vλ in the quotient V (λ)/W for brevity. Then L0 (λ) = spanF {1 ⊗ vλ } is one-dimensional. 3.2. Generalized restricted irreducible representations of g. In this subsection, we exploit the structure of generalized restricted simple g-modules based on the known information on restricted simple g[0] -modules presented in § 3.1. Recall that g+ = g[0] + g1 , and g1 = spanF {xi ξ j D, xi ξ j ∂ | i + j ≥ 2, 1 ≤ i ≤ pn − 1, j = 0, 1}.
6
YU-FENG YAO, BIN SHU AND YI-YANG LI
We first need the following result which implies that simple restricted modules for g[0] and g+ coincide. Lemma 3.2. Each simple restricted g+ -module is simple as a g[0] -module with trivial g1 action. Conversely, each simple restricted g[0] -module can be extended to a simple restricted g+ -module with trivial g1 -action. Proof. Since g1 is an ideal of g+ , the second statement is obvious. We only need to show the first statement. Let M be a simple restricted g+ -module. Note that g1 ∩g¯0 has an ordered basis {X1 , · · · , Xs } [p]2
such that Xi
= 0, ∀ 1 ≤ i ≤ s. Since M is a restricted g+ -module, 2
[p]2
Xip · m = Xi
· m = 0, ∀ 1 ≤ i ≤ s, m ∈ M.
On the other hand, 1 Y 2 = [Y, Y ] ∈ g1 ∩ g¯0 , ∀ Y ∈ g1 ∩ g¯1 . 2 This implies that each element of a basis of g1 acts on M nilpotently. It follows from the superversion of Jacobson’s weakly nilpotent theorem (see [18]) that M g1 := {m ∈ M | z · m = 0, ∀ z ∈ g1 } = 0. It is easy to check that M g1 is a g+ -submodule, since g1 is an ideal of g+ . The simplicity of M as a g+ -module implies that M g1 = M . Consequently, g1 acts on M trivially.
By Lemma 3.2, each simple restricted g[0] -module L0 (λ) can be regarded as a simple restricted g+ -module, where λ = (λ1 , λ2 ) ∈ Λ. Then one can define the generalized restricted Kac g-module K(λ) as the induced g-module u(g) ⊗u(g+ ) L0 (λ). We have the following description for generalized restricted Kac g-modules and simple generalized restricted gmodules. Proposition 3.3. The following statements hold. (i) Each generalized restricted Kac g-module K(λ) has a unique simple submodule, i.e., K(λ) has a simple socle. (ii) Each generalized restricted Kac g-module K(λ) has a unique maximal submodule J(λ). Hence, K(λ) has a unique simple quotient, denoted by L(λ). (iii) Each simple generalized restricted g-module is of the form L(λ) for some λ ∈ Λ. (iv) Let λ, μ ∈ Λ, then K(λ) ∼ = K(μ) if and only if λ = μ.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
7
(v) Let λ, μ ∈ Λ, then L(λ) ∼ = L(μ) if and only if λ = μ. Proof. (i) Let S be an arbitrary simple g-submodule of K(λ). Take a nonzero element pn −1
v=
1
Di ∂ j ⊗ vij ∈ S,
i=0 j=0
where vij ∈ L0 (λ), ∀ 0 ≤ i ≤ pn − 1, j = 0, 1. Let i0 = min{i | vij = 0 for some j} and j0 = min{j | vi0 j = 0}. Then Dp ξD · Dp
n −1
∂ ⊗ vi0 j0 = −Dp
n −1
n −1−i
0
∂ 1−j0 v = Dp
n −1
∂ ⊗ vi0 j0 ∈ S. Note that
∂ ⊗ ξD · vi0 j0 , x∂ · Dp
n −1
∂ · ⊗vi0 j0 = −Dp
n −1
∂ ⊗ x∂ · vi0 j0
and h · Dp
n −1
∂ ⊗ vi0 j0 ∈ FDp
n −1
∂ ⊗ vi0 j0 + FDp
n −1
∂ ⊗ h · vi0 j0 , ∀ h ∈ h.
Since L0 (λ) is a simple g[0] -module, L0 (λ) = u(g[0] )vi0 j0 . Hence, u(g[0] )Dp
n −1
∂ ⊗ vi0 j 0 = D p
Consequently, S = u(g)Dp
n −1
n −1
∂ ⊗ L0 (λ) ∼ = L0 (λ + (1, −1)) as u(g[0] )-modules.
∂ ⊗ L0 (λ).
(ii) Let N − = FξD + g[−1] which is a generalized restricted subalgebra of g. Note that n
n
Dp = 0 and (ξD)2 = ∂ 2 = 0 in u(N − ). Then for each f ∈ u(N − )N − , we have f 2p = 0 in u(N − ). We claim that each nonzero proper submodule U of K(λ) is included in u(N − )N − ⊗ vλ . Indeed, suppose U u(N − )N − ⊗ vλ , then there exists a nonzero element ν = (1 + f ) ⊗ vλ ∈ U for some f ∈ u(N − )N − . Consequently, (1−f +f 2 −f 3 +· · ·−f 2p
n −1
)ν = 1⊗vλ ∈ U .
Hence, U = K(λ), a contradiction. Now let J(λ) be the sum of all proper submodules of K(λ). Then J(λ) ⊂ u(N − )N − ⊗ vλ K(λ). Therefore, J(λ) is the unique maximal submodule, and K(λ) has a unique simple quotient L(λ) = K(λ)/J(λ). (iii) Let M be a simple generalized restricted g-module. Then M contains a simple restricted g[0] -submodule L0 (λ) for some λ ∈ Λ. We then have the following natural g-module homomorphism ϕ : K(λ) = u(g) ⊗u(g+ ) L0 (λ) −→ M u ⊗ v −→ u · v. The simplicity of M as a g-module implies that ϕ is surjective. Hence M is isomorphic to a quotient of K(λ). Now it follows from the statement (ii) that M ∼ = L(λ). (iv) It suffices to show that λ = μ if K(λ) ∼ = K(μ). For that, let
K(λ)D,∂ := {w ∈ K(λ) | D · w = ∂ · w = 0} and K(μ)D,∂ := {w ∈ K(μ) | D · w = ∂ · w = 0}.
8
YU-FENG YAO, BIN SHU AND YI-YANG LI
Then K(λ)D,∂ and K(μ)D,∂ are g[0] -submodules of K(λ) and K(μ), respectively. Moreover, it is easy to check that K(λ)D,∂ = Dp
n −1
∂ ⊗ L0 (λ) ∼ = L0 (λ + (1, −1))
and K(μ)D,∂ = Dp
n −1
∂ ⊗ L0 (μ) ∼ = L0 (μ + (1, −1)).
Since K(λ) ∼ = K(μ) as g-modules, it follows that K(λ)D,∂ ∼ = K(μ)D,∂ as g[0] -modules, i.e., we get an isomorphism of g[0] -modules: L0 (λ + (1, −1)) ∼ = L0 (μ + (1, −1)). It follows from the discussion in §3.1 that λ + (1, −1) = μ + (1, −1). Consequently, λ = μ, as desired. (v) It suffices to show that λ = μ if L(λ) = L(μ). Indeed, suppose ψ : L(λ) = K(λ)/J(λ) −→ L(μ) = K(μ)/J(μ) is an isomorphism between L(λ) and L(μ). Assume ψ(v) = 1 ⊗ vμ for some v ∈ K(λ) \ J(λ). Then y · v ∈ J(λ), ∀ y ∈ n+ + g1 , and h · v − μ(h)v ∈ J(λ), ∀ h ∈ h.
(3.3.1) We can write v = a ⊗ vλ +
ui ⊗ fi vλ where a ∈ F \ {0} and
ui ⊗ fi ∈ u(g[−1] )g[−1] ⊗ u(n− ) + u(g[−1] ) ⊗ u(n− )n− , ∀ i. Hence, (3.3.2)
h · v = aλ(h) ⊗ vλ +
hui ⊗ fi vλ , ∀ h ∈ h.
It follows from (3.3.1) and (3.3.2) that a(λ(h) − μ(h)) ⊗ vλ ∈ u(g[−1] )g[−1] ⊗ vλ + u(g[−1] ) ⊗ u(n− )n− vλ , ∀ h ∈ h. This implies that λ(h) = μ(h), ∀ h ∈ h, i.e., λ = μ, as desired.
Definition 3.4. Let M be a g-module and λ ∈ h∗ . A nonzero vector v ∈ M is said to be a singular vector of weight λ if h · v = λ(h)v, ∀ h ∈ h and y · v = 0, ∀ y ∈ n+ + g1 .
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
9
In the following, we will precisely determine the structure of the simple generalized restricted g-modules L(λ) for λ = (λ1 , λ2 ) ∈ Λ. Type I: λ1 = −λ2 . Let λ = (λ1 , −λ1 ) ∈ Λ. Then the simple restricted g[0] -module L0 (λ) is one-dimensional with a basis vλ such that x∂ · vλ = ξD · vλ = 0, and xD · vλ = λ1 vλ , ξ∂ · vλ = −λ1 vλ .
(3.4.1)
The generalized restricted Kac g-module K(λ) has a basis {Di ∂ j ⊗ vλ | 0 ≤ i ≤ pn − 1, j = 0, 1}. Let M be a nonzero submodule of K(λ). It follows from Proposition 3.3 (i) that M contains the vector Dp (3.4.2)
x
pn −1
D·D
n −1
∂ ⊗ vλ . Hence,
pn −1
pn −1
∂ ⊗ vλ =
(−1)
i
i=0
pn − 1 n n Dp −1−i xp −1−i D∂ ⊗ vλ i
= D∂ ⊗ vλ + DxD∂ ⊗ vλ = D∂ ⊗ vλ + D∂ ⊗ xD · vλ = (1 + λ1 )D∂ ⊗ vλ ∈ M. Case (i): λ1 = −1, 0. In this case, it follows from (3.4.2) that D∂ ⊗ vλ ∈ M . Moreover, we have x2 D · D∂ ⊗ vλ = [x2 D, D]∂ ⊗ vλ = −xD∂ ⊗ vλ = −∂ ⊗ xD · vλ = −λ1 ∂ ⊗ vλ ∈ M. Since λ1 = 0, we have ∂ ⊗ vλ ∈ M . Furthermore, xξD · ∂ ⊗ vλ = [xξD, ∂] ⊗ vλ = xD ⊗ vλ = 1 ⊗ xD · vλ = λ1 ⊗ vλ ∈ M. Hence 1 ⊗ vλ ∈ M due to the assumption that λ1 = 0. Consequently, M = K(λ). We get the following result. Proposition 3.5. Let λ = (λ1 , −λ1 ) ∈ Λ with λ1 = −1, 0. Then the generalized restricted Kac g-module K(λ) is irreducible. Case (ii): λ1 = λ2 = 0. Let λ = (0, 0). It follows from (3.4.2) that D∂ ⊗ vλ ∈ M . Since ξD · D∂ ⊗ vλ = DξD∂ ⊗ vλ = D2 ⊗ vλ ∈ M,
10
YU-FENG YAO, BIN SHU AND YI-YANG LI
it follows that x2 D · D2 ⊗ vλ = D ⊗ vλ ∈ M . Furthermore, (−x∂) · D ⊗ vλ = ∂ ⊗ vλ ∈ M . Consequently, I := spanF {Di ∂ j ⊗ vλ | 0 ≤ i ≤ pn − 1, j = 0, 1, and i + j ≥ 1} ⊆ M. It is easy to check that I is a submodule. Hence, M = I or K(λ). In conclusion, I is the unique maximal submodule of K(λ) and L(λ) = K(λ)/I which is a one-dimensional trivial g-module. We get the following result. Proposition 3.6. Let λ = (0, 0). Then the irreducible g-module L(λ) is a one-dimensional trivial g-module. Case (iii): λ1 = −1, λ2 = 1. Let λ = (−1, 1). Note that Dp
n −1
∂ ⊗ vλ ∈ M . Set M1 := spanF {Dp
n −1
∂ ⊗ vλ }. We claim
that M1 is a g-submodule. Indeed, D · (Dp x∂ · (Dp xD · (D and for i ≥ 1, we have i
xD·D
pn −1
∂ ⊗ vλ
n −1
n −1
∂ ⊗ vλ ) = ∂ · (Dp
n −1
n −1
∂ ⊗ vλ ) = 0,
pn −1
∂ ⊗ vλ ) = 0,
∂ ⊗ vλ ) = ξD · (Dp
pn −1
∂ ⊗ vλ ) = ξ∂ · (D
∂ ⊗ vλ ) = 0,
pn − 1 n Dp −1−j xi−j D∂ ⊗ vλ = (−1) j j=0 n n n i p −1 pn −i i−1 p − 1 = (−1) ∂ ⊗ vλ + (−1) D Dp −i xD∂ ⊗ vλ i i−1 i
j
= Dp
n −i
∂ ⊗ vλ + D p
n −i
∂ ⊗ xD · vλ
= Dp
n −i
∂ ⊗ vλ − D p
n −i
∂ ⊗ vλ
= 0, i
x∂·D
pn −1
∂ ⊗ vλ = = = =
pn − 1 n Dp −1−j xi−j ∂ ∂ ⊗ vλ (−1) j j=0 n n i−1 p − 1 (−1) Dp −i x∂ ∂ ⊗ vλ i−1 n n i−1 p − 1 −(−1) Dp −i ∂ ⊗ x∂ · vλ i−1 0, i
j
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS i
x ξD · D
pn −1
11
pn − 1 n = (−1) Dp −1−j xi−j ξD∂ ⊗ vλ j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) ξD∂ ⊗ vλ + (−1) D Dp −i xξD∂ ⊗ vλ i i−1 n n p −1 p −1 n n = (−1)i Dp −i ⊗ vλ + (−1)i−1 Dp −i xD ⊗ vλ i i−1 i
∂ ⊗ vλ
= Dp
j
n −i
⊗ vλ − D p
n −i
⊗ vλ
= 0, and i
x ξ∂ · D
pn −1
∂ ⊗ vλ
pn − 1 n Dp −1−j xi−j ξ∂ ∂ ⊗ vλ = (−1) j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) ξ∂ ∂ ⊗ vλ + (−1) D Dp −i xξ∂ ∂ ⊗ vλ i i−1 n n n i p −1 pn −1−i i p −1 D Dp −1−i ∂ ⊗ ξ∂ · vλ = −(−1) ∂ ⊗ vλ + (−1) i i n p −1 n −(−1)i−1 Dp −i x∂ ⊗ vλ i−1 i
= −Dp
j
n −1−i
∂ ⊗ vλ + D p
n −1−i
∂ ⊗ vλ − D p
n −i
∂ ⊗ x∂ · vλ
= 0. Hence, M1 is a g-submodule. Furthermore, we claim that M = M1 or K(λ). In fact, suppose M1 M .Then there exists v=
aij Di ∂ j ⊗ vλ ∈ M \ M1 .
Let i0 = min{i | aij = 0 for some j}. If i0 = pn − 1, then v = aDp
n −1
⊗ vλ + bDp
with a = 0. We have pn −1
x
pn − 1 n n ∂·v = (−1) a Dp −1−i xp −1−i ∂ ⊗ vλ i i=0 n pn −1 p −1 n n i + (−1) b Dp −1−i xp −1−i ∂ ∂ ⊗ vλ i i=0 pn −1
i
= a∂ ⊗ vλ ∈ M.
n −1
∂ ⊗ vλ
12
YU-FENG YAO, BIN SHU AND YI-YANG LI
Hence ∂ ⊗ vλ ∈ M . Moreover, xξD · ∂ ⊗ vλ = xD ⊗ vλ = 1 ⊗ xD · vλ = −1 ⊗ vλ ∈ M, so that M = K(λ). If i0 < pn − 1. Let
⎧ ⎨1, if a = 0, i0 0 j0 = ⎩0, if a = 0. i0 0
A direct computation implies that Dp
n −2−i
0
∂ j0 · v = ai0 ,1−j0 Dp
where ai0 ,1−j0 = 0. Hence Dp
n −2
∂ ⊗ vλ + ai0 +1,0 Dp
n −1
∂ ⊗ vλ ∈ M
n −2
∂ ⊗ vλ ∈ M . We then have n pn −2 n n pn −1 pn −2 i p −2 D·D ∂ ⊗ vλ = (−1) Dp −2−i xp −1−i D∂ ⊗ vλ x i i=0 = −xD∂ ⊗ vλ = −∂ ⊗ xD · vλ = ∂ ⊗ vλ ∈ M.
Moreover, xξD · ∂ ⊗ vλ = xD ⊗ vλ = 1 ⊗ xD · vλ = −1 ⊗ vλ ∈ M. Therefore, M = K(λ). In conclusion, M1 is the unique maximal submodule of K(λ). Consequently, L(λ) = K(λ)/M1 . We get the following result. Proposition 3.7. Let λ = (−1, 1). Then the irreducible generalized restricted g-module L(λ) = K(λ)/M1 , where M1 = spanF {Dp
n −1
∂ ⊗ vλ }. In particular, L(λ) has dimension
2pn − 1. Type II: λ1 = −λ2 . Let λ = (λ1 , λ2 ) ∈ Λ with λ1 = −λ2 . Then by Lemma 3.1, the simple restricted g[0] module L0 (λ) coincides with the baby Verma module V (λ) which has a basis {v1 , v2 } with the action of g[0] defined as follows: (3.7.1)
xD · v1 = λ1 v1 , ξ∂ · v1 = λ2 v1 , xD · v2 = (λ1 − 1)v2 , ξ∂ · v2 = (λ2 + 1)v2 ,
and (3.7.2)
x∂ · v1 = 0, ξD · v1 = v2 , x∂ · v2 = (λ1 + λ2 )v1 , ξD · v2 = 0.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
13
The generalized restricted Kac g-module K(λ) has a basis {Di ∂ j ⊗ v1 , Di ∂ j ⊗ v2 | 0 ≤ i ≤ pn − 1, j = 0, 1}. Let M be a nonzero submodule of K(λ). It follows from Proposition 3.3 (i) that M contains Dp
n −1
∂ ⊗ L0 (λ).
Case (i): λ1 = −1, 0. In this case, since Dp
n −1
∂ ⊗ v1 ∈ M , we can replace Dp
n −1
∂ ⊗ vλ by Dp
n −1
∂ ⊗ v1 in (3.4.2).
A similar argument as in the situation of Type I yields that 1 ⊗ v1 ∈ M . Consequently, M = K(λ). We get the following result. Proposition 3.8. Let λ = (λ1 , λ2 ) ∈ Λ with λ1 = −λ2 and λ1 = −1, 0. Then the generalzied restricted Kac g-module K(λ) is irreducible. Case (ii): λ1 = 0 and λ2 = ±1. In this case, λ2 = 0. Set W = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}. We have the following result on the simple generalized restricted g-module L (0, λ2 ) . Proposition 3.9. Let λ = (0, λ2 ) ∈ Λ with λ2 = 0, ±1. Then W is the unique maximal submodule of K(λ), and L(λ) = K(λ)/W . In particular, dim L(λ) = 2pn . Proof. Let M be a nonzero submodule of K(λ). It follows from Proposition 3.3 (i) that Dp D
n −1
pn −1
∂ ⊗ v1 ∈ M . A similar computation as in (3.4.2) with Dp
n −1
∂ ⊗ vλ replaced by
k
∂ ⊗ v1 yields that D∂ ⊗ v1 ∈ M . Hence D ∂ ⊗ v1 ∈ M for 1 ≤ k ≤ pn − 1. Moreover,
since xξ∂ · D∂ ⊗ v1 = [xξ∂, D]∂ ⊗ v1 + D[xξ∂, ∂] ⊗ v1 = −ξ∂ ∂ ⊗ v1 − D ⊗ x∂ · v1 = −∂ ⊗ ξ∂ · v1 − [ξ∂, ∂] ⊗ v1 = −λ2 ∂ ⊗ v1 + ∂ ⊗ v1 = (1 − λ2 )∂ ⊗ v1 ∈ M and λ2 = 1, it follows that ∂ ⊗ v1 ∈ M . Furthermore, ξD · ∂ ⊗ v1 = [ξD, ∂] ⊗ v1 − ∂ ⊗ ξD · v1 = D ⊗ v1 − ∂ ⊗ v2 ∈ M. Hence, Dl · (D ⊗ v1 − ∂ ⊗ v2 ) = Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ∈ M for 0 ≤ l ≤ pn − 1.
14
YU-FENG YAO, BIN SHU AND YI-YANG LI
In particular, Dp
n −1
∂ ⊗ v2 ∈ M . Consequently, W ⊆ M . What remain to show are the
following two statements. (i) W is a g-submodule. (ii) W is maximal. To show (i), we need to check that W is invariant under the action of g. Indeed, it is obvious that W is invariant under the action of g[−1] . Moreover, for i ≥ 0, we have k (−1) Dk−j (xi−j D)∂ ⊗ v1 ∈ W for 0 ≤ k ≤ pn − 1, j
min{k,i} i
k
x D · D ∂ ⊗ v1 =
j
j=0
min{k,i} i
k
x ξD · D ∂ ⊗ v1 =
j=0
=
k Dk−j (xi−j ξD)∂ ⊗ v1 (−1) j j
⎧ ⎨(−1)i−1 ⎩0,
k
x ∂ · D ∂ ⊗ v1 =
j=0
(Dk+2−i ⊗ v1 − Dk+1−i ∂ ⊗ v2 ) ∈ W, if k ≥ i − 1, if k < i − 1,
min{k,i} i
k i−1
k Dk−j (xi−j ∂)∂ ⊗ v1 = 0 for 0 ≤ k ≤ pn − 1, (−1) j j
and k (−1) Dk−j (xi−j ξ∂)∂ ⊗ v1 j
min{k,i} i
k
x ξ∂ · D ∂ ⊗ v1 =
j=0
j
⎧ ⎨(−1)i (λ − 1)kDk−i ∂ ⊗ v ∈ W, if k ≥ i, 2 1 i = ⎩0, if k < i.
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
For 0 ≤ l ≤ pn − 2, we have
= =
=
= =
= =
=
= =
xi D · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l D Dl−j (xi−j D)∂ ⊗ v2 (−1) x D ⊗ v1 − (−1) j j j=0 j=0 l+1 l (−1)i Dl+2−i ⊗ v1 − (−1)i Dl+1−i ∂ ⊗ v2 i i l Dl+1−i (xD)∂ ⊗ v2 −(−1)i−1 i−1 i l+1 l+2−i i l D Dl+1−i ∂ ⊗ v2 (−1) ⊗ v1 − (−1) i i l Dl+1−i ∂ ⊗ v2 +(−1)i−1 i−1 l l i l+1 l+2−i i D + Dl+1−i ∂ ⊗ v2 (−1) ⊗ v1 − (−1) i i i−1 i l+1 Dl+2−i ⊗ v1 − Dl+1−i ∂ ⊗ v2 ∈ W, (−1) i
xi ξD · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l (−1) x ξD ⊗ v1 − (−1) D Dl−j (xi−j ξD)∂ ⊗ v2 j j j=0 j=0 l+1 l Dl+1−i ⊗ ξD · v1 − (−1)i Dl−i (ξD)∂ ⊗ v2 (−1)i i i l i−1 Dl+1−i (xξD)∂ ⊗ v2 −(−1) i−1 i l+1 l+1−i i l D Dl+1−i ⊗ v2 (−1) ⊗ v2 − (−1) i i l i−1 Dl+1−i ⊗ xD · v2 −(−1) i−1 l i l+1 l+1−i i l l+1−i i−1 (−1) ⊗ v2 − (−1) ⊗ v2 + (−1) D D Dl+1−i ⊗ v2 i i i−1 0,
15
16
YU-FENG YAO, BIN SHU AND YI-YANG LI
xi ∂ · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l D Dl−j (xi−j ∂)∂ ⊗ v2 (−1) x ∂ ⊗ v1 − (−1) = j j j=0 j=0 l+1 l = (−1)i Dl+1−i ∂ ⊗ v1 + (−1)i−1 Dl+1−i ∂ ⊗ x∂ · v2 i i−1 l+1 l = (−1)i + (−1)i−1 λ2 Dl+1−i ∂ ⊗ v1 ∈ W, i i−1
and
= =
=
=
= =
xi ξ∂ · (Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ) min{i, l+1} min{i, l} j l+1 l+1−j i−j j l (−1) x ξ∂ ⊗ v1 − (−1) D Dl−j (xi−j ξ∂)∂ ⊗ v2 j j j=0 j=0 l+1 l Dl+1−i ⊗ ξ∂ · v1 − (−1)i Dl−i (ξ∂)∂ ⊗ v2 (−1)i i i l i−1 Dl+1−i (xξ∂)∂ ⊗ v2 −(−1) i−1 l+1 i l+1−i i l D Dl−i ∂ ⊗ v2 (−1) λ2 ⊗ v1 + (−1) i i l i l l−i i−1 D ∂ ⊗ ξ∂ · v2 + (−1) Dl+1−i ⊗ x∂ · v2 −(−1) i i−1 l+1 i l+1−i i l (−1) λ2 ⊗ v1 + (−1) D Dl−i ∂ ⊗ v2 i i l l i l−i i−1 −(−1) (λ2 + 1) D ∂ ⊗ v2 + (−1) λ2 Dl+1−i ⊗ v1 i i−1 l+1 l l i l+1−i i (−1) λ2 ⊗ v1 − (−1) λ2 − D Dl−i ∂ ⊗ v2 i i−1 i l l+1−i (−1)i λ2 ⊗ v1 − Dl−i ∂ ⊗ v2 ∈ W. D i
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
Moreover, we have
n
xi D · (Dp −1 ∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j D)∂ ⊗ v2 j j=0 n n n i p −1 pn −i i−1 p − 1 D Dp −i ∂ ⊗ xD · v2 = (−1) ∂ ⊗ v2 + (−1) i i−1 = Dp
n −i
∂ ⊗ v2 − D p
n −i
∂ ⊗ v2
= 0,
n
xi ξD · (Dp −1 ∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j ξD)∂ ⊗ v2 j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) (ξD)∂ ⊗ v2 + (−1) D Dp −i (xξD)∂ ⊗ v2 i i−1 = Dp
n −i
⊗ v2 + D p
n −i
⊗ xD · v2
= Dp
n −i
⊗ v2 − D p
n −i
⊗ v2
= 0,
xi ∂ · (Dp
n −1
∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j ∂)∂ ⊗ v2 j j=0 n n i−1 p − 1 = (−1) Dp −i (x∂)∂ ⊗ v2 i−1 = −Dp
n −i
= −λ2 Dp
∂ ⊗ x∂ · v2
n −i
∂ ⊗ v1 ∈ W
17
18
YU-FENG YAO, BIN SHU AND YI-YANG LI
and n
xi ξ∂ · (Dp −1 ∂ ⊗ v2 ) n i n j p −1 = (−1) Dp −1−j (xi−j ξ∂)∂ ⊗ v2 j j=0 n n n i p −1 pn −1−i i−1 p − 1 = (−1) (ξ∂)∂ ⊗ v2 + (−1) D Dp −i (xξ∂)∂ ⊗ v2 i i−1 = −Dp
n −1−i
∂ ⊗ v2 + D p
= −Dp
n −1−i
∂ ⊗ v2 + (λ2 + 1)Dp
= −λ2 (Dp
n −i
⊗ v1 − D p
n −1−i
n −1−i
∂ ⊗ ξ∂ · v2 − Dp n −1−i
n −i
⊗ x∂ · v2
∂ ⊗ v2 − λ2 Dp
n −i
⊗ v1
∂ ⊗ v2 ) ∈ W.
Therefore, W is a g-submodule. To show the statement (ii), it suffices to prove that K(λ)/W is irreducible. Indeed, we can take {Di ⊗ v1 , Di ⊗ v2 | 0 ≤ i ≤ pn − 1} as a basis of K(λ)/W . Let M be a nonzero n −1 p 2 submodule of K(λ)/W . Let 0 = v = aij Di ⊗ vj ∈ M . Set i0 = min{i | aij = 0 for some j}. Then Dp
n −1−i
i=0 j=1
0
· v = aDpn −1 ⊗ v1 + bDpn −1 ⊗ v2 ∈ M for some a, b ∈ F with
a = 0 or b = 0. If a = 0, then xp
n −1
D · (aDpn −1 ⊗ v1 + bDpn −1 ⊗ v2 ) n n pn −1 pn −1 p −1 p − 1 n n i i (−1) a Dp −1−i xp −1−i D ⊗ v1 + (−1) b Dpn −1−i xpn −1−i D ⊗ v2 = i i i=0 i=0 = aD ⊗ v1 + bD ⊗ v2 + bD ⊗ xD · v2 = aD ⊗ v1 ∈ M . Hence D ⊗ v1 ∈ M . Furthermore, xξ∂ · D ⊗ v1 = [xξ∂, D] ⊗ v1 = −ξ∂ ⊗ v1 = −λ2 ⊗ v1 ∈ M . Since λ2 = 0, it follows that 1 ⊗ v1 ∈ M . Therefore, M = K(λ)/W . If a = 0, then b = 0 and Dpn −1 ⊗ v2 ∈ M . Consequently, x
pn −1
∂·D
⊗ v2 =
i=0
pn − 1 Dpn −1−i xpn −1−i ∂ ⊗ v2 i
pn −1 pn −1
(−1)
i
= ∂ ⊗ v2 + D ⊗ x∂ · v2 = ∂ ⊗ v2 + λ2 D ⊗ v1 ∈ M
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
19
and xξD · (∂ ⊗ v2 + λ2 D ⊗ v1 ) = [xξD, ∂] ⊗ v2 + λ2 [xξD, D] ⊗ v1 = xD ⊗ v2 − λ2 ξD ⊗ v1 = −(λ2 + 1) ⊗ v2 ∈ M . Since λ2 = −1, it follows that 1 ⊗ v2 ∈ M . Consequently, we also get M = K(λ)/W . Therefore, K(λ)/W is simple, so that W is maximal. We complete the proof.
Case (iii): λ1 = 0 and λ2 = 1. Let W1 = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 1 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}
and W = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}
We then have the following result. Proposition 3.10. Let λ = (0, 1) ∈ Λ. Then W1 and W are the unique minimal and maximal submodules of K(λ), respectively. Consequently, L(λ) = K(λ)/W . In particular, dim L(λ) = 2pn . Proof. Let M be a nonzero submodule of K(λ). A similar argument as in the proof of Proposition 3.9 implies that Dk ∂ ⊗ v1 ∈ M for 1 ≤ k ≤ pn − 1. Moreover, since xξD · D∂ ⊗ v1 = [xξD, D]∂ ⊗ v1 + D(xξD)∂ ⊗ v1 = −ξD∂ ⊗ v1 + D(xD) ⊗ v1 = −[ξD, ∂] ⊗ v1 + ∂ ⊗ ξD · v1 + D ⊗ xD · v1 = −D ⊗ v1 + ∂ ⊗ v2 ∈ M, it follows that Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 ∈ M for 0 ≤ l ≤ pn − 1. In particular Dp
n −1
∂ ⊗ v2 ∈ M .
Consequently, W1 ⊆ M . It is a routine to check that W1 and W are g-submodules by a similar computation as in the proof of Proposition 3.9. Hence, W1 is the unique minimal submodule of K(λ). To show that W is a maximal submodule, it suffices to prove that K(λ)/W is irreducible. We can take {Di ⊗ v1 , Di ⊗ v2 | 0 ≤ i ≤ pn − 1} as a basis of K(λ)/W . Let M n −1 p 2 aij Di ⊗ vj in M . be a nonzero submodule of K(λ)/W . Take a nonzero element v = i=0 j=1
20
YU-FENG YAO, BIN SHU AND YI-YANG LI
Set i0 = min{i | aij = 0 for some j}. Then Dp
n −1−i
0
· v = aDpn −1 ⊗ v1 + bDpn −1 ⊗ v2 ∈ M
with a = 0 or b = 0. If a = 0, then M = K(λ)/W by a similar computation as in the proof of Proposition 3.9. If a = 0, then Dpn −1 ⊗ v2 ∈ M . Since n pn −1 p − 1 n pn −1 j ξ∂·Dp −1 ⊗ v2 = (−1) Dpn −1−j (xpn −1−j ξ∂) ⊗ v2 = ξ∂ ⊗ v2 = 2 ⊗ v2 ∈ M x j j=0 and char F = p > 2, it follows that 1 ⊗ v2 ∈ M . Hence, we also get M = K(λ)/W . Consequently, K(λ)/W is irreducible, i.e., W is maximal. We complete the proof.
Case (iv): λ1 = 0 and λ2 = −1. Let W2 = spanF {Dk ∂⊗v1 , Dl+1 ⊗v1 −Dl ∂⊗v2 , Dp
n −1
∂⊗v2 , Dp
n −1
⊗v2 | 0 ≤ k ≤ pn −1, 0 ≤ l ≤ pn −2}
and W = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2}.
We then have the following result. Proposition 3.11. Let λ = (0, −1) ∈ Λ. Then W and W2 are the unique minimal and maximal submodules of K(λ), respectively. Consequently, L(λ) = K(λ)/W2 . In particular, dim L(λ) = 2pn − 1. Proof. Let M be a nonzero submodule of K(λ). A similar argument as in the proof of Proposition 3.9 implies that W ⊆ M , and W is a g-submodule. Hence, W is the unique minimal submodule of K(λ). Moreover, for i ≥ 0, since n i n n n i pn −1 j p −1 ⊗ v2 = (−1) x D·D Dp −1−j xi−j D ⊗ v2 = Dp −i ⊗ v2 + Dp −i ⊗ xD · v2 = 0, j j=0 i
x ξD · D
pn −1
⊗ v2 =
i j=0
i
x∂·D
pn −1
(−1)
j
pn − 1 n n Dp −1−j xi−j ξD ⊗ v2 = Dp −1−i ⊗ ξD · v2 = 0, j
⊗ v2 =
i
(−1)
j=0
j
pn − 1 n Dp −1−j xi−j ∂ ⊗ v2 j
= Dp
n −1−i
∂ ⊗ v2 + D p
n −i
⊗ x∂ · v2
= Dp
n −1−i
∂ ⊗ v2 − D p
n −i
⊗ v1 ∈ W2 ,
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
and i
x ξ∂ · D
pn −1
⊗ v2 =
i
21
(−1)
j
j=0
pn − 1 n n Dp −1−j xi−j ξ∂ ⊗ v2 = Dp −1−i ⊗ ξ∂ · v2 = 0, j
it follows that W2 is also a g-submodule. To show that W2 is maximal, it suffices to prove that K(λ)/W2 is irreducible. Indeed, we can take {Di ⊗ v1 , Dj ⊗ v2 | 0 ≤ i ≤ pn − 1, 0 ≤ j ≤ pn − 2} as a basis of K(λ)/W2 . Let M be a nonzero submodule of K(λ)/W2 and let pn −1
0 = v =
pn −2
ai Di ⊗ v1 +
i=0
bj D j ⊗ v2 ∈ M .
j=0
Set i0 = min{i | ai = 0 or bi = 0} and
⎧ ⎨0, if a = 0, i0 l= ⎩1, if a = 0. i0
Then Dp
n −1−l−i
0
· v = aDpn −1 ⊗ v1 + bDpn −2 ⊗ v2 + cDpn −1 ⊗ v2 ∈ M
where a, b, c ∈ F, a = 0 or b = 0. Hence aDpn −1 ⊗ v1 + bDpn −2 ⊗ v2 ∈ M . If a = 0, then xp
n −1
D · (aDpn −1 ⊗ v1 + bDpn −2 ⊗ v2 ) n pn −1 p −1 j (−1) a Dpn −1−j (xpn −1−j D) ⊗ v1 = j j=0
pn − 2 + (−1) b Dpn −2−j (xpn −1−j D) ⊗ v2 j j=0
pn −2
j
= aD ⊗ v1 + aD ⊗ xD · v1 − b ⊗ xD · v2 = aD ⊗ v1 + b ⊗ v2 ∈ M , and xξ∂ · (aD ⊗ v1 + b ⊗ v2 ) = a[xξ∂, D] ⊗ v1 = −aξ∂ ⊗ v1 = a ⊗ v1 ∈ M . It follows that M = K(λ)/W2 . If a = 0, then Dpn −2 ⊗ v2 ∈ M . Since n pn −2 n n −2 p −1 j p −2 p D·D ⊗ v2 = (−1) xpn −1−j D ⊗ v2 = −xD ⊗ v2 = 1 ⊗ v2 ∈ M , x j j=0
22
YU-FENG YAO, BIN SHU AND YI-YANG LI
we also get M = K(λ)/W2 . Consequently, K(λ)/W2 is irreducible. Therefore, W2 is the unique maximal submodule
of K(λ), and L(λ) = K(λ)/W2 . Case (v): λ1 = −1. In this case, λ2 = 1. Let M be a nonzero submodule of K(λ). Since Dp Proposition 3.3 (i), we can replace D
pn −1
∂ ⊗ vλ by D
pn −1
n −1
∂ ⊗ v2 ∈ M by
∂ ⊗ v2 in (3.4.2). It follows from a
similar computation as in (3.4.2) that D∂ ⊗ v2 ∈ M . Furthermore, we have x∂ · D∂ ⊗ v2 = D(x∂)∂ ⊗ v2 − ∂ ∂ ⊗ v2 = −D∂(x∂) ⊗ v2 = (1 − λ2 )D∂ ⊗ v1 ∈ M. Since λ2 = 1, it follows that D∂ ⊗v1 ∈ M . Moreover, we have x2 D·D∂ ⊗v1 = −(xD)∂ ⊗v1 = ∂ ⊗ v1 ∈ M , so that xξD · ∂ ⊗ v1 = xD ⊗ v1 = −1 ⊗ v1 ∈ M . Consequently, M = K(λ). We get the following result. Proposition 3.12. Let λ = (−1, λ2 ) ∈ Λ with λ2 = 1. Then the generalized restricted Kac g-module K(λ) is irreducible. We are now in the position to present the following main result on irreducible generalized restricted representations of the Zassenhaus superalgebra. Theorem 3.13. Let g = Z(n) be the Zassenhaus superalgebra over an algebraically closed field F of characteristic p > 2. Let λ = (λ1 , λ2 ) ∈ Λ, and J(λ), R(λ) the maximal and minimal submodules of the generalized restricted Kac g-module K(λ), respectively. Set L(λ) = K(λ)/J(λ) be the irreducible generalized restricted g-module with a singular vector of weight λ. Then the following statements hold. Case I: λ1 = −λ2 . (I-1) λ1 ∈ {−1, 0}. J(λ) = R(λ) = 0 and L(λ) = K(λ). Consequently, dim L(λ) = 2pn . (I-2) λ1 = 0.
J(λ) = R(λ) = spanF {Di ∂ j ⊗ v0 | 0 ≤ i ≤ pn − 1, 0 ≤ j ≤ 1, i + j ≥ 1} ∼ = L (0, −1) , where v0 is a basis of the one-dimensional trivial g[0] -module L0 (0, 0) . Hence, L(λ) = K(λ)/J(λ) = spanF {1 ⊗ v0 } is one-dimensional.
(I-3) λ1 = −1. J(λ) = R(λ) = FDp
n −1
∂ ⊗ vλ ∼ = L (0, 0) , where vλ is a basis of the one-dimensional
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
23
n restricted g[0] -module L0 (λ) = L0 (−1, 1) . Hence, L(λ) = K(λ)/FDp −1 ∂ ⊗ vλ and dim L (−1, 1) = 2pn − 1.
Case II: λ1 = −λ2 . (II-1) λ1 = 0. J(λ) = R(λ) = 0 and L(λ) = K(λ). Consequently, dim L(λ) = 4pn . (II-2) λ1 = 0. (i) λ2 ∈ {±1}. n
J(λ) = R(λ) = spanF {Dk ∂ ⊗v1 , Dl+1 ⊗v1 −Dl ∂ ⊗v2 , Dp −1 ∂ ⊗v2 | 0 ≤ k ≤ pn −1, 0 ≤ l ≤ pn − 2} ∼ = L (0, λ2 − 1) , where {v1 , v2 } is a basis of the two-dimensional simple restricted g[0] -module L0 (λ) with xD · v1 = 0, ξ∂ · v1 = λ2 v1 , xD · v2 = −v2 , ξ∂ · v2 = (λ2 + 1)v2 , and x∂ · v1 = ξD · v2 = 0, ξD · v1 = v2 , x∂ · v2 = λ2 v1 . Hence, L(λ) = K(λ)/J(λ) = spanF {Di ⊗ vj | 0 ≤ i ≤ pn − 1, j = 1, 2} has dimension 2pn . (ii) λ2 = 1. J(λ) = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 | 0 ≤ k ≤ pn − 1, 0 ≤ l ≤ n
pn − 2} and R(λ) = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp −1 ∂ ⊗ v2 | 1 ≤ k ≤ pn − 1, 0 ≤ l ≤ pn − 2} ∼ = L (−1, 1) where v1 and v2 are defined as in the case (i). Hence, L(λ) = K(λ)/J(λ) = spanF {Di ⊗ vj | 0 ≤ i ≤ pn − 1, j = 1, 2} has dimension 2pn . (iii) λ2 = −1. J(λ) = spanF {Dk ∂ ⊗ v1 , Dl+1 ⊗ v1 − Dl ∂ ⊗ v2 , Dp
n −1
∂ ⊗ v2 , D p
n −1
⊗ v2 | 0 ≤ k ≤ pn − n
1, 0 ≤ l ≤ pn −2} and R(λ) = spanF {Dk ∂ ⊗v1 , Dl+1 ⊗v1 −Dl ∂ ⊗v2 , Dp −1 ∂ ⊗v2 | 0 ≤ k ≤ pn −1, 0 ≤ l ≤ pn −2} ∼ = L (0, −2) where v1 and v2 are defined as in the case (i). Hence, L(λ) = K(λ)/J(λ) = spanF {Di ⊗ v1 , Dj ⊗ v2 | 0 ≤ i ≤ pn −1, 0 ≤ j ≤ pn −2} has dimension 2pn − 1. Moreover, 0, J(λ) and R(λ) are all the proper submodules of K(λ) in any case mentioned above. Proof. It suffices to show that R (0, λ2 ) ∼ = L (0, λ2 − 1) for λ2 = 1, R (0, 1) ∼ = L (−1, 1) and R (−1, 1) ∼ = L (0, 0) . Other statements follow from Propositions 3.5–3.12 directly.
24
YU-FENG YAO, BIN SHU AND YI-YANG LI
Define the following linear maps: Φ1 : R (0, 0)
−→
L (0, −1) = K (0, −1) /J (0, −1) = spanF {Di ⊗ v1 , Dj ⊗ v2 | 0 ≤ i ≤ pn − 1, 0 ≤ j ≤ pn − 2}
D i ⊗ v0
−→
Di−1 ⊗ v2
D i ∂ ⊗ v0
−→
D i ⊗ v1 ,
Φ2 : R (0, λ2 ) −→
L (0, λ2 − 1) = K (0, λ2 − 1) /J (0, λ2 − 1) = spanF {Di ⊗ vj | 0 ≤ i ≤ pn − 1, j = 1, 2}
D k ∂ ⊗ v1
−→
D k ⊗ v1
(0 ≤ k ≤ pn − 1)
Dl+1 ⊗ v1 − Dl ∂ ⊗ v2
−→
D l ⊗ v2
(0 ≤ l ≤ pn − 2)
∂ ⊗ v2
−→
Dpn −1 ⊗ v2 ,
Φ3 : R (0, 1)
−→
L (−1, 1) = K (−1, 1) /J (−1, 1)
Dp
n
−1
= spanF {Di ∂ j ⊗ v((−1,1)) | 0 ≤ i ≤ pn − 1, j = 0, 1, i + j < pn } D k ∂ ⊗ v1
−→
Dk−1 ∂ ⊗ v((−1,1))
Dl+1 ⊗ v1 − Dl ∂ ⊗ v2
−→
Dl ⊗ v((−1,1))
∂ ⊗ v2
−→
Dpn −1 ⊗ v((−1,1)) ,
Φ4 : R (−1, 1)
−→
L (0, 0) = K (0, 0) /J (0, 0) = spanF {1 ⊗ v0 }
−→
1 ⊗ v0 ,
Dp
Dp
n
−1
n
−1
∂ ⊗ v((−1,1))
(1 ≤ k ≤ pn − 1) (0 ≤ l ≤ pn − 2)
where λ2 = 0, 1. It follows from a direct computation that these Φi ’s are isomorphisms for 1 ≤ i ≤ 4, as desired.
Definition 3.14. Let λ = (λ1 , λ2 ) ∈ Λ. The weight λ is called typical (resp. atypical) if L(λ) = K(λ) (resp. L(λ) = K(λ)). As a direct consequence of Theorem 3.13, we have Corollary 3.15. The following statements hold. (i) The weight λ = (λ1 , λ2 ) is atypical if and only if λ1 = 0, λ2 ∈ Fp or λ1 = −1, λ2 = 1. (ii) We have the following non-split short exact sequences: 0 −→ L (0, −1) −→ K (0, 0) −→ L (0, 0) −→ 0;
GENERALIZED RESTRICTED REPRESENTATIONS OF THE ZASSENHAUS SUPERALGEBRAS
0 −→ J (0, 1) −→ K (0, 1) −→ L (0, 1) −→ 0; 0 −→ L (−1, 1) −→ J (0, 1) −→ L (0, 0) −→ 0; 0 −→ J (0, −1) −→ K (0, −1) −→ L (0, −1) −→ 0; 0 −→ L (0, −2) −→ J (0, −1) −→ L (0, 0) −→ 0; 0 −→ L (0, i − 1) −→ K (0, i) −→ L (0, i) −→ 0 (2 ≤ i ≤ p − 2); 0 −→ L (0, 0) −→ K (−1, 1) −→ L (−1, 1) −→ 0.
25
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YU-FENG YAO, BIN SHU AND YI-YANG LI
[16] L. S. Zheng and B. Shu, On representations of gl(m|n) and infinitesimal subgroups of GL(m|n), Chinese J. Contemp. Math. 31 (2), 107-122, 2010. [17] C. W. Zhang, On the simple modules for the restricted Lie superalgebra sl(n|1), J. Pure Appl. Algebra 213 (5), 756-765, 2009. [18] Y. Z. Zhang and W. D. Liu, Modular Lie superalgebras (in Chinese), Scientific Press, Beijing, 2001. Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China. E-mail address: [email protected] Department of Mathematics, East China Normal University, Shanghai, 200241, China. E-mail address: [email protected] School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China. E-mail address: yiyang [email protected]