Algebr Represent Theor https://doi.org/10.1007/s10468-017-9753-5

On Irreducible Representations of the Zassenhaus Superalgebras with p-Characters of Height 0 Yu-Feng Yao1 · Temuer Chaolu1

Received: 16 June 2017 / Accepted: 22 November 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017

n

Abstract Let n be a positive integer, and A(n) = F[x]/(x p ), the divided power algebra over an algebraically closed field F of prime characteristic p > 2. Let (n) be the tensor  product of A(n) and the Grassmann superalgebra (1) in one variable. The Zassenhaus superalgebra Z (n) is defined to be the Lie superalgebra of the special super derivations of the superalgebra (n). In this paper we study simple modules over the Zassenhaus superalgebra Z (n) with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained. Keywords The Zassenhaus superalgebra · Irreducible module · p-character · Height Mathematics Subject Classification 2010 17B10 · 17B50 · 17B70

1 Introduction Representations of simple Lie (super)algebras over C have been extensively studied. They are essentially interchangeable with the corresponding Lie groups. Over the past decades, the study of modular representations of restricted Lie (super)algebras in prime characteristic has made significant progress (see [1, 2, 4, 8, 10, 13–17, 19]). The modular representations Presented by Peter Littelmann. This work is supported by National Natural Science Foundation of China (Grant Nos. 11771279 and 11571008) and Natural Science Foundation of Shanghai (Grant No. 16ZR1415000).  Yu-Feng Yao

[email protected] Temuer Chaolu [email protected] 1

Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China

Y.-F. Yao, T. Chaolu

of restricted Lie (super)algebras have intimate connection to algebraic (super)groups (cf. [2, 4, 7]). Another motivation for the modular representation theory arises from the study of quantum groups at a root of unity. The tools of algebraic (super)groups are not easily available in the study of representations of non-restricted Lie (super)algebras. A natural question is how to develop the modular representation theory for non-restricted Lie (super)algebras. The first author and his collaborators initiated the study of generalized restricted representations in [18] for the simplest non-restricted simple Lie superalgebras, the Zassenhaus superalgebras. This paper is a sequel to [18]. We continue the investigations by studying representations with p-character of height 0 for the Zassenhaus superalgebras. We briefly introduce the contents of this paper as follows. In Section 2, we recall the definition of Zassenhaus superalgebras and some of their fundamental properties. Zassenhaus superalgebras are examples of generalized restricted Lie superalgebras. We also review general generalized restricted Lie superalgebras and their generalized χ -reduced representations. As a preparation for the study of representations of Zassenhaus superalgebras, we recall in this section the structure of restricted simple modules over the zero graded components of Zassenhaus superalgebras. Section 3 is devoted to studying simple modules of Zassenhaus superalgebras with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is provided.

2 Preliminaries In this paper, we keep the same notations and conventions as in [18] for consistency. Throughout we work with an algebraically closed field F of prime characteristic p > 2 as the ground field. We exclude the case p = 2, since Lie superalgebras in characteristic 2 coincide with Z2 -graded Lie algebras. Unless otherwise stated, all modules (algebras) M are over F, and are Z2 -graded. We write v¯ to denote the parity of a homogeneous element v ∈ M. Let A = A0¯ ⊕ A1¯ be a superalgebra. A homogeneous super derivation D of A is a homogenous linear tranformation of A such that D (ab) = D (a)b + (−1)D a¯ a D (b) for any a, b ∈ A0¯ ∪ A1¯ .

2.1 Zassenhaus Superalgebras n

Let n be a positive integer, and A(n) = F[x]/(x p ), the divided power algebra in the variable x with multiplication defined via   i + j (i+j ) x , 0 ≤ i, j ≤ pn − 1, x (i) x (j ) = i where we use the notation x (i) to denote the imageof x i under the quotient map from F[x] to A(n). We also write x (1) as x for brevity.  Let (1) be the Grassmann superalgebra in one variable ξ . Set (n) = A(n) ⊗F (1), which has an F-basis {x (i) ξ j | 0 ≤ i ≤ p n − 1, j = 0, 1}. Then (n) is an associative superalgebra with the Z2 -gradation defined as (n)0¯ = spanF {x (i) | 0 ≤ i ≤ pn − 1}, (n)1¯ = spanF {x (i) ξ | 0 ≤ i ≤ pn − 1}. The multiplication in (n) is defined as   i + s (i+s) j +t x ξ , 0 ≤ i, s ≤ pn − 1, j = 0, 1, t = 0, 1 x (i) ξ j · x (s) ξ t = i

On Irreducible Representations of the Zassenhaus Superalgebras...

with the convention that x (k) ξ l = 0 for k ≥ p n or l > 1. A super derivation D of (n) is called special if for 0 ≤ i ≤ pn − 1 and j = 0, 1,

D (x (i) ξ j ) = x (i−1) D (x)ξ j + j x (i) D (ξ ). Let D and ∂ be two linear super derivations on (n) defined via D(x (i) ξ j ) = x (i−1) ξ j , ∂(x (i) ξ j ) = j x (i) , 0 ≤ i ≤ pn − 1, j = 0, 1. Then both D and ∂ are special super derivations of (n). Let Z (n) be the set of all special super derivations of (n). It follows from a straightforward calculation that

Z (n) = spanF {x (i) ξ j D, x (i) ξ j ∂ | 0 ≤ i ≤ pn − 1, j = 0, 1} = Z (n)0¯ ⊕ Z (n)1¯ , where

Z (n)0¯ = spanF {x (i) D, x (i) ξ ∂ | 0 ≤ i ≤ pn − 1} and

Z (n)1¯ = spanF {x (i) ∂, x (i) ξ D | 0 ≤ i ≤ pn − 1}. Moreover, Z (n) is a Lie subsuperalgebra of the derivation superalgebra of (n) under the usual Lie bracket (cf. [21]). Furthermore, it can be shown that Z (n) is a simple Lie superalgebra, which will be referred to as a Zassenhaus superalgebra. The even part of Z (n) contains a subalgebra spanF {x (i) D | 0 ≤ i ≤ pn − 1} that is isomorphic to the usual Zassenhaus algebra (cf. [11, §7.6]). Furthermore, the Zassenhaus superalgebra Z (n) is a restricted Lie superalgebra if and only if n = 1 (cf. [21, Lemma 1.12 and Proposition 1.13 in Chapter 5]). The Zassenhaus superalgebra Z (n) admits a natural Z-grading: Z (n) =

n −1 p

Z (n)[i] ,

i=−1

where

Z (n)[i] = spanF {x (k) ξ l D, x (k) 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:

Z (n) = Z (n)−1 ⊃ Z (n)0 ⊃ · · · ⊃ Z (n)pn −1 , where

Z (n)i =



(2.0.1)

Z (n)[j ] for − 1 ≤ i ≤ pn − 1.

j ≥i

The homogeneous component Z (n)[0] of degree zero is isomorphic to gl(1|1) via the map φ : Z (n)[0] −→ gl(1|1), xD → E11 , ξ ∂ → E22 , x∂ → E12 , ξ D → E21 . Moreover, we have a triangular decomposition Z (n)[0] = n− ⊕ h ⊕ n+ , where n− = Fξ D, n+ = Fx∂, and h = FxD + Fξ ∂.

2.2 Generalized Restricted Lie Superalgebras and Their Generalized χ-Reduced Modules The following definition of generalized restricted Lie superalgebras is a natural generalization of the notion of restricted Lie superalgebras. Definition 2.1 A Lie superalgebra g = g0¯ ⊕ g1¯ is called generalized restricted if the even part g0¯ is a generalized restricted Lie algebra in the sense of [5, Definition 1.1], and the odd

Y.-F. Yao, T. Chaolu

part g1¯ as an adjoint g0¯ -module is a generalized restricted module. More precisely, there exist an ordered basis E = {ei | i ∈ I } of g0¯ , a non-negative |I |-tuple s = (si )i∈I and s ϕ a generalized restricted map ϕs : E → g0¯ sending ei to ei s such that (ad(ei ))p i (y) = ϕ ad(ei s )(y) for any i ∈ I , y ∈ g. Remark 2.2 A restricted Lie superalgebra (L, [p]) is a generalized restricted Lie superalgebra with an arbitrary basis E of L0¯ and the generalized restricted map ϕs = [p]|E and s = (1, 1, · · · , 1). Example 2.3 The Zassenhaus superalgebra Z (n) is a generalized restricted Lie superalgebra in the sense of Definition 2.1, where there is an ordered basis {x (i) D, x (i) ξ ∂ | 0 ≤ i ≤ p n − 1} of g0¯ , and the generalized restricted map ϕs is defined as t

ϕs (x (p ) D) = −x (p

t+1 −p+1)

D for 1 ≤ t ≤ n − 1,

ϕs (xD) = xD, ϕs (ξ ∂) = ξ ∂, and ϕs (x (j ) D) = ϕs (x (k) ξ ∂) = 0 for other j and 1 ≤ k ≤ pn − 1, n

where s = (n, 1, 1, · · · , 1). Indeed, since D p (x (i) ξ j ) = 0 for any 0 ≤ i ≤ p n − 1, j = n 0, 1, it follows that D p = 0. While (ξ ∂ ◦ ξ ∂ · · · ◦ ξ ∂ )(ξ ) = ξ   p times

and (ξ ∂ ◦ ξ ∂ · · · ◦ ξ ∂ )(x) = 0,   p times

= ξ ∂. Moreover, for any k with 1 ≤ k ≤ p n − 1, we have it follows that

(k)  (k) (x ξ ∂) ◦ (x (k) ξ ∂) · · · ◦ (x (k) ξ ∂) (ξ ) = x (k) · x (x) · · · · x (ξ ) = 0   (ξ ∂)p

p times

p times

and



 (x (k) ξ ∂) ◦ (x (k) ξ ∂) · · · ◦ (x (k) ξ ∂) (x) = 0.   p times

Hence = 0. The generalized restricted structure for other basis elements follows from [6, (3.2.2), Page 560] (see also [9, (8), Page 61]). (x (k) ξ ∂)p

Note that any irreducible representation of a generalized restricted Lie superalgebra (g, E, ϕs ) is attached with a unique p-character χ ∈ g∗0¯ . Such a module is indeed a mod

 ule over the generalized χ -reduced enveloping superalgebra Uχ (g, E, ϕs ) of g, where  Uχ (g, E, ϕs ) = U (g)/Jχ , Jχ is the ideal of the universal enveloping superalgebra U (g) psi

ϕ

s

generated by ei − ei s − χ (ei )p i , i ∈ I . The generalized χ -reduced enveloping superalgebra Uχ (g, E, ϕs ) is usually denoted by Uχ (g) for brevity. In the case χ = 0, we call U0 (g) the generalized restricted enveloping superalgebra of g, denoted by u(g) for brevity. s psi ϕ A g-module M is said to be a generalized χ -reduced module if ei · v − ei s · v = χ (ei )p i v for any 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. For convenience, we always regard χ ∈ g∗0¯ as an element in g∗

On Irreducible Representations of the Zassenhaus Superalgebras...

with χ (g1¯ ) = 0. We also make the convention that any linear function χ ∈ g∗ in this paper satisfies χ (g1¯ ) = 0. From now on, we always assume that g = Z (n) is a Zassenhaus superalgebra, and G = Aut(g) is the automorphism group of g. Each automorphism of g is induced from an automorphism of the underlying superalgebra (n) (cf. [3, Theorem 4.2]). For a given pcharacter χ ∈ g∗ , define the height ht(χ ) of χ as ht(χ ) = min{i ≥ −1 | χ (gi ) = 0}. Since the filtration (2.0.1) is invariant under the action of G (cf. [20, Theorem 1], [21, Theorem 3.2.10], [3, Lemma 3.1]), the height of a p-character χ is invariant under the coadjoint action

 ∼ (g, E, ϕ of G. Moreover, each σ ∈ G induces an isomorphism of superalgebras: U χ s) =  σ σ σ Uσ ·χ (g, E , ϕs ) , where E = {σ (x) | x ∈ E} is another ordered basis of g0¯ , and ϕsσ : E σ −→ g0¯ with ϕsσ (σ (x)) = σ ϕs σ −1 ((σ (x))) = σ (ϕs (x)), ∀ x ∈ E. As a result, the generalized χ -reduced representation theory of g depends only on the G-orbit of χ . That is to say that the study of the generalized χ -reduced representations of g coincides with the study of the generalized (σ · χ )-reduced representations for any σ ∈ G. Note that each simple g-module is attached with a unique p-character χ ∈ g∗ , all simple gmodules split into different classes, and all simple objects in each class admit p-characters of the same height. As a sequel to our previous work [18], we classify in this paper the isomorphism classes of simple g-modules with p-characters χ of height 0, and determine their dimensions. Moreover, we give a sufficient and necessary condition for a Kac module to be irreducible.

2.3 Irreducible Restricted Representations of g[0] Since each element h in h is semisimple, h acts semisimply on each object in the category of finite dimensional restricted representations of g[0] (cf. [12, Proposition 3.3(2), Chapter 2]). That is to say that each finite dimensional restricted g[0] -module M is a weight module, i.e., M = λ∈h∗ Mλ , where Mλ = {m ∈ M | h · m = λ(h)m, ∀ h ∈ h}. Moreover, since M is a restricted module, we have λ(h)p = λ(h[p] ) for any weight λ of M and h ∈ h. Hence, λ(xD) ∈ Fp and λ(ξ ∂) ∈ Fp . We regard such weights as restricted weights. Let

:= {λ = (λ1 , λ2 ) | λi ∈ Fp , i = 1, 2} be the set of restricted weights, which coincides with F2p . Then the isomorphism classes of restricted simple modules of g[0] , denoted by L0 (λ), are parameterized by (cf. [14, 22]). More precisely, when λ = (λ1 , −λ1 ) ∈ , the simple restricted g[0] -module L0 (λ) is one-dimensional with a basis vλ such that x∂ · vλ = ξ D · vλ = 0, and xD · vλ = λ1 vλ , ξ ∂ · vλ = −λ1 vλ .

(2.3.1)

While when λ = (λ1 , λ2 ) ∈ with λ1 = −λ2 , the simple restricted g[0] -module L0 (λ) is two-dimensional, and has a basis {v1 , v2 } with the action of g[0] defined as follows: xD · v1 = λ1 v1 , ξ ∂ · v1 = λ2 v1 , xD · v2 = (λ1 − 1)v2 , ξ ∂ · v2 = (λ2 + 1)v2 ,

(2.3.2)

and x∂ · v1 = 0, ξ D · v1 = v2 , x∂ · v2 = (λ1 + λ2 )v1 , ξ D · v2 = 0.

(2.3.3)

3 Representations of Zassenhaus Superalgebras with p-Characters of Height 0 In this section we study the structure of simple modules of the Zassenhaus superalgebra g = Z (n) with p-characters of height 0. We will classify the isomorphism classes of these simple modules. Let χ ∈ g∗ with ht(χ ) = 0, i.e., χ (g0 ) = 0 and χ (g[−1] )  = 0. Without loss

Y.-F. Yao, T. Chaolu

of generality (see the last paragraph in Section 2.2), under the action of the automorphism group of g, we can assume that χ (D) = 1 and χ (∂) = χ (g0 ) = 0. We always make this assumption in this section. Then each simple Uχ (g0 )-module is a restricted g[0] -module L0 (λ) with trivial g1 -action for some λ = (λ1 , λ2 ) ∈ . We define the generalized χ reduced Kac g-module Kχ (λ) as the induced Uχ (g)-module Uχ (g) ⊗u(g0 ) L0 (λ). We have the following easy observation. Lemma 3.1 Each simple Uχ (g)-module is a quotient of a generalized χ -reduced Kac gmodule Kχ (λ) for some λ ∈ . Proof Let N be a simple Uχ (g)-module. Regard N as a module over Uχ (g0 ) = u(g0 ), and take a simple u(g0 )-submodule N0 . Then we have an isomorphism ϕ : L0 (λ) → N0 for some λ ∈ . The universal property of tensor products implies that ϕ induces the following g-module homomorphism  : Kχ (λ) = Uχ (g) ⊗u(g0 ) L0 (λ) −→ N D i ∂ j ⊗ v −→ D i ∂ j · ϕ(v). Since N is simple,  is surjective, i.e., N is a quotient of Kχ (λ), as desired. For λ ∈ , a nonzero vector w in a g-module W is said to be a singular vector of weight λ if h · w = λ(h)w and u · w = 0 for any h ∈ h and u ∈ n+ + g1 . In the following, for λ = (λ1 , λ2 ) ∈ , we determine when a generalized χ -reduced Kac g-module Kχ (λ) is irreducible by analyzing the structure of weight vectors.

3.1 Type I: λ1 = −λ2 In this case, L0 (λ) = Fvλ is one-dimensional, and Kχ (λ) has a basis {D i ∂ j ⊗ vλ | 0 ≤ i ≤ p n − 1, j = 0, 1}. Let M be a nonzero submodule of Kχ (λ). Let v ∈ M be a nonzero weight vector with respect to the Cartan subalgebra h. Note that ξ ∂ · D i ⊗ vλ = λ2 D i ⊗ vλ

for 0 ≤ i ≤ pn − 1

and ξ ∂ · D j ∂ ⊗ vλ = (λ2 − 1)D j ∂ ⊗ vλ for 0 ≤ j ≤ pn − 1.

We can assume that v = ai D i ∂ ⊗ vλ without loss of generality. Let = {i | ai  = 0} and i0 = max{i | i ∈ }. Since v is a weight vector, each summand D i ∂ ⊗ vλ in the expression of v has the same weight. Consequently, i ≡ j (mod p), ∀ i, j ∈ . Note that  1 pn −1−i0 n D · v = D p −1 ∂ ⊗ vλ + bj D j ∂ ⊗ vλ ∈ M. ai0 n j ≤p −p−1

We can assume i0 = may occurs.

p n − 1 and ai0

= 1 without loss of generality. The following two cases

Case (i): | | = 1. n n n In this case, v = D p −1 ∂ ⊗ vλ . Then D · v = D p ∂ ⊗ vλ = χ (D)p ∂ ⊗ vλ = ∂ ⊗ vλ ∈ n n M. Hence, ξ D ·∂ ⊗vλ = [ξ D, ∂]⊗vλ = D ⊗vλ ∈ M and D p −1 ·D ⊗vλ = D p ⊗vλ = n χ (D)p ⊗ vλ = 1 ⊗ vλ ∈ M. Consequently, M = Kχ (λ). Case (ii): | | > 1.

On Irreducible Representations of the Zassenhaus Superalgebras...

In this case, x (p = x = = = =

n −1)

(pn −1)

D·v n −1

∂ ⊗ vλ pn −1 (pn −1) RD (x D)∂ ⊗ vλ pn −1 (pn −1) (LD − adD) (x D)∂ ⊗ vλ n −1   n p p − 1 pn −1−i n LD (−1)i (adD)i (x (p −1) D)∂ i i=0 n −1 p

D · Dp

 (−1)

i

i=0

⊗ vλ

 pn − 1 n n D p −1−i (x (p −1−i) D)∂ ⊗ vλ i

= (1 + λ1 )D∂ ⊗ vλ ∈ M, where LD : Uχ (g) −→ Uχ (g) y −→ Dy, ∀ y ∈ Uχ (g) and RD : Uχ (g) −→ Uχ (g) y −→ yD, ∀ y ∈ Uχ (g) are the left and right multiplications by D, respectively. If λ1 = −1, then D∂ ⊗ vλ ∈ M, so that D p −1 · D∂ ⊗ vλ = ∂ ⊗ vλ ∈ M. A similar argument as in Case (i) yields that 1 ⊗ vλ ∈ M. Hence, M = Kχ (λ). If λ1 = −1, let

n n i1 = max{i | i ∈ \i0 }. Then a1i D p −2−i1 ·v = D p −2 ∂ ⊗vλ + cj D j ∂ ⊗vλ ∈ M. n

j ≤pn −p−2

1

It follows that x (p

n −1)

D·(

1 pn −2−i1 n n D · v) = x (p −1) D · (D p −2 ∂ ⊗ vλ ) ai1 n −2   n p n n i p −2 D p −2−i (x (p −1−i) D)∂ ⊗ vλ (−1) = i i=0

= −∂ ⊗ xD · vλ = ∂ ⊗ vλ ∈ M. By a similar argument as in Case (i), we also get M = Kχ (λ). Consequently, the generalized χ -reduced Kac g-module Kχ (λ) is irreducible for all λ = (λ1 , −λ1 ) ∈ . Moreover, for λ = (λ1 , −λ1 ), μ = (μ1 , −μ1 ) ∈ , we claim that Kχ (λ) is isomorphic to Kχ (μ) if and only if λ = μ. Indeed, suppose ϕ : Kχ (λ) −→ Kχ (μ) is an

isomorphism map, and ϕ(1 ⊗ vλ ) = ai D i ⊗ vμ + bi D i ∂ ⊗ vμ . Then 0 = ϕ(1 ⊗ ξ D · vλ ) = ϕ(ξ D · (1 ⊗ vλ )) = ξ D · ϕ(1 ⊗ vλ ) = bpn −1 ⊗ vμ +

n −2 p

bi D i+1 ⊗ vμ .

i=0

It follows that bi = 0 for 0 ≤ i ≤

pn

− 1. Furthermore, since

0 = ϕ(1 ⊗ x∂ · vλ ) = ϕ(x∂ · (1 ⊗ vλ )) = x∂ · ϕ(1 ⊗ vλ ) =

n −1 p

i=1

−iai D i−1 ∂ ⊗ vμ ,

Y.-F. Yao, T. Chaolu

it follows that ai = 0 for 1 ≤ i ≤ p n − 1 and i ≡ 0 (mod p). Hence, ϕ(1 ⊗ vλ ) =

pn−1 −1

apj D pj ⊗ vμ .

j =0

Let l = max{pj | apj = 0}. We claim that l = 0. Otherwise, suppose l > 0, then 0 = ϕ(1⊗x D ·vλ ) = x D ·ϕ(1⊗vλ ) = (l)

(l)

pn−1 −1

apj (x (l) D)·D pj ⊗vμ = (−1)l al D ⊗vμ ,

j =0

which is a contradiction. Therefore, l = 0, i.e., ϕ(1 ⊗ vλ ) = a0 1 ⊗ vμ . By comparing the weights of 1 ⊗ vλ and 1 ⊗ vμ , we get λ = μ, as desired. In conclusion, we obtain the following result. Proposition 3.2 The generalized χ -reduced Kac g-module Kχ (λ) is irreducible for λ = (λ1 , −λ1 ) ∈ . Moreover, for μ = (μ1 , −μ1 ) ∈ , the two generalized χ -reduced Kac g-modules Kχ (λ) and Kχ (μ) are isomorphic if and only if λ = μ.

3.2 Type II: λ1 = −λ2 In this case, the simple restricted g[0] -module L0 (λ) is two-dimensional, and has a basis {v1 , v2 } with the g[0] -action defined as in Eqs. 2.3.2 and 2.3.3. Then the generalized χ reduced Kac g-module Kχ (λ) has a basis {D i ∂ j ⊗ vk | 0 ≤ i ≤ pn − 1, j = 0, 1, k = 1, 2}. Let M be a nonzero submodule of Kχ (λ). Let 0 = w ∈ M be a weight vector. Then by a direct analysis of the weight of D i ∂ j ⊗ vk and by applying appropriate action of D s and ∂ t to w, we can assume that  n ai D i ∂ ⊗ v1 w = D p −1 ∂ ⊗ v1 + i≤pn −p−1

or w = Dp

n −1



∂ ⊗ v2 +

bi D i ∂ ⊗ v2 .

i≤pn −p−1

We divide the following discussion into three cases. Case (i): If

λ1 = −1, 0. w = Dp

n −1



∂ ⊗ v1 +

ai D i ∂ ⊗ v1 ∈ M,

i≤pn −p−1

then x (p

n −1)

D · D p −1 ∂ ⊗ v1 n   n p −1 n n j p −1 D p −1−j (x (p −1−j ) D)∂ ⊗ v1 (−1) = j

D · w = x (p

n −1)

n

j =0

= (1 + λ1 )D∂ ⊗ v1 ∈ M. Since λ1 = −1, we have D∂ ⊗ v1 ∈ M. If n w = D p −1 ∂ ⊗ v2 +

 i≤pn −p−1

bi D i ∂ ⊗ v2 ∈ M,

(3.2.1)

On Irreducible Representations of the Zassenhaus Superalgebras...

then by a similar computation as in Eq. 3.2.1, we have x (p −1) ∂ ·w = (λ1 +λ2 )D∂ ⊗v1 ∈ M. Since λ1 = −λ2 , we also get D∂ ⊗ v1 ∈ M. n Hence, D p −1 · D∂ ⊗ v1 = ∂ ⊗ v1 ∈ M, and xξ D · ∂ ⊗ v1 = xD ⊗ v1 = λ1 ⊗ v1 ∈ M. Since λ1 = 0, it follows that 1 ⊗ v1 ∈ M. This implies that M = Kχ (λ). Consequently, Kχ (λ) is irreducible. Case (ii): λ1 = −1. If  n ai D i ∂ ⊗ v1 ∈ M with ai = 0, ∀ i ≤ pn − p − 1, w = D p −1 ∂ ⊗ v1 + n

i≤pn −p−1

then D · w = D · D D

pn −1

pn −2−i

0

∂ ⊗ v1 = ∂ ⊗ v1 ∈ M. Otherwise, let i0 = max{i | ai  = 0}. Then  n · w = ai0 D p −2 ∂ ⊗ v1 + cj D j ∂ ⊗ v1 ∈ M. j ≤pn −p−2

Hence, x (p

n −1)

D · (D p

n −2−i

0

D · D p −2 ∂ ⊗ v1 n   n p −2 p −2 n n D p −2−k (x (p −1−k) D)∂ ⊗ v1 = ai0 (−1)k k

· w) = ai0 x (p

n −1)

n

k=0

= ai0 ∂ ⊗ v1 ∈ M. Consequently, we also get ∂ ⊗ v1 ∈ M. Moreover, since xξ D · ∂ ⊗ v1 = [xξ D, ∂] ⊗ v1 = xD ⊗ v1 = −1 ⊗ v1 ∈ M, it follows that M = Kχ (λ). For the case that  n w = D p −1 ∂ ⊗ v2 +

bi D i ∂ ⊗ v2 ∈ M,

i≤pn −p−1

by a similar argument, we also get M = Kχ (λ). Therefore, Kχ (λ) is irreducible. Case (iii): λ1 = 0. If  n bi D i ∂ ⊗ v2 ∈ M with bi = 0, ∀ i ≤ pn − p − 1, w = D p −1 ∂ ⊗ v2 + i≤pn −p−1

then D · w = D · D D

pn −1

pn −2−i0

∂ ⊗ v2 = ∂ ⊗ v2 ∈ M. Otherwise, let i0 = max{i | bi  = 0}. Then  n · w = bi0 D p −2 ∂ ⊗ v2 + dj D j ∂ ⊗ v2 ∈ M. j ≤pn −p−2

Hence, x (p

n −1)

D · (D p

n −2−i

0

D · D p −2 ∂ ⊗ v2 n −2   n p p −2 n n D p −2−j (x (p −1−j ) D)∂ ⊗ v2 = bi0 (−1)j j

· w) = bi0 x (p

n −1)

n

j =0

= bi0 ∂ ⊗ v2 ∈ M. Consequently, we also get ∂ ⊗ v2 ∈ M. Moreover, since xξ D · ∂ ⊗ v2 = [xξ D, ∂] ⊗ v2 = xD ⊗ v2 = −1 ⊗ v2 ∈ M, it follows that M = Kχ (λ).

Y.-F. Yao, T. Chaolu

If w = Dp

n −1

∂ ⊗ v1 +



ai D i ∂ ⊗ v1 ∈ M

with ai = 0, ∀ i ≤ pn − p − 1,

i≤pn −p−1

then D · w = D · D D

pn −1

pn −2−j

0

∂ ⊗ v1 = ∂ ⊗ v1 ∈ M. Otherwise, let j0 = max{i | ai  = 0}. Then  n · w = aj0 D p −2 ∂ ⊗ v1 + lj D j ∂ ⊗ v1 ∈ M. j ≤pn −p−2

Hence, x (p

n −2)

D · (D p

n −2−j

0

D · D p −2 ∂ ⊗ v1 n −2   n p p −2 n n D p −2−j (x (p −2−j ) D)∂ ⊗ v1 = a j0 (−1)j j

· w) = aj0 x (p

n −2)

n

j =0

Consequently,

− a1j 0

n D p −1

= −aj0 D∂ ⊗ v1 ∈ M.

(pn −2)  n · x D · (D p −2−j0 · w) = ∂ ⊗ v1 ∈ M. This implies

that D k ∂ ⊗ v1 ∈ M for 0 ≤ k ≤ pn − 1. Moreover, since ξ D · (∂ ⊗ v1 ) = [ξ D, ∂] ⊗ v1 − ∂ ⊗ ξ D · v1 = D ⊗ v1 − ∂ ⊗ v2 ∈ M, it follows that D l+1 ⊗ v1 − D l ∂ ⊗ v2 ∈ M for 0 ≤ l ≤ p n − 1. In particular, 1 ⊗ v1 − n D p −1 ∂ ⊗ v2 ∈ M. Let U = spanF {D k ∂ ⊗ v1 , D l+1 ⊗ v1 − D l ∂ ⊗ v2 , 1 ⊗ v1 − D p

n −1

∂ ⊗ v2 | 0 ≤ k ≤ p n − 1, 0 ≤ l ≤ p n − 2}. (3.2.2)

Then U ⊆ M. By a similar argument as in the proof of [18, Proposition 3.9], one can show that the subspace U is a g-submodule. Moreover, it is the unique minimal and the unique maximal submodule of Kχ (λ). Hence, Kχ (λ) has a unique simple quotient Kχ (λ)/U . For λ = (λ1 , λ2 ) ∈ , it follows from the above discussion that the generalized χ -reduced Kac g-module Kχ (λ) has a unique maximal submodule, and we obtain the following result. Theorem 3.3 For λ = (λ1 , λ2 ) ∈ , let  Kχ (λ)/U , if λ1 = 0 and λ2  = 0, Lχ (λ) = Kχ (λ), otherwise, where U is defined in Eq. 3.2.2. Then Lχ (λ) is an irreducible Uχ (g)-module with a singular vector of weight λ.

3.3 Isomorphism Classes of Simple Modules with p-Character of Height 0 In this subsection, we determine the isomorphism classes of the simple modules Lχ (λ) for λ = (λ1 , λ2 ) ∈ . Lemma 3.4 Let λ = (λ1 , λ2 ), μ = (μ1 , μ2 ) ∈ with λ1  = −λ2 , μ1  = −μ2 , and λ1 = 0, μ1 = 0. Then Kχ (λ) ∼ = Kχ (μ) if and only if λ = μ.

On Irreducible Representations of the Zassenhaus Superalgebras...

Proof Note that Kχ (λ) = spanF {D i ∂ j ⊗ vk | 0 ≤ i ≤ pn − 1, j = 0, 1, k = 1, 2} and Kχ (μ) = spanF {D i ∂ j ⊗ wk | 0 ≤ i ≤ p n − 1, j = 0, 1, k = 1, 2}, where {v1 , v2 } and {w1 , w2 } are bases of the simple restricted g[0] -modules L0 (λ) and L 0 (μ), respectively. Suppose ϕ : Kχ (λ) −→ Kχ (μ) is an isomorphism and ϕ(1 ⊗ v1 ) = aij k D i ∂ j ⊗ wk . is a weight vector and ∂ · (1 ⊗ v1 ) = ∂ ⊗ v1  = 0, we can assume that Since 1 ⊗ v1 i i i−1 ϕ(1 ⊗ v1 ) = bi D

⊗ wi2 or ϕ(1 ⊗ v1 ) = (ci D ⊗ w1 + di D ∂ ⊗ w2 ). bi D ⊗ w2 . Suppose that there exists some i0 with 1 < i0 < p n − 1 If ϕ(1 ⊗ v1 ) = such that bi0 = 0, then 0 = ϕ(x (i0 +1) D · (1 ⊗ v1 ))  b i D i ⊗ w2 = x (i0 +1) D ·



= bi0 x (i0 +1) D · D i0 ⊗ w2 +

bj x (i0 +1) D · D j ⊗ w2

j ≥i0 +p

=

i bi0 RD0 (x (i0 +1) D) ⊗ w2



+

bj RD (x (i0 +1) D) ⊗ w2 j

j ≥i0 +p

= bi0 (LD − adD) (x i0

(i0 +1)

D) ⊗ w2 +



bj (LD − adD)j (x (i0 +1) D) ⊗ w2

j ≥i0 +p

= bi0

  i0 i0 −k LD (adD)k (x (i0 +1) D) ⊗ w2 (−1)k k

i0  k=0

  j   j j −k (−1)k bj LD (adD)k (x (i0 +1) D) ⊗ w2 k j ≥i0 +p k=0    j = (−1)i0 bi0 xD ⊗ w2 + bj ((−1)i0 D j −i0 xD ⊗ w2 i0 j ≥i0 +p   j D j −i0 ⊗ w2 ) +(−1)i0 +1 i0 + 1      j +1 j i0 i0 D j −i0 ⊗ w2 . = (−1) bi0 (μ1 − 1) ⊗ w2 + (−1) bj μ1 − i0 + 1 i0 +

j ≥i0 +p

Hence, μ1 = 1. Moreover, by a similar argument, we have 0 = ϕ(x (i0 ) D · (1 ⊗ v1 ))  b i D i ⊗ w2 = x (i0 ) D · = bi0 x (i0 ) D · D i0 ⊗ w2 +



bj x (i0 ) D · D j ⊗ w2

j ≥i0 +p

= (−1) bi0 D ⊗ w2 + (−1)i0 −1 i0 bi0 D(xD) ⊗ w2 + i0



bj D j −i0 +1 ⊗ w2

j ≥i0 +p

= (−1) bi0 D ⊗ w2 + (−1) i0

= (−1) bi0 D ⊗ w2 + i0

i0 −1

 j ≥i0 +p

i0 bi0 (μ1 − 1)D ⊗ w2 +



j ≥i0 +p

bj D j −i0 +1

⊗ w2 ,

bj D j −i0 +1 ⊗ w2

Y.-F. Yao, T. Chaolu

which implies that bi0 = 0, a contradiction. Hence, ϕ(1 ⊗ v1 ) = b0 1 ⊗ w2 , b1 D ⊗ w2 , or n bpn −1 D p −1 ⊗ w2 . If ϕ(1 ⊗ v1 ) = b0 1 ⊗ w2 , then 0 = ϕ(x∂ · (1 ⊗ v1 )) = x∂ · ϕ(1 ⊗ v1 ) = b0 ⊗ x∂ · w2 = b0 (μ1 + μ2 ) ⊗ w1 , which is a contradiction. If ϕ(1 ⊗ v1 ) = b1 D ⊗ w2 , then 0 = ϕ(x∂ · 1 ⊗ v1 ) = x∂ · ϕ(1 ⊗ v1 ) = b1 x∂ · D ⊗ w2 = b1 (μ1 + μ2 )D ⊗ w1 − b1 ∂ ⊗ w2 , which is a contradiction. If ϕ(1 ⊗ v1 ) = bpn −1 D p 0 = ϕ(x (p = x

=

⊗ w2 , then

D · (1 ⊗ v1 ))

D · ϕ(1 ⊗ v1 )

(pn −1)

pn −1

= bpn −1

x

n −1 p

D · Dp

n −1

⊗ w2 pn −1 (pn −1) bpn −1 RD (x D) ⊗ w2 n −1 n p bpn −1 (LD − adD) (x (p −1) D) ⊗ w2

= b =

n −1)

(pn −1)

n −1



(−1)j

j =0

 p n − 1 pn −1−j n LD (adD)j (x (p −1) D) ⊗ w2 j

= bpn −1 D ⊗ w2 + bpn −1 D(xD) ⊗ w2 = bpn −1 μ1 D ⊗ w2 , which is a contradiction. Therefore ϕ(1 ⊗ v1 ) = c0 1 ⊗ w1 + d0 D

pn −1

∂ ⊗ w2 +

n −1 p

(ci D i ⊗ w1 + di D i−1 ∂ ⊗ w2 ).

i=1

We first claim that there exists some i such that ci = 0. Otherwise, ϕ(1 ⊗ v1 ) = d0 D p

n −1

∂ ⊗ w2 +

n −1 p

di D i−1 ∂ ⊗ w2 ,

i=1

then ϕ(∂ ⊗v1 ) = ϕ(∂ ·(1⊗v1 )) = ∂ ·ϕ(1⊗v1 ) = ∂ ·(d0 D p

n −1

∂ ⊗w2 +

n −1 p

di D i−1 ∂ ⊗w2 ) = 0,

i=1

which is absurd. Let j0 = max{i | ci = 0}. If 0 < j0 < p n − 1, then 0 = ϕ(x (j0 +1) D · (1 ⊗ v1 )) = x (j0 +1) D · ϕ(1 ⊗ v1 ) = cj0 x (j0 +1) D · D j0 ⊗ w1 + d0 x (j0 +1) D · D p  = (−1)j0 cj0 λ1 ⊗ w1 + dj D j ∂ ⊗ w2 ,

n −1

∂ ⊗ w2 +

which is absurd, since λ1 = 0. If j0 = p n − 1, then ϕ(1⊗v1 ) = (cpn −1 D p

n −1

⊗w1 +dpn −1 D p

n −2

∂⊗w2 )+





i≤pn −p−1

di x (j0 +1) D ·D i−1 ∂ ⊗ w2

(ci D i ⊗w1 +di D i−1 ∂⊗w2 ).

On Irreducible Representations of the Zassenhaus Superalgebras...

We have 0 = ϕ(x (p = x

n −1)

(pn −1)

D · (1 ⊗ v1 ))

D · ϕ(1 ⊗ v1 ))

= cpn −1 x (p

n −1)

D · Dp

n −1

⊗ w1 + dpn −1 x (p

n −1)

D · Dp

n −2

∂ ⊗ w2

= cpn −1 (μ1 + 1)D ⊗ w1 + dpn −1 (1 − μ1 )∂ ⊗ w2 . This implies that μ1 = −1 and dpn −1 = 0. Moreover, since 0 = ϕ(x∂ · (1 ⊗ v1 )) = x∂ · ϕ(1 ⊗ v1 ) = cpn −1 x∂ · D p −1 ⊗ w1 + dpn −1 x∂ · D p −2 ∂ ⊗ w2  + (ci x∂ · D i ⊗ w1 + di x∂ · D i−1 ∂ ⊗ w2 ) n

n

i≤pn −p−1

= (cpn −1 − dpn −1 (μ1 + μ2 ))D p −2 ∂ ⊗ w1  − (ici + di (μ1 + μ2 ))D i−1 ∂ ⊗ w1 , n

i≤pn −p−1

it follows that cpn −1 = dpn −1 (μ1 + μ2 ) = 0, a contradiction. Hence, j0 = 0, i.e., ϕ(1 ⊗ n v1 ) = c0 1 ⊗ w1 + d0 D p −1 ∂ ⊗ w2 . Note that 0 = ϕ(x∂ · (1 ⊗ v1 )) = x∂ · ϕ(1 ⊗ v1 ) = c0 ⊗ x∂ · w1 + d0 x∂ · D p = −d0 (μ1 + μ2 )D

pn −1

n −1

∂ ⊗ w2

∂ ⊗ w1 ,

which implies that d0 = 0, since μ1 = −μ2 . Hence, ϕ(1 ⊗ v1 ) = c0 1 ⊗ w1 . Now by comparing the weights of 1 ⊗ v1 and 1 ⊗ w1 , we get λ = μ. Lemma 3.5 Let λ = (0, λ2 ), μ = (0, μ2 ) ∈ with λ2  = 0, μ2  = 0. Then Lχ (λ) ∼ = Lχ (μ) if and only if λ = μ. Proof We note that Lχ (λ) = Kχ (λ)/U , where U is defined as in Eq. 3.2.2. Similarly, Lχ (μ) is a quotient of Kχ (μ). Then by the definition of U in Eq. 3.2.2, we see Lχ (λ) = spanF {D i ⊗ vj | 0 ≤ i ≤ p n − 1, j = 1, 2} and Lχ (μ) = spanF {D i ⊗ wj | 0 ≤ i ≤ p n − 1, j = 1, 2}, where {v1 , v2 } and {w1 , w2 } are bases of the simple restricted g[0] modules L0 (λ) and L0 (μ), respectively. Suppose ϕ : Lχ (λ) −→ Lχ (μ) is an isomorphism

and ϕ(1 ⊗ v1 ) = aij D i ⊗ wj . Since ∂ · 1 ⊗ v1 = ∂ ⊗ v1 = 0, we can assume that

i ϕ(1 ⊗ v1 ) = bi D ⊗ w1 . Let i0 = max{i | bi = 0}. If i0 > 1, we have 0 = ϕ(x (i0 ) D · 1 ⊗ v1 ) = x (i0 ) D · ϕ(1 ⊗ v1 ) = bi0 x (i0 ) D · D i0 ⊗ w1 = (−1)i0 bi0 D ⊗ w1 which is a contradiction. This implies that i0 = 0 or 1. If i0 = 1, i.e., ϕ(1 ⊗ v1 ) = b0 1 ⊗ w1 + b1 D ⊗ w1 with b1 = 0, then 0 = ϕ(xξ D · 1 ⊗ v1 ) = xξ D · ϕ(1 ⊗ v1 ) = b1 xξ D · D ⊗ w1 = −b1 1 ⊗ w2 which is a contradiction. Hence, i0 = 0, i.e., ϕ(1 ⊗ v1 ) = b0 1 ⊗ w1 . By comparing the weights of 1 ⊗ v1 and 1 ⊗ w1 , we get λ = μ, as desired.

Y.-F. Yao, T. Chaolu

Lemma 3.6 Let λ = (0, λ2 ), μ = (μ1 , −μ1 ) ∈ with λ2  = 0. Then Lχ (λ) ∼ = Lχ (μ) if and only if λ2 = −1 and μ1 = 0. Proof Note that Lχ (λ) = spanF {D i ⊗ wj | 0 ≤ i ≤ p n − 1, j = 1, 2} and Lχ (μ) = Kχ (μ) = spanF {D i ⊗ vμ , D i ∂ ⊗ vμ | 0 ≤ i ≤ pn − 1} where {w1 , w2 } and {vμ } are bases of the simple restricted g[0] -modules L0 (λ) and L0 (μ), respectively. Suppose Lχ (μ) ∼ = Lχ (λ) and ϕ : Lχ (μ) −→ Lχ (λ) is an isomorphism. Since 1 ⊗ vμ is a weight vector

and ∂ · (1 ⊗ vμ ) = ∂ ⊗ vμ = 0, we can assume that ϕ(1 ⊗ vμ ) = ai D i ⊗ w2 . Let i0 = max{i | ai = 0}. If 0 < i0 < p n − 1, we have 0 = ϕ(x (i0 +1) D · 1 ⊗ vμ ) = x (i0 +1) D · ϕ(1 ⊗ vμ ) = ai0 x (i0 +1) D · D i0 ⊗ w2 = (−1)i0 +1 ai0 1 ⊗ w2

which is a contradiction. If i0 = 0, then ϕ(1 ⊗ vμ ) = a0 1 ⊗ w2 . By comparing the weights of 1 ⊗ vμ and 1 ⊗ w2 , we get (μ1 , −μ1 ) = (−1, λ2 + 1). It follows that λ2 = 0, a contradiction. Hence, i0 = p n − 1, i.e., ϕ(1 ⊗ vμ ) = apn −1

n D p −1

⊗ w2 +

pn−1 −1

apn −pj −1 D p

n −pj −1

⊗ w2 .

j =1

By comparing the weights of 1 ⊗ vμ and ϕ(1 ⊗ vμ ), we get (μ1 , −μ1 ) = (0, λ2 + 1). It follows that μ1 = 0 and λ2 = −1. Moreover, it is easy to check by a direct computation that x (i) D · D p

n −1

⊗ w2 = x (i) ξ ∂ · D p

n −1

⊗ w2 = 0 for i > 1

and ξ D · Dp

n −1

⊗ w2 = x (j ) ξ D · D p

n −1

⊗ w2 = x (j ) ∂ · D p

n −1

⊗ w2 = 0 for j ≥ 1.

p −1 ⊗ w gives a g -module homomorphism between This 2 0

implies  that ψ : cvμ − → cD it induces a nontrivial g-module homomorL0 (0, 0) and Lχ (0, −1) . Consequently,

 phism  : Lχ (0, 0) −→ Lχ  (0, −1) by the universal property of tensor products.

 ∼ Since

Lχ (0,  0) and Lχ (0, −1) are irreducible,  is an isomorphism, i.e., Lχ (0, 0) = Lχ (0, −1) , as desired. This completes the proof. n

3.4 Main Results We conclude our paper by summarizing our results on the irreducible Z (n)-modules with a p-character of height zero in the following theorem. Its proof is an immediate consequence of Lemma 3.1, Proposition 3.2, and Lemmas 3.4–3.6. Theorem 3.7 Let g = Z (n) be a Zassenhaus superalgebra over an algebraically closed field F of characteristic p > 3. Let χ ∈ g∗ with ht(χ ) = 0 and χ (D) = 1. For λ = (λ1 , λ2 ) ∈ , the generalized χ -reduced Kac g-module Kχ (λ) is irreducible if and only if λ1 = −λ2 , or λ1 = −λ2 and λ1 = 0. Moreover, U = spanF {D k ∂ ⊗v1 , D l+1 ⊗v1 −D l ∂ ⊗v2 , 1⊗v1 −D p

n −1

∂ ⊗v2 | 0 ≤ k ≤ pn −1, 0 ≤ l ≤ pn −2}

On Irreducible Representations of the Zassenhaus Superalgebras...

is the unique nonzero proper submodule of Kχ (λ) for λ = (0, λ2 ) ∈ with λ2  = 0. Furthermore, {Lχ (λ) | λ ∈ \ {(0, 0)}} exhausts all non-isomorphic irreducible generalized χ -reduced g-modules. The dimensions of the simple modules are listed as follows:  n 2p , if λ1 = −λ2 or λ1 = 0, dim Lχ (λ) = 4p n , otherwise. Acknowledgements The authors would like to express their sincere gratitude to the referee for his/her valuable suggestions and comments which help us to improve the exposition of this paper.

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