Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.


Introduction
Let F be an arbitrary eld of characteristic p > and Z = {¯ ,¯ } be the residue class ring mod . Throughout this paper we have assumed that all vector spaces, linear mappings and tensor products are over the underlying base eld F. Assume that x is a Z -homogeneous element and d(x) is the Z -degree of x, if d(x) occurs in an expression.
In 1967, Rudakov and Shafarevich [1] described all the irreducible representations of sl( ) over an algebraically closed eld F of characteristic p > . They demonstrated that in addition to the p-representations known since 1930s, all of which possess a highest and lowest weight and are labeled by one integer, there are other representations that form a variety of dimension . They described the g-modules not possessing a p-structure for Lie algebras g with Cartan matrix. In 1974, Rudakov [2] described irreducible g-modules, where g is a simple Lie algebras of vector elds over C, for modules dual to modules of (formal) tensor elds. For a review of similar results and the importance of this particular type of module, we refer the readers to the papers [3,4]. In the 1980s, Krylyuk [5,6] studied the highest weight modules over the algebras of vector elds of series W and S possessing a p-structure. Shu [7] discussed the representations of Cartan type Lie algebras in characteristic p > from the viewpoint of reducing rank. Zhang [8] constructed the simple L-modules with nonsingular characters and some simple modules with singular characters, where L is a restricted simple Lie algebra of Cartan type.
Since the classi cation of all the nite-dimensional simple complex Lie superalgebras was done by Kac [9], the problems of constructing a uni ed representation theory for all the types of simple Lie superalgebras has become more important than ever. Kac obtained essential results for the highest weight representations of classical Lie superalgebras [10,11]. Most of Kac's results can also be extended to the remaining classical series of Lie superalgebras [12][13][14], while the representations of Lie superalgebras of Cartan type have been studied in [15,16]. Recent work on the representation theory of modular Lie superalgebras of Cartan type can also be found in [17][18][19].
The structure of gradation plays a critical role in the research of Lie algebras and superalgebras. Shen [20][21][22] introduced an important notion which is called the mixed product and realized the graded modules over Lie algebras of Cartan type. The method of the mixed product can also be applied to Lie superalgebras of Cartan type over elds of characteristic zero [23]. In the case of modular Lie superalgebras, Zhang [24] has obtained the Z-graded modules over nite-dimensional Hamiltonian Lie superalgebras.
This paper generalizes some of Shen's results in [20][21][22]. A brief summary of the relevant concepts in generalized Witt and special modular Lie superalgebras is presented in Section 2. Section 3 gives some properties of the graded modules over modular Lie superalgebras. In Section 4, the certain connection which is called a P-expansion between irreducible highest weight representations of generalized Witt and special modular Lie superalgebras, and the same irreducible highest weight representations of general linear Lie superalgebras gl(m, n) and special linear Lie superalgebras sl(m, n), is established.

Generalized Witt and special modular Lie superalgebras
In addition to the standard notation Z, N and N is used for the set of positive integers and the set of nonnegative integers, respectively. Generally, let m, n denote xed integers in N \ { , }. For α = (α , . . . , αm) ∈ N m , we put |α| := m i= α i . Following [25], let O(m) denote the divided power algebra over F with an F-basis x i x j = −x j x i for i, j = m + , . . . , m + n, where ∂/∂xr is the superderivation of Λ(n) such that ∂xs /∂xr = δrs for r, s ∈ Y . For more details on superderivations for Lie superalgebras, the reader is referred to [9,26] An easy veri cation shows that , n)). In particular, the following formula holds in W(m, n): Then W(m, n, t) is called the generalized Witt modular Lie superalgebra and it is a subalgebra of W(m, n). In particular, it is a nite-dimensional simple Lie superalgebra (see [27]). Clearly, W(m, n, t) is a free O(m, n, t)-module with basis {Dr | r ∈ Y}. Note that W(m, n, t) possesses a standard F-basis Let r, s ∈ Y and Drs : O(m, n, t) → W(m, n, t) be a linear mapping such that where f ∈ O(m, n, t) and r, s ∈ Y. Then the following equation holds: Then S(m, n, t) is called the special modular Lie superalgebra. S(m, n, t) is also a nite-dimensional simple Lie superalgebra (see [27]).
Then S(m, n, t) is contained in S(m, n, t) and S(m, n, t) is a subalgebra of W(m, n, t).
In addition, S(m, n, t) is also a Z-graded subalgebra of W(m, n, t). For convenience, W(m, n, t), S(m, n, t), S(m, n, t) and O(m, n, t) will be denoted by W, S, S and O, respectively.

Graded modules over modular Lie superalgebras
Let gl(m, n) = gl(m, n)¯ ⊕ gl(m, n)¯ be the general linear Lie superalgebra of all s × s matrices over F (see [9]), De ne the operation [ , ] in gl(m, n, t) as follows: where where E kj is an s × s matrix whose (i, l)-entry is δ ki δ jl . Then A ∈ gl(m, n, t)α. By virtue of the de nition of superderivation we have Using the formulae from (2) to (6), a direct calculation shows the following proposition.
Suppose that L is a subalgebra of gl(m, n) and The formula (7) shows that Ω is a subalgebra of modular Witt Lie superalgebras W. The subalgebra Ω is called the P-expansion of L into W. Then the P-expansion of gl(m, n) into W is exactly W.
The special linear Lie superalgebra sl(m, n) = {A ∈ gl(m, n) | str(A) = } is a subalgebra of gl(m, n) (see [9]). Let Ω is the P-expansion of sl(m, n) into W. If A = s i= a i D i ∈ W, then, for α ∈ Z , Hence Ωα = Sα. It follows that Ω = S. Let ρ be a representation of L(P) on Z -graded space V. Then ρ can be expanded to a representation ρ where

Proposition 3.2 ([24], Proposition 2).
Let Ω be the P-expansion of L into W. Then de nes a representation ρ of Ω on Z -graded space O ⊗ V.
By Proposition 3.2, O ⊗ V which will be denoted by V is a Ω-module. In [20] the module V is called the mixed product of O and the module V.
Hence V is a positively graded Ω-module. Since the P-expansion of gl(m, n) and sl(m, n)) into W are ,respectively, W and S, Proposition 3.2 shows the following corollary.

Proposition 3.4. Suppose that V is an irreducible L(P)-module, v λ is the weight vector associated with the highest weight λ and V is the unique irreducible submodule of V(λ), where L = gl(m, n) or sl(m, n), then (i) V contains ⊗ V as a submodule. (ii) If x (π) x E ⊗ v λ ∈ V, then V = V(λ), that is V(λ) is an irreducible Ω-module, where Ω is the P-expansion of L(P) into W.
Proof. Recall that Ω = W if L = gl(m, n) and Ω = S if L = sl(m, n).

Irreducibility of module V(λ) over W and S
Let V be an irreducible sl(m, n)-module with the highest weight λ. Proposition 3.2 shows that V(λ) is an Smodule. Clearly, it is also an S-module. Assuming that V is irreducible as an S-module and if x (π) x E ⊗ v λ ∈ V, then it follows that V = V(λ) from the similar methods used in Proposition 3.4 (ii). Therefore, V(λ) is an irreducible S-module. We know that the standard Cartan subalgebra H of S is h i | i = , , . . . , s − , where   8) and (9) and where v λ is a weight vector associated with the highest weight λ, Applying the formulae (8), (9) and (10) and If c i ≠ or , then x (π) x E ⊗ v λ ∈ V and V(λ) is an irreducible S-module. So it was assumed that c k = or , where k = , , . . . , m − . Since λ|−m ≠ and λ|−m ≠ λ i , there exist at least two k ∈ { , , . . . , m − } such that c k = . Without loss of generality we assumed that c i and c j , i < j, are the rst and second coe cients which are equal to . Then Therefore, x (π) x E ⊗ v λ ∈ V and V(λ) is an irreducible S-module. Proof. If c m+ ≠ cm + , then cm( + cm − c m+ ) ≠ . A direct computation shows that Since cm ≠ − , we have + cm ≠ . Therefore, As c k c k+i ≠ , we have x (π) x E ⊗ v λ ∈ V. But this conclusion confutes that V(λ) is a reducible S-module. As c k+i = , i > and λ = c k λ k + c k+ λ k+ . Then Since V(λ) is a reducible S-module, we have + c k + c k+ = . Therefore, c k+ = −(c k + ) and λ = c k λ k − (c k + )λ k+ .
The proof is completed.
The W-module V(λ) will be discussed in the following theorem. In conclusion, the proof is completed.