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

1 Introduction

Let 𝔽 be an arbitrary field of characteristic p > 2 and ℤ₂ = {0̄, 1̄} be the residue class ring mod 2. Throughout this paper we have assumed that all vector spaces, linear mappings and tensor products are over the underlying base field 𝔽. Assume that x is a ℤ₂-homogeneous element and d(x) is the ℤ₂-degree of x, if d(x) occurs in an expression.

In 1967, Rudakov and Shafarevich [1] described all the irreducible representations of 𝔰𝔩(2) over an algebraically closed field 𝔽 of characteristic p > 2. 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 3. 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 fields over ℂ, for modules dual to modules of (formal) tensor fields. 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 fields of series W and S possessing a p-structure. Shu [7] discussed the representations of Cartan type Lie algebras in characteristic p > 2 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 classification of all the finite-dimensional simple complex Lie superalgebras was done by Kac [9], the problems of constructing a unified 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 fields of characteristic zero [23]. In the case of modular Lie superalgebras, Zhang [24] has obtained the ℤ-graded modules over finite-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.

2 Generalized Witt and special modular Lie superalgebras

In addition to the standard notation ℤ, ℕ and ℕ₀ is used for the set of positive integers and the set of nonnegative integers, respectively. Generally, let m, n denote fixed integers in ℕ ∖ {1, 2}. For α = (α₁, …, α_m) ∈ N0m , we put |α| := ∑i=1m α_i. Following [25], let 𝓞(m) denote the divided power algebra over 𝔽 with an 𝔽-basis {x^(α) | α ∈ N0m }. For ε_i := (δ_i1, …, δ_im), we abbreviate {x^(ε_i)} to x_i, i = 1, 2, …, m, where δ_ij is Kronecker delta. Let Λ(n) be the exterior superalgebra over 𝔽 in n variables x_m+1, x_m+2, …, x_m+n and 𝓞(m, n) denote the tensor product 𝓞(m) ⊗_𝔽 Λ(n) . Clearly, 𝓞(m, n) is an associative superalgebra with a ℤ₂-gradation induced by the trivial ℤ₂-gradation of 𝓞(m) and the natural ℤ₂-gradation of Λ(n). Moreover, 𝓞(m, n) is super-commutative. For g ∈ 𝓞(m), f ∈ Λ(n), we write gf for g ⊗ f. The following formulae hold in 𝓞(m, n):

where α+βα:=∏i=1mαi+βiαi. Put Y₀ := {1, 2, …, m}, Y₁ := {m + 1, m + 2, …, m + n} and Y := Y₀ ∪ Y₁. Set

and B:=⋃k=0nBk, where 𝔹₀ := ∅. For u = 〈i₁, i₂, …, i_k〉 ∈ 𝔹_k, set |u| := k, |∅| := 0, x^∅ := 1, x^u := x_i1 x_i2 ⋯ x_ik and x^E := x_m+1 x_m+2 ⋯ x_m+n. Clearly, {x^(α)x^u | α ∈ N0m , u ∈ 𝔹} constitutes an 𝔽-basis of 𝓞(m, n). Let D₁, D₂, …, D_m+n be the linear transformations of 𝓞(m, n) such that

where ∂/∂ x_r is the superderivation of Λ(n) such that ∂ x_s/∂ x_r = δ_rs for r, s ∈ Y₁. For more details on superderivations for Lie superalgebras, the reader is referred to [9, 26]. D₁, D₂, …, D_m+n are superderivations of the superalgebra 𝓞(m, n). Let

Then W(m, n) is a Lie superalgebra, which is contained in Der(𝓞(m, n)), where Der(𝓞(m, n)) denotes the superderivation space of 𝓞(m, n). Obviously, d(D_i) = τ(i), where

An easy verification shows that

for f, g ∈ 𝓞(m, n), D, E ∈ Der(𝓞(m, n)). In particular, the following formula holds in W(m, n):

for f, g ∈ 𝓞(m, n), r, s ∈ Y.

Let

where π_i := p^t_i − 1, i ∈ Y₀. Let 𝔸 := {α ∈ N0m | α_i ⩽ π_i, i = 1, 2, …, m}. Then

is a finite-dimensional subalgebra of 𝓞(m, n) with a natural ℤ-gradation 𝓞(m, n, t) = ⊕r=1ξ 𝓞(m, n, t)_r by putting

Set

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 finite-dimensional simple Lie superalgebra (see [27]). Clearly, W(m, n, t) is a free 𝓞(m, n, t)-module with basis {D_r | r ∈ Y}. Note that W(m, n, t) possesses a standard 𝔽-basis {x^(α) x^u D_r | α ∈ 𝔸, u ∈ 𝔹, r ∈ Y}.

Let r, s ∈ Y and D_rs : 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:

Put

Then S(m, n, t) is called the special modular Lie superalgebra. S(m, n, t) is also a finite-dimensional simple Lie superalgebra (see [27]).

Let div : W(m, n, t) → 𝓞(m, n, t) be the divergence such that

It follows that

Then div is a superderivation from W(m, n, t) to 𝓞(m, n, t). Following [1], put

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). The ℤ-gradation of 𝓞(m, n, t) induces naturally ℤ-gradation structures of W(m, n, t) = ⊕i=−1ξ−1 W(m, n, t)_i and S(m, n, t) = ⊕i=−1ξ−2 S(m, n, t)_i, where

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.

3 Graded modules over modular Lie superalgebras

Let 𝔤𝔩(m, n) = 𝔤𝔩(m, n)_0̄ ⊕ 𝔤𝔩(m, n)_1̄ be the general linear Lie superalgebra of all s × s matrices over 𝔽 (see [9]), where s = m + n. Set 𝔤𝔩(m, n, t) = 𝓞 ⊗ 𝔤𝔩(m, n). Define the operation [ , ] in 𝔤𝔩(m, n, t) as follows:

where a, b ∈ 𝓞, x, y ∈ 𝔤𝔩(m, n). Then 𝔤𝔩(m, n, t) is a Lie superalgebra. For A ∈ W, define A ⊗ 1 ∈ End(𝔤𝔩(m, n, t)) by

Let P ∈ 𝔤𝔩(m, n)_0̄ be an s × s invertible matrix. Suppose that A = ∑i=1s a_i D_i ∈ W_α, where α ∈ ℤ₂. Let

where E_kj is an s × s matrix whose (i, l)-entry is δ_kiδ_jl. Then A͠ ∈ 𝔤𝔩(m, n, t)_α. By virtue of the definition of superderivation we have

where a, b ∈ 𝓞, k = 1, …, s.

Let A = ∑i=1s a_i D_i ∈ W_α and B = ∑i=1s b_j D_j ∈ W_β, where α, β ∈ ℤ₂. Then

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 L(P) = {P^–1 AP | A ∈ L}. Then L(P) is a subalgebra of 𝔤𝔩(m, n). Let Ω = Ω_0̄ ⊕ Ω_1̄, where

If A, B ∈ Ω, then [A͠, B͠] ∈ Ω. 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) = 0} is a subalgebra of 𝔤𝔩(m, n) (see [9]). Let Ω is the P-expansion of 𝔰𝔩(m, n) into W. If A = ∑i=1s a_i D_i ∈ W, then, for α ∈ ℤ₂,

Hence Ω_α = S_α. It follows that Ω = S.

Let ρ be a representation of L(P) on ℤ₂-graded space V. Then ρ can be expanded to a representation ρ₁ of 𝓞 ⊗ L(P) on the space 𝓞 ⊗ V, defined by

where a, b ∈ 𝓞, x ∈ L(P), v ∈ V.

By Proposition 3.2, 𝓞 ⊗ V which will be denoted by V͠ is a Ω-module. In [20] the module V͠ is called the mixed product of 𝓞 and the module V.

A ℤ-graded module V of X is called positively graded if V = ⨁i≥0 V_i and L_j ⋅ V_i ⊆ V_i+j, where X is a ℤ-graded Lie superalgebra.

Let V͠_i = 〈a ⊗ v | a ∈ 𝓞_i, v ∈ V〉. Then V͠ = ⨁i=0ξ V͠_i, where ξ = ∑i=1m π_i + n. Put V͠_i = 0 for i > ξ. A direct verification shows that Ω_i ⋅ V͠_j ⊆ V͠_i+j. Hence V͠ 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.

If V is an irreducible L(P)-module with a highest weight λ, then denote V͠ by V͠(λ). Furthermore, the weight vector associated with the highest weight λ is denoted by v_λ, where v_λ ∈ V͠(λ). Similar to [21, Theorem 1.2] one may obtain that Ṽ(λ) has an unique irreducible submodule U(L)(1 ⊗ V), where U(L) is the universal enveloping algebra of L. We customarily denote the unique irreducible submodule by V. Then x^(α) x^u ⊗ v_λ ∈ V for all α = (α₁, …, α_m) ∈ 𝔸, u = (i₁, …, i_r) ∈ 𝔹.

Suppose that π = (π₁, …, π_m), E = (m + 1, …, s), then π ∈ 𝔸, E ∈ 𝔹. Since Lie superalgebra W₀ is isomorphic to 𝔤𝔩(m, n), the element ∑i,j=1s a_ij E_ij of L(P) can be identified as the element ∑i,j=1s a_ijx_iD_j of W, where a_ij ∈ 𝔽. Hence x^(π) x^E ⊗ V can be regarded as an L(P)-module. Similarly, by [21, Proposition 2.4], we can prove that V is an irreducible L(P)-module if and only if x^(π) x^E ⊗ V is an irreducible L(P)-module. The following proposition is the analogue of [21, Proposition 2.1].

4 Irreducibility of module V͠(λ) over W and S

Let V be an irreducible 𝔰𝔩(m, n)-module with the highest weight λ. Proposition 3.2 shows that V͠(λ) is an S-module. 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 = V͠(λ) from the similar methods used in Proposition 3.4 (ii). Therefore, V͠(λ) is an irreducible S-module.

We know that the standard Cartan subalgebra H of S is 〈h_i | i = 1, 2, …, s – 1〉, where

Let Λ_i be the linear function on 〈E₁₁, …, E_ss〉 such that Λ_i(E_jj) = δ_ij, where i, j = 1, 2, …, s. Set

Then λ_i, i = 1, 2, …, s – 1, is a fundamental weight of 𝔰𝔩(m, n) and λ_i(h_j) = δ_ij. We know that Λ_i – Λ_j is a positive root of 𝔰𝔩(m, n) and the corresponding vectors of the positive root are E_ij, where 1 ≤ i < j ≤ s.

If V is a finite-dimensional irreducible 𝔰𝔩(m, n)-module, then λ = ∑i=1s−1 c_iλ_i, where c_i ∈ 𝔽. Let λ|_–m = ∑i=1m−1 c_iλ_i and λ|_+m = ∑i=m+1s−1 c_iλ_i. Then λ = λ|_–m + c_m λ_m + λ|_+m. Also, assuming that ρ and ρ͠ are representations corresponding with the modules V and V͠(λ), respectively.

The W-module V͠(λ) will be discussed in the following theorem.