Lie n superderivations and generalized Lie n superderivations of superalgebras

1 Introduction

Let A be an associative algebra. A linear mapping d : A → A is called a derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ A. A Lie derivation δ of A is a linear mapping from A into itself satisfying δ[x, y] = [δ(x), y] + [x,δ(y)] for all x, y ∈ A. A Lie triple derivation is a linear mapping ψ : A → A which satisfies ψ[[x, y], z] = [[ψ(x), y], z] + [[x, ψ(y)], z]+[[x, y], ψ(z)] for all x, y, z ∈ A. Obviously, each derivation is a Lie derivation and each Lie derivation is a Lie triple derivation. Brešar [1] described the structure of Lie derivations and Lie triple derivations on prime rings and obtained that each Lie derivation or Lie triple derivation of a prime ring is the sum of a derivation and an additive mapping. Wang [2] studied the structure of Lie superderivations of superalgebras in 2016. Lie n-derivations were introduced by Abdullaev [3], where the form of Lie n-derivations of a certain von Neumann algebra was described. In 2012, Benkovič and Eremita [4] gave the form of Lie n-derivations on triangular rings, which has been generalized to generalized matrix algebras in [5].

The concept of a generalized derivation was introduced by Brešar [6] and generalized by Hvala [7], who has proved in [8] that each generalized Lie derivation of a prime ring is the sum of a generalized derivation and a central mapping which vanishes on all commutators.

A functional identity can be described as an identical relation involving elements in a ring together with functions. The goal when studying a functional identity is to describe the form of these functions or to determine the structure of the ring admitting the functional identity in question. The theory of functional identities in rings originated from the results of commuting mappings [9]. The name “functional identity” was introduced by Brešar in [10]. The crucial tool in the theory of functional identities in rings is the d-free set, which was developed by Beidar and Chebotar in [11, 12]. Making use of the theory of functional identities in rings, Herstein’s conjectures on Lie mappings in rings have been settled [13,14,15]. After this, Wang [16] established the theory of functional identities in superalgebras and gave the definition of d-superfree sets. As an application, Wang [17] described Lie superhomomorphisms from the set of skew elements of a superalgebra with superinvolution into a unital superalgebra. The knowledge of functional identities and d-superfree sets of superalgebras refer to [16], [18] and [19].

In the paper, our purpose is to study Lie n superderivations and generalized Lie n superderivations of superalgebras, using the theory of functional identities in superalgebras. Section 2 presents some preliminaries. In the third section we discuss the structure of Lie n superderivations. In Section 4 the results of generalized Lie n superderivations are stated and proved.

2 Preliminaries

Throughout the paper, by an algebra we shall mean an algebra over a fixed unital commutative ring Φ. We assume without further mentioning that 12∈Φ.

An associative algebra A over Φ is said to be an associative superalgebra if there exist two Φ-submodules A₀ and A₁ of A such that A = A₀ ⊕ A₁ and A_iA_j ⊆ A_i+j, i, j ∈ Z₂. We call A₀ the even and A₁ the odd part of A. The elements of A_i are homogeneous of degreei and we write |A_i| = i for all A_i ∈ A_i. For a superalgebra A, we define σ:A → A by (A₀ + A₁)^σ = A₀ – A₁, then σ is an automorphism of A such that σ² = 1. On the other hand, for an algebra A, if there exists an automorphism σ of A such that σ² = 1, then A becomes a superalgebra A = A₀ ⊕ A₁, where Ai = {x ∈ A|x^σ = (–1)ⁱx, i = 0, 1. A superalgebra A is called a prime superalgebra if and only if aAb = 0 implies a = 0 or b = 0, where at least one of the elements a and b is homogeneous.

On a superalgebra A, define for any x, y ∈ A₀ ∪ A₁ the Lie superproduct

Thus

where a = a₀ + A₁, b = b₀ + b₁.

In [20] Montaner obtained that a prime superalgebra A is not necessarily a prime algebra but a semiprime algebra. Hence one can define the maximal right ring of quotients Q_mr of A, and the useful properties of Q_mr can be found in [21]. By [21, proposition 2.5.3] σ can be uniquely extended to Q_mr. Therefore, Q_mr is also a superalgebra. Moreover, we can get that Q_mr is a prime superalgebra.

On the other hand, we will introduce some important concepts of the theory of functional identities in superalgebras.

Let Q = Q₀ ⊕ Q₁ be a unital superalgebra with grading automorphism σ and center C = C₀ ⊕ C₁ satisfying [C, Q] = 0. Fix an element ω ∈ Q as follows : If either σ = 1 or σ is outer, we set ω = 0. Otherwise, we denote ω as an invertible element in Q such that σ(x) = ωxω^–1 for all x ∈ Q. It is easy to check that ω ∈ Q₀, ω² ∈ C₀, ωx₀ = x₀ω for all x₀ ∈ Q₀, and ωx₁ = –x₁ω for all x₁ ∈ Q₁. We shall call the ω the grading element of Q. If A = b + cω, b, c ∈ C, we set ā = b – cω.

Let m ∈ N^*, 𝓤₁,𝓤₂, …, 𝓤_m are subsets of Q such that either 𝓤_i ⊆ Q₀ or 𝓤_i ⊆ Q₁ for every 1 ≤ i ≤ m. Set ε_i = ± 1, where either ε_i = 1 if 𝓤_i ⊆ Q₀ or ε_i = – 1 if 𝓤_i ⊆ Q₁. 𝓢₁, 𝓢₂,…, 𝓢_m are nonempty sets, 𝓘, 𝓙 ⊆ {1, 2, …, m} and δ_l : 𝓢_l → 𝓤_l, l ∈ 𝓘 ∪ 𝓙, are surjective maps. Set S^=∏k=1mSk,U^=∏k=1mUk and Δ = {δ_l|l ∈ 𝓘 ∪ 𝓙}.

We shall consider functional identities on Ŝ of the following form

for all X̅_m ∈ Ŝ, where E_i : ∏_k≠i𝓢_k → Q and F_j : ∏_k≠j𝓢_k → Q.

Suppose that ω = 0 or each 𝓤_i ⊆ Q₀. There exist maps

such that

for all X̅_m ∈ Ŝ, where λ_l = 0 if l ∉ 𝓘 ∩ 𝓙.

Otherwise, there exist maps

such that

for all X̅_m ∈ Ŝ, where λ_l = 0 = μ_l if l ∉ 𝓘 ∩ 𝓙. We shall refer to (3) and (4) as a standard solution of (1) and (2).

If each 𝓢_k = 𝓤_k and each δ_l = id_𝓤l, then the 𝓤̂ is said to be d-superfree provided that (𝓢̂;δ; 𝓤̂) is so. Let R = R₀ ⊕ R₁ be a graded Φ-submodule of Q. For every 1 ≤ i ≤ m, either 𝓤_i = R₀ or 𝓤i = R₁. Then R is said to be d-superfree provided that each 𝓤̂ is d-superfree. And, we can get the following result.

Let {x₁, x₂, …, x_m} be a finite set of variables and k be a nonnegative integer such that k ≤ m. We denote by Mmk the set of all multilinear monomials of degree k in the variables {x₁, x₂, …, x_m}. It is understood that M10 = {1}. We write Mm=⋃k=0mMmk. For a given monomial M = x_i1 … x_iu ∈ 𝓜_m where u ≤ m – k. We denote by Mmk(M) the set of all multilinear monomials of degree k in the variables {x₁, …, x_m}\{x_i1, …, x_iu} and write Mm(M)=⋃k=0m−uMmk(M).

For every 1 ≤ t ≤ m, let 𝓢_t and 𝓡_t be two sets and let δ_t : 𝓢_t → 𝓡_t be a surjective mapping. We set

where s_it ∈ 𝓢_it.

We set 𝓢̂ = S^=∏i=1mSi and S^(M)=∏t=1m−uSjt. For any given F : 𝓢̂(M) → Q we introduce a mapping F^M : 𝓢̂ → Q by the rule

for any s̄_m ∈ 𝓢̂.

Let M∈Mmk and let λ_M : 𝓢̂(M) → C + Cω. A mapping 𝓢̂ → Q defined by the rule s¯m→λMM(s¯m)M(s¯m) for any s̄_m ∈ 𝓢̂ is called a superquasi-monomial and is denoted by λ_MM. A sum ∑_{L ∈ 𝓜m} λ_LL of different superquasi-monomials will be called a superquasi-polynomial.

An element x ∈ A₀ ∪ A₁ is said to be algebraic over C of degree ≤ n if there exist C₀, C₁, …, c_n ∈ C, not all zero and such that ∑i=0ncixn−i=0. The element x is said to be algebraic over C of degree n if it is algebraic over C of degree ≤ n and is not algebraic over C of degree ≤ n – 1. By deg(x) we shall mean the degree of x over C (if x is algebraic over C) or ∞ (if x is not algebraic over C). Given a nonempty subset S ⊆ A₀ ∪ A₁, we set

Let A be a superalgebra. For i ∈ {0, 1}, a superderivation of degreei is actually a Φ-linear mapping d_i : A → A which satisfies d_i(A_j) ⊆ A_{i + j}, j ∈ Z₂, and d_i(ab) = d_i(a)b + (–1)^i|a|ad_i(b) for all A, b ∈ A₀ ∪ A₁. If d = d₀ + d₁, then d is called a superderivation.

Let A be a superalgebra. For i ∈ {0, 1}, a Φ-linear mapping g_i : A → A is called a generalized superderivation of degreei if g_i(A_j) ⊆ A_{i + j}, j ∈ Z₂, and g_i(xy) = g_i(x)y + (–1)^i|x|xd_i(y) for all x, y ∈ A₀ ∪ A₁, where di is a superderivation of degree i. If g = g_{0}+g₁, then g is called a generalized superderivation.

The following identity will be used frequently,

where i, j, k ∈ {0, 1}.

3 Lie n superderivations of superalgebras

In the section, we describe the structure of Lie n superderivations on a superalgebra.

Obviously, each superderivation is a Lie superderivation on A.

Let us define the following sequence of polynomials: p₁(x) = x and

Thus, p₂ (x₁, x₂) = [x₁, x₂]_s, p₃ (x₁, x₂, x₃) = [[x₁, x₂]_s, x₃]_s, etc.

Comparing the above expressions, we get

for all x, y, z ∈ A₀ ∪ A₁.

When |x| = |y| = |z| = 0, it follows from (10) that

An easy computation shows that:

1. The coefficient of zxy is λ1′;

2. The coefficient of xz is μ1′(y)−μ1(y);

3. The coefficient of z is μ₁(xy) + ν₁(x, y).

By [16, Theorem 3.7], we have

When |x| = |z| = 0 and |y| = 1, it follows from (10) that

An easy computation shows that:

1. The coefficient of zxy is λ3′;

2. The coefficient of yzx is −λ2′;

3. The coefficient of xz is μ2′(y)−μ3(y);

4. The coefficient of x is −μ2′(yz)−ν3(y,z);

5. The coefficient of z is μ₃(xy) + ν₂(x, y)

By [16, Theorem 3.7], we have

When |x| = 0 and |y| = |z| = 1, it follows from (10) that

An easy computation shows that:

1. The coefficient of xyz is λ₂ – λ₁;

2. The coefficient of zxy is λ4′;

3. The coefficient of yz is μ₂(x) – μ₁(x);

4. The coefficient of xz is μ2′(y) – μ₄(y);

5. The coefficient of x is – μ1′(yz) – ν₄(y, z).

By [16, Theorem 3.7], we have

By the definition of 𝓑, we have

for all x₁, x₂, …, x_n, ω₁ ∈ A₀. Since the coefficient of x₁x₂ … x_nω₁ is (n−1)λ₁, [16, Theorem 3.7] yields λ₁ = 0.

On the other hand, we have