Abstract
We study a twisted generalization of Lie superalgebras, called Hom-Lie superalgebras. It is obtained by twisting the graded Jacobi identity by an even linear map. We give a complete classification of the complex multiplicative Hom-Lie superalgebras of low dimensions.
1 Introduction
In recent years, Hom-Lie algebras and other Hom-algebras have been extensively studied. They first arised in quasideformation of Lie algebras of vector fields. The Hom-Lie and quasi-Hom-Lie structures are obtained from twisted derivations of discrete modifications of vector fields. For those algebras, the Jacobi condition is twisted [1]. Homalgebras were studied in [2–10]. Furthermore, Hom-Lie algebras on some q-deformations of Witt and Virasoro algebras were considered in [11–13].
Hom-Lie algebras were generalized to Hom-Lie superalgebras by Ammar and Makhlouf [14] and to Hom-Lie color algebras by Yuan [15]. In particular, Cao and Luo [16] proved that there were only the trivial Hom-Lie superalgebras on the finite-dimensional simple Lie superalgebras. The representation theory and the cohomology theory of Hom-Lie superalgebras were studied in [17]. In [18], some infinite-dimensional Hom-Lie superalgebras were constructed, induced by affinizations of the Hom-Balinskii-Novikov superalgebras and Hom-Novikov superalgebras. Furthermore, the central extensions of the infinite-dimensional Hom-Lie superalgebras were studied.
During the past few years, the theories of Hom-Lie superalgebras make great advances. But, only few people study the classification of low dimensional Hom-Lie superalgebras. There are few results about it. Zhang [19] and Li [20] studied the classifications of low dimensional Lie superalgebras and low dimensional multiplicative Hom-Lie algebras, respectively. In this paper, we will discuss the classification of low dimensional multiplicative Hom-Lie superalgebras. For the pure even Hom-Lie superalgebras, they are Hom-Lie algebras. Their classifications have been obtained. It can be found in [20], so we only discuss the odd part, which is nontrivial. We know that derived algebras of non-abelian Hom-Lie superalgebras are non-zero. We note that the properties of the derived algebras can be hold up to isomorphisms. Thus, it is crucial to determine the multiplication tables by using the dimension and properties of derived algebras. Finally, we can obtain the classifications of the low dimensional multiplicative Hom-Lie superalgebras.
The paper is organized as follows: In Section 2, we give some notations and some basic properties of Hom-Lie superalgebras. We give a method to determine whether two different multiplicative Hom-Lie superalgebras are isomorphism or not, and then we obtain a complete classification of multiplicative Hom-Lie superalgebras up to isomorphism. In Section 3 and 4, we get the classifications of the low dimensional multiplicative Hom-Lie superalgebras.
In this paper, we discuss all algebras and vector spaces over a complex field ℂ. The parity of the homogeneous element x is denoted by |x|. hg(L) denotes the set of all homogeneous elements of Hom-Lie superalgebras (L, [, ], σ).
2 Preliminaries
In this section, we recall the notions of Hom-Lie superalgebras. Then we can get some basic properties of Hom-Lie superalgebras. We aim to give a method to determine whether two different multiplicative Hom-Lie superalgebras are isomorphism or not.
A Hom-Lie superalgebra is a triple (L, [, ], σ) consisting of a super space L, an even bilinear [, ] : L × L → Land an even linear mapσ : L → L satisfying
Let (L, [, ], σ) and (L′, [, ]′, σ′) be two Hom-Lie superalgebras. An even linear map ϕ : L → L′ is said to be a homomorphism of Hom-Lie superalgebras if the following conditions are satisfied:
for any x, y ∈ hg(L). If ϕ is a bijection, then ϕ is called an isomorphism of Hom-Lie superalgebras.
A Hom-Lie superalgebra (L, [, ], σ) is called a multiplicative Hom-Lie superalgebra if the even linear map σ satisfiesσ[x, y] = [σ(x, σ(y)] for any x, y ∈ hg(L). A Hom-Lie superalgebra (L, [, ], σ) is called a classical Hom-Lie superalgebra if (L, [, ], σ) is a Lie superalgebra.
In the following paper, Hom-Lie superalgebras always denote to be the multiplicative Hom-Lie superalgebras.
Let (L, [, ], σ) be a Hom-Lie superalgebra and σ be an even invertible linear map. Define [, ]1 : L × L → L by [x, y]1 = [σ-1(x), σ-1(y)], for any x, y ∈ hg(L). Then (L, [, ]1) is a Lie superalgebra and σ is a Lie superalgebra endomorphism.
It is clear that [, ]1 is an even bilinear map and is super skew-symmetric. Since
we have, for any x, y, z ∈ hg(L),
Obviously, σ is an even endomorphism of the Lie superalgebra (L, [, ]1).
Let L be a linear super space and [, ] : L × L → L be an even bilinear map. Mor(L, [, ]) denotes the set of all even linear maps σ such that (L, [, ], σ) is a multiplicative Hom-Lie superalgebra.
Let L be a linear super space, [, ]1and [, ]2be two even bilinear maps on L. If there exists an even invertible linear map ϕ of L satisfying
then
Since ϕ is invertible, we have, for any σ1∈Mor(L, [, ]1), x, y, z ∈ hg(L)
and
So
Conversely, for any σ2 ∈ Mor(L, [, ]2), similar to above, we have ϕ-1σ2ϕ ∈ Mor(L, [, ]1). Set ϕ-1σ2ϕ = σ1. Then σ2 = ϕσ1ϕ-1. Thus, we have
By Lemma 2.5, we know that there exists a one-to-one correspondence between the multiplicative Hom-Lie superalgebras (L, [, ]1, σ1) and the multiplicative Hom-Lie superalgebras (L, [, ]2, σ2). In order to simplify the classification of multiplicative Hom-Lie superalgebras, we can simplify [, ]1to [, ]2by an even invertible linear map ϕ. Then we determine the multiplicative Hom-Lie superalgebra structure by means of [, ]2.
Let (L, [, ]1, σ1) and (L, [, ]2, σ2) be two Hom-Lie superalgebras with the same underlying vector super space L. (L, [, ]1, σ1) is isomorphic to (L, [, ]2, σ2) if and only if there exists an invertible matrix T such that T[ei, ej]1 = [T(ei), T(ej)]2andA2 = TA1T-1, where A1, A2andTdenote matrices corresponding to σ1, σ2and the homomorphism ϕ : L → Lwith respect to the basis {e1, e2, ..., en}, respectively.
We denote the derived algebra of Hom-Lie superalgebras (L, [, ], σ) by L′ = [L, L]. It consists of all linear combinations of commutators [x, y] for x, y ∈ hg(L). The Hom-Lie superalgebra (L, [, ], σ) is called an abelian Hom-Lie superalgebra if [L, L] = 0.
Let (L1, [, ]1, σ1) and (L2, [, ]2, σ2) be two abelian Hom-Lie superalgebras. If (L1, [, ]1, σ1) is isomorphic to (L2, [, ]2, σ2), then
An isomorphism from (L1, [, ]1, σ1) to (L2, [, ]2, σ2) is an isomorphism of their underlying vector super space. So there exists an even invertible linear map ϕ :
Let (L, [, ]1, σ1) and (L, [, ]2, σ2) be two abelian Hom-Lie superalgebras with the same underlying vector super space L. (L, [, ]1, σ1) is isomorphic to (L, [, ]2, σ2) if and only if the matrices corresponding to σ1 and σ2 have the same elementary divisors.
It is straightforward.
Let (L1, [, ]1, σ1) and (L2, [, ]2, σ2) be two Hom-Lie superalgebras. If
For any x, y ∈ hg(L1), we have
3 A classification of 2-dimensional Hom-Lie superalgebras
In this section, we are going to give the classification of 2-dimensional Hom-Lie superalgebras.
Every 2-dimensional Hom-Lie superalgebra is isomorphic to one of the following nonisomorphic Hom-Lie superalgebra: each algebra is denoted by
(1)
(2)
(3)
(4)
(5)
In order to prove the above theorem, we give some lemmas.
Let (L, [, ]1, σ) be a 2-dimensional Hom-Lie superalgebra and {e0, e1} be a basis of (L, [, ], σ).
If e0, e1 ∈
If e0 ∈
In the following, we will discuss the situation:
Let
If α ≠ 0, β = 0, by Lemma 2.5, then there exists an even invertible linear map
Similarly, if α = 0, β ≠ 0, the bracket can be simplified as
If α ≠ 0, β ≠ 0, the bracket can be simplified as
However, there does not exist an even invertible linear map such that formulas (5), (6) and (7) are equivalent.
For the bracket (5), Hom-Lie superalgebras are (L, [, ], σi)(i = 1, 2) up to isomorphism, where
Let σ : L → L be an even linear map of Hom-Lie superalgebras and σ(e0) = ae0, σ(e1) = be1. The Hom-Jacobi identity is satisfied for any a, b ∈ ℂ. Since σ is multiplicative, we have a = 1 or b = 0.
For the bracket (6), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where
It is similar to Lemma 3.2.
For the bracket (7), there is not a non-zero even linear map σ such that it is multiplicative and satisfies Hom-Jacobi identity.
From Lemmas 3.2 and 3.3, we obtain Theorem 3.1.
(1) All 2-dimensional Hom-Lie superalgebras are classical Lie superalgebras.
(2) From Lemma 2.8, Hom-Lie superalgebras
4 A classification of 3-dimensional Hom-Lie superalgebras
By Lemma 2.10, we note that it can keep the structures of the derived algebras with the isomorphism. Thus, we can use the properties of the derived algebras to determine the structures of 3-dimensional Hom-Lie superalgebras.
Every 3-dimensional Hom-Lie superalgebra is isomorphic to one of the following non-isomorphic Hom-Lie superalgebra: each algebra is denoted by
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(46)
(47)
(48)
(49)
(50)
In order to prove the above theorem, we give some lemmas.
Let (L, [, ], σ) be a 3-dimensional Hom-Lie superalgebra and {e1e2e3} be a basis of (L, [, ], σ).
If
Now we consider the 3-dimensional Hom-Lie superalgebra (L, [, ], σ), where dim
(1) If [L, L] = 0, then (L, [, ], σ) is an abelian Hom-Lie superalgebra.
(2) If dim [L, L] = 1, we consider two cases as follows.
Case I:
If α = 0, β ≠ 0, by Lemma 2.5, there exists an even invertible linear map
Similarly, if α ≠ 0, β = 0, the bracket can be simplified as
If α ≠ 0, β = 0, the bracket can be simplified as
However, there is not an even invertible linear map such that formulas (8), (9) and (10) are equivalent.
For the bracket (8), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where
Let σ : L → L be an even linear map of the Hom-Lie superalgebras and σ(e1) = a1e1 + a2e2, σ(e2) = b1e1 + b2e2, σ(e3) = ce3. For any
For the bracket (9), the Hom-Lie superalgebra is (L, [, ], σi)(i = 1, 2) up to isomorphism, where
It is similar to Lemma 4.2.
For the bracket (10), the Hom-Lie superalgebra is (L, [, ], σi up to isomorphism, where
Let σ : L → L be an even linear map of the Hom-Lie superalgebras and σ(e1) = a1e1 + a2e2, σ(e2) = b1e1 + b2e2, σ(e3) = ce3. Since σ is multiplicative, we have a1 = c2 = a1b2 - a2b1 and a2 = 0. By Hom-Jacobi identity, we obtain b2 = 0, then c = 0. Set
such that
Case II:
If α = 0, β ≠ 0, by Lemma 2.5, there exists an even invertible linear map
For the bracket (11), the Hom-Lie superalgebras are (L, [, ], σi)(i = 1, 2) up to isomorphism, where
It is similar to Lemma 4.4.
If α ≠ 0, β = 0 or α ≠ 0, β ≠, 0 by Lemma 2.5, we know that Hom-Lie superalgebras are isomorphic to those Hom-Lie superalgebras in Lemma 4.5.
All Hom-Lie superalgebras in Theorem 4.2-4.5 are classical Lie superalgebras.
(3) If dim [L, L] = 2, we will consider two cases as follows.
Case I:
The matrix
If β1 = α2 = 0, we have α1 ≠ 0, β2 ≠ 0. By Lemma 2.5, there exists an even invertible linear map
Other cases can be simplified as (12) by Lemma 2.5.
For the bracket (12), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ =
It is similar to Lemma 4.4.
Case II: [L, L] = ℂe1 + ℂe3. Let [e1, e2] = αe1, [e1, e3] = βe3, [e2, e3] = λe3, [e3, e3] = μe1. We will consider situations as follows.
If α ≠ 0, β ≠ 0, λ = 0, μ = 0, by Lemma 2.5, there exists an even invertible linear map
Moreover, for α ≠ 0, β ≠ 0, λ ≠ 0, μ = 0, the bracket can be simplified as (13).
For the bracket (13), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 3) up to isomorphism, where
It is similar to Lemma 4.4.
If α = 0, β = 0, λ ≠ 0, μ ≠ 0, then the bracket can be simplified as
For the bracket (14), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 3) up to isomorphism, where
It is similar to Lemma 4.4.
If α ≠ 0, β = 0, λ ≠ 0, μ = 0, then
.
For the bracket (15), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 5) up to isomorphism, where
It is similar to Lemma 4.4.
If α = 0, β ≠ 0, λ = 0, μ ≠ 0, then the bracket can be simplified as
Moreover, for α = 0, β ≠ 0, λ ≠ 0, μ ≠ 0, the bracket can be simplified as (16).
For the bracket (16), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 2) up to isomorphism, where
It is similar to Lemma 4.4.
If α ≠ 0, β ≠ 0, λ = 0, μ ≠ 0, then the bracket can be simplified as
Moreover, for α ≠ 0, β ≠ 0, λ ≠ 0, μ ≠ 0, the bracket can be simplified as (17). However, for formula (17), there are not non-zero even linear maps σ satisfying σ[ei, ej] = [σ(ei), σ(ej)] such that (L, [, ], σ) is a Hom-Lie superalgebra.
If α ≠ 0, β = 0, λ ≠ 0, μ ≠ 0, then
For the bracket (17), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ =
It is similar to Lemma 4.4.
(4) If dim [L, L] = 3, then [L, L] = L. Let [e1, e2] = α1e1 + α2e2, [e1, e3] = μe3, [e3, e3] = β1e1 + β2e2. From dim [L, L] = 3, it follows that λ2 + μ2 ≠ 0 and the matrix
If μ ≠ 0, α1 ≠ 0, β2 ≠ 0, λ = β1 = α2 = 0, then [e1, e2] = α1e1, [e1, e3] = 0, [e2, e3] = μe3, [e3, e3] = β2e2.
If λ ≠ 0, α1 ≠ 0, β2 ≠ 0, μ = β1 = α2 = 0, then [e1, e2] = α1e1, [e1, e3] = λe3, [e2, e3] = 0, [e3, e3] = β2e2.
If λ ≠ 0, α1 ≠ 0, β2 ≠ 0, μ ≠ 0, β1 = α2 = 0, then [e1, e2] = α1e1, [e1, e3] = λe3, [e2, e3] = μe3, [e3, e3] = β2e2.
For the above formulas, there are not non-zero even linear maps σ satisfying σ[ei, ej] = [σ(ei), σ(ej)] such that (L, [, ], σ) is a Hom-Lie superalgebra.
In the following paragraph, we will consider the 3-dimensional Hom-Lie superalgebra (L, [, ], σ), where dim
(1) If dim [L, L] = 0, then (L, [, ], σ) is an abelian Hom-Lie superalgebra.
(2) If dim [L, L] = 1, we consider the two cases as follows.
Case I: [L, L] = ℂe1 =
Similar to above, we have
For the bracket (19), Hom-Lie superalgebras are (L, [, ], σi) (1 ≥ i ≥ 2) up to isomorphism, where
It is similar to Lemma 4.4.
For the bracket (20), Hom-Lie superalgebras are (L, [, ], σi)(1 ≥ i ≥ 5) up to isomorphism, where
It is similar to Lemma 4.4.
Case II: [L, L] = ℂe2 ⊆
Similar to above, we have
.
For the bracket (21), Hom-Lie superalgebras are (L, [, ], σi)(1 ≥ i ≥ 5) up to isomorphism, where
It is similar to Lemma 4.4.
For the bracket (22), Hom-Lie superalgebras are (L, [, ], σi)(1 ≥ i ≥ 2) up to isomorphism, where
It is similar to Lemma 4.4.
Hom-Lie superalgebras in Lemma 4.15 and Lemma 4.16 are classical Lie superalgebras.
(3) If dim [L, L] = 2, we consider the two cases as follows.
Case I:
For the bracket (23), Hom-Lie superalgebras are (L, [, ], σi) (1 ≤ i ≤ 4) up to isomorphism, where
It is similar to Lemma 4.4.
Case II:
where α2 + β2 ≠ 0, λ2 + μ2 + γ2 ≠ 0.
Similar to above, we have
For the bracket (24), Hom-Lie superalgebras are (L, [, ], σi)(1 ≤ i ≤ 2) up to isomorphism, where
It is similar to Lemma 4.4.
For the bracket (25), the Hom-Lie superalgebra is (L, [, ]σ) up to isomorphism, where σ =
It is similar to Lemma 4.4.
For the bracket (26), Hom-Lie superalgebras are (L, [, ], σi)(1 ≤ i ≤ 2) up to isomorphism, where
It is similar to Lemma 4.4.
For the bracket (27), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ =
It is similar to Lemma 4.4.
For the bracket (28), the Hom-Lie superalgebra is (L, [, ], σ) up to isomorphism, where σ =
It is similar to Lemma 4.4.
For the bracket (29), Hom-Lie superalgebras are (L, [, ], σ) up to isomorphism,, where
It is similar to Lemma 4.4.
(4) If dim [L, L] = 3, then [L, L] = L. Let [e1, e2] = α1e2 + β1e3, [e1, e3] = α2e2 + β2e3, [e2, e2] = λe1, [e2, e3] = μe1, [e3, e3] = γe1. From dim[L, L] = 3, it follows that λ2 + μ2 + γ2 ≠ 0 and the matrix
For the above formulas, there are not non-zero even linear maps σ satisfying σ [ei, ej] = [σ(ei), σ(ej)] such that (L, [, ], σ) is a Hom-Lie superalgebra.
From Lemmas 4.2-4.24, we obtain Theorem 4.1.
By Lemma 2.8, Hom-Lie superalgebras
Acknowledgement
This work was supported by National Natural Science Foundation of China(11471090) and Science and Technology Research Fund of Jilin Province (2016111).
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