#### Rota–Baxter Operators on Pre-Lie Superalgebras

Rota-Baxter Operators on Pre-Lie Superalgebras
El-Kadri Abdaoui 0 1 2
Sami Mabrouk 0 1 2
Abdenacer Makhlouf 0 1 2
Communicated by See Keong Lee. 0 1 2
B Sami Mabrouk Mabrouksami 0 1 2
@yahoo.fr 0 1 2
El-Kadri Abdaoui 0 1 2
Abdenacer Makhlouf 0 1 2
0 Université de Haute Alsace , 4 rue des frères Lumière, 68093 Mulhouse , France
1 Faculté des Sciences, Université de Gafsa , Gafsa , Tunisia
2 Faculté des Sciences Sfax, Université de Sfax , BP 1171, 3038 Sfax , Tunisia
In this paper, we study Rota-Baxter operators and super O-operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L -dendriform superalgebras. Then we give some properties of pre-Lie superalgebras constructed from associative superalgebras, Lie superalgebras and L -dendriform superalgebras. Moreover, we provide all Rota-Baxter operators of weight zero on complex pre-Lie superalgebras of dimensions 2 and 3. Mathematics Subject Classification 17A30 · 17A36 · 17B70
Rota-Baxter operator; Super O-operator; Associative superalgebra; Lie superalgebra; Pre-Lie superalgebra; L -dendriform superalgebras
Introduction
Rota–Baxter operators of weight λ ∈ K fulfil the so-called Rota–Baxter relation which
may be regarded as one possible generalization of the standard shuffle relation [
36,51
].
They appeared for the first time in the work of the mathematician Baxter [
7
] in 1960
and were then intensively studied by Atkinson [
6
], Miller [47], Rota [50], Cartier
[
18
], and more recently, they reappeared in the work of Guo [37] and Ebrahimi-Fard
[
27
].
Pre-Lie algebras (called also left-symmetric algebras, Vinberg algebras,
quasiassociative algebras) are a class of a natural algebraic systems appearing in many
fields in mathematics and mathematical physics. They were first mentioned by Cayley
in 1890 [
20
] as a kind of rooted tree algebra and later arose again from the study
of convex homogeneous cones [53], affine manifold and affine structures on Lie
groups [40], and deformation of associative algebras [
34
]. They play an important
role in the study of symplectic and complex structures on Lie groups and Lie algebras
[
5,22,24,25,44
], phases spaces of Lie algebras [8,42], certain integrable systems [
16
],
classical and quantum Yang–Baxter equations [
26
], combinatorics [
27
], quantum field
theory [
23
] and operads [
19
]. See [
17
] for a survey. Recently, pre-Lie superalgebras,
the Z2-graded version of pre-Lie algebras also appeared in many others fields; see,
for example, [
19,34,52
]. To our knowledge, they were first introduced by
Gerstenhaber in 1963 to study the cohomology structure of associative algebras [34]. They
are a class of natural algebraic appearing in many fields in mathematics and
mathematical physics, especially in supersymplectic geometry, vertex superalgebras and
graded classical Yang–Baxter equation. Recently, classifications of complex pre-Lie
superalgebras in dimensions two and three were given by Zhang and Bai [
15
]. See
[
3,21,38,39,55
] about further results.
It turns out that the construction of pre-Lie superalgebras from associative
superalgebras uses Rota–Baxter operators. Let A be an associative superalgebra (product of
x and y is denoted by x y) and R be a Rota–Baxter operator of weight λ on A, which
means that it satisfies, for any homogeneous elements x , y in A, the identity
(0.1)
(0.2)
(0.3)
R(x )R(y) = R R(x )y + x R(y) + λx y .
If λ = 0 (resp. λ = −1), the product
resp.
x ◦ y = R(x )y − (−1)|x||y| y R(x ) − x y, ∀ x , y ∈ H(A)
defines a pre-Lie superalgebra (see Theorem 1.2).
The notion of dendriform algebras was introduced in 1995 by Loday [45].
Dendriform algebras are algebras with two operations, which dichotomize the notion of
associative algebras. The motivation came from algebraic K-theory, and they have been
studied quite extensively with connections to several areas in mathematics and physics,
including operads, homology, Hopf algebras, Lie and Leibniz algebras, combinatorics,
arithmetic and quantum field theory (see [
30
] and the references therein). The
relationship between dendriform algebras, Rota–Baxter algebras and pre-Lie algebras
was given by Aguiar and Ebrahimi-Fard [
2,27,28
]. Bai, Liu, Guo and Ni generalized
the concept of Rota–Baxter operator and introduced a new class of algebras, namely
L-dendriform algebras, in [
12–14
]. Moreover, a close relationship among associative
superalgebras, Lie superalgebras, pre-Lie superalgebras and dendriform superalgebras
is given as follows in the sense of commutative diagram of categories:
Lie superalgebra
←−
pre-Lie superalgebra
↑ ↑
associative superalgebra ←− dendriform superalgebra
Recently, the notion of Rota–Baxter operator on a bimodule was introduced by
Aguiar [1]. The construction of associative, Lie, pre-Lie and L-dendriform
superalgebras is extended to the corresponding categories of bimodules. See [
9,29,31–33,43,46
]
about further results and [
10,11,41,48,49
] about relationships with Yang–Baxter
equation.
The main purpose of this paper is to study, through Rota–Baxter operators and
O-operators, the relationship between associative superalgebras, Lie superalgebras,
pre-Lie superalgebras and L-dendriform superalgebras. Moreover, we classify Rota–
Baxter operators of weight zero on the complex pre-Lie superalgebras of dimensions
2 and 3.
This paper is organized as follows. In Sect. 1, we recall some definitions of
associative superalgebras, Lie superalgebras and pre-Lie superalgebras and we introduce
the notion of super O-operator of these superalgebras that generalizes the notion of
Rota–Baxter operators. We show that every Rota–Baxter associative superalgebra of
weight λ = −1 gives rise to a Rota–Baxter Lie superalgebra. Moreover, a super
Ooperator on a Lie superalgebra (of weight zero) gives rise to a pre-Lie superalgebra.
As an Example of computations, we provide all Rota–Baxter operators (of weight
zero) on the orthosymplectic Lie superalgebra osp(1, 2). In Sect. 2, we introduce the
notion of L-dendriform superalgebra and then study some fundamental properties of
L-dendriform superalgebras in terms of super O-operator of pre-Lie superalgebras.
Their relationship with associative superalgebras is also described. Sections 3 and 4
are devoted to classification of all Rota–Baxter operators (of weight zero) on the
complex pre-Lie superalgebras of dimension 2 and 3 with one-dimensional even part and
with two-dimensional even part, respectively.
Throughout this paper, all superalgebras are finite-dimensional and are over a field
K of characteristic zero. Let (A, ◦) be a superalgebra, then L◦ and R◦ denote the even
left and right multiplication operators L◦, R◦ : A → E nd(A) defined as L◦(x )(y) =
(−1)|x||y| R◦(y)(x ) = x ◦ y for all homogeneous element x , y in A. In particular,
when (A, [ , ]) is a Lie superalgebra, we let ad(x ) denote the adjoint operator, that
is, ad(x )(y) = [x , y] for all homogeneous element x , y in A.
Let (A, ◦) be an algebra over a field K. It is said to be a superalgebra if the underlying
vector space of A is Z2-graded, that is, A = A0 ⊕ A1, and Ai ◦ A j ⊂ Ai+ j , for
i, j ∈ Z2. An element of A0 is said to be even and an element of A1 is said to be odd.
The elements of A j , j ∈ Z2, are said to be homogenous and of parity j . The parity of
a homogeneous element x is denoted by |x |, and we refer to the set of homogeneous
elements of A by H(A).
We extend to graded case the concepts of A-bimodule K-algebra, O-operator and
extended O-operator introduced in [
13
].
Definition 1.1 (1) An associative superalgebra is a pair (A, μ) consisting of a
Z2graded vector space A and an even bilinear map μ : A ⊗ A −→ A, (Ai A j ⊆
Ai+ j , ∀ i, j ∈ Z2) satisfying for all x , y, z ∈ H(A)
x (yz) = (x y)z.
(2) Let (A, μ) be an associative superalgebra and V be a Z2-graded vector space.
Let l, r : A −→ E nd(V ) be two even linear maps. A triple (V , l, r ) is called an
A-bimodule if for all x , y ∈ H(A) and v ∈ H(V )
l(x y)(v) = l(x )l(y)(v), r (x y)(v) = r (y)r (x )(v), l(x )r (y)(v) = r (y)l(x )(v).
Moreover, the quadruple (V , μV , l, r ) is said to be an A-bimodule K-superalgebra
if (V , l, r ) is an A-bimodule compatible with the multiplication μV on V , that is,
for all x , y ∈ H(A) and v, w ∈ H(V ),
l(x )(μV (v, w)) = μV (l(x )(v), w), r (x )(μV (v, w)) = μV (v, r (x )(w)),
μV (r (x )(v), w) = μV (v, l(x )(w)).
(3) Fix λ ∈ K, a pair (T , T ) of even linear maps T , T : V −→ A is called an
extended super O-operator with modification T of weight λ associated with the
bimodule (V , l, r ) if T satisfies
λl(T (u))v = λr (T (v))u,
T (u)T (v) = T l(T (u))v + (−1)|u||v|r (T (v))u
+λT (u)T (v), ∀ u, v ∈ H(V ).
(1.1)
(1.2)
(4) An even linear map T : V −→ A is called a super O-operator of weight λ
associated with the bimodule K-superalgebra (V , μV , l, r ) if it satisfies
T (u)T (v) = T l(T (u))v + (−1)|u||v|r (T (v))u + λμV (u, v) ,
∀ u, v ∈ H(V ).
Notice that the notions of super O-operator and extended super O-operator coincide
when λ = 0.
In particular, a super O-operator of weight λ ∈ K associated with the bimodule
K-algebra (A, μA, Lμ, Rμ) is called a Rota–Baxter operator of weight λ on A, that
is, R satisfies the identity (0.1). We denote by a triple (A, μ, R) the Rota–Baxter
associative superalgebra.
We define now Rota–Baxter operators on A-bimodules.
Definition 1.2 Let (A, μ, R) be a Rota–Baxter associative superalgebra of weight
zero. A Rota–Baxter operator on an A-bimodule V (relative to R) is a map RV :
V −→ V such that for all x ∈ H(A) and v ∈ H(V )
R(x )RV (v) = RV R(x )v + x RV (v) ,
RV (v)R(x ) = RV RV (v)x + v R(x ) .
We have similar definitions on Lie superalgebras.
Definition 1.3 (1) A Lie superalgebra is a pair (A, [ , ]) consisting of a Z2-graded
vector space A, and an even bilinear map [ , ] : A ⊗ A −→ A, ([Ai , A j ] ⊆
Ai+ j , ∀ i, j ∈ Z2) satisfying for all x , y, z ∈ H(A),
[x , y] = −(−1)|x||y|[y, x ], (super-skew-symmetry)
[x , [y, z]] = [[x , y], z] + (−1)|x||y|[y, [x , z]], (super-Jacobi identity).
(2) Let (A, [ , ]) be a Lie superalgebra, V be a Z2-graded vector space, and ρ :
A −→ E nd(V ) be an even linear map. The pair (V , ρ) is said to be an A-module
or a representation of (A, [ , ]) if for all x , y ∈ H(A) and v ∈ H(V ),
ρ([x , y])(v) = ρ(x )ρ(y)v − (−1)|x||y|ρ(y)ρ(x )v.
The triple (V , [ , ]V , ρ), where [ , ]V is a super-skew-symmetric bracket, is said
to be an A-module K-superalgebra if, for x ∈ H(A) and v, w ∈ H(V ),
ρ(x )[v, w]V = [ρ(x )(v), w]V + (−1)|v||w|[v, ρ(x )(w)]V .
(3) Let (A, [ , ]) be a Lie superalgebra and (V , ρ) be a representation of A. An even
linear map T : V −→ A is called a super O-operator of weight λ ∈ K associated
with an A-module K-superalgebra (V , [ , ]V , ρ) if T satisfies:
[T (u), T (v)] = T ρ(T (u))v −(−1)|u||v|ρ(T (v))u +λ[u, v]V , ∀ u, v ∈ H(V ).
(1.3)
(1.4)
(1.5)
(1.6)
In particular, a super O-operator of weight λ ∈ K associated with the bimodule
(A, L◦, R◦) is called a Rota–Baxter operator of weight λ ∈ K on (A, [ , ]), that is,
R satisfies for all x , y, z in H(A)
[R(x ), R(y)] = R [R(x ), y] − (−1)|x||y|[R(y), x ] + λ[x , y] .
(1.7)
The triple (A, [ , ], R) refers to a Rota–Baxter Lie superalgebra, see [54].
Definition 1.4 Let (A, [ , ], R) and (A , [ , ] , R ) be two Rota–Baxter Lie
superalgebras. An even homomorphism f : (A, [ , ], R) −→ (A , [ , ] , R ) is said
to be a morphism of two Rota–Baxter Lie superalgebras if, for all x , y ∈ H(A),
f ([x , y]) = [ f (x ), f (y)] and f ◦ R = R ◦ f.
Proposition 1.1 Let (A, μ, R) be a Rota–Baxter associative superalgebra of weight
λ ∈ K. Then the triple (A, [ , ], R), where [x , y] = x y − (−1)|x||y| yx , is a Rota–
Baxter Lie superalgebra of weight λ ∈ K.
We introduce the notion of super O-operators of pre-Lie superalgebras and study
some properties over Lie superalgebras and pre-Lie superalgebras.
Definition 1.5 Let A be a Z2-graded vector space and ◦ : A ⊗ A −→ A be an even
binary operation. The pair (A, ◦) is called a pre-Lie superalgebra if, for x , y, z in
H(A), the associator
as(x , y, z) = (x ◦ y) ◦ z − x ◦ (y ◦ z)
is super-symmetric in x and y, that is, as(x , y, z) = (−1)|x||y|as(y, x , z), or
equivalently
(x ◦ y) ◦ z − x ◦ (y ◦ z) = (−1)|x||y| (y ◦ x ) ◦ z − y ◦ (x ◦ z) .
(1.8)
The identity (1.8) is called pre-Lie super-identity.
Definition 1.6 Let (A, ◦) be a pre-Lie superalgebra.
(1) Let V be a Z2-graded vector space and l, r : A −→ E nd(V ) be two even linear
maps. The triple (V , l, r ) is said to be an A-bimodule of (A, ◦) if, for x , y ∈ H(A)
and v ∈ H(V ),
l(x )l(y)v − l(x ◦ y)v = (−1)|x||y| l(y)l(x )v − l(y ◦ x )v ,
l(x )r (y)v − r (y)l(x )v = (−1)|x||v|(r (x ◦ y)v − r (y)r (x )v).
(1.9)
(1.10)
Moreover, the quadruple (V , ◦V , l, r ) is said to be an A-bimodule K-superalgebra
if (V , l, r ) is an A-bimodule compatible with the multiplication ◦V on V , that is,
for x , y ∈ H(A) and v, w ∈ H(V ),
l(x )(v ◦V w) − l(x )(v) ◦V w = (−1)|x||v|(v ◦V l(x )(w)) − r (x )(v) ◦V w),
r (x )(v ◦V w) − v ◦V r (x )(w) = (−1)|v||w|(r (x )(w ◦V v) − w ◦V r (x )(v)).
(2) Let (V , ◦V , l, r ) be an A-bimodule K-superalgebra. An even linear map T : V −→
A is called a super O-operator of weight λ ∈ K associated with (V , ◦V , l, r ) if it
satisfies:
T (u) ◦ T (v) = T l(T (u))v + (−1)|u||v|r (T (v))u + λu ◦V v , ∀ u, v ∈ H(V ).
(1.11)
In particular, a super O-operator of weight λ ∈ K associated with the A-bimodule
(A, L◦, R◦) is called a Rota–Baxter operator of weight λ on (A, ◦), that is, R satisfies
R(x ) ◦ R(y) = R R(x ) ◦ y + x ◦ R(y) + λx ◦ y
(1.12)
for all x , y, z in H(A).
Proposition 1.2 Let (A, ◦) be a pre-Lie superalgebra.
(1) The commutator
[x , y] = x ◦ y − (−1)|x||y| y ◦ x
defines a Lie superalgebra (A, [ , ]) which is called the sub-adjacent Lie
superalgebra of A and A is also called a compatible pre-Lie superalgebra structure on
the Lie superalgebra.
(2) The map L◦ gives a representation of the Lie superalgebra (A, [ , ]), that is,
L◦([x , y]) = L◦(x )L◦(y) − (−1)|x||y| L◦(y)L◦(x ).
Corollary 1.1 Let (A, ◦) be a pre-Lie superalgebra and (V , l, r ) be an A-bimodule.
Let (A, [ , ]) be the sub-adjacent Lie superalgebra. If T is a super O-operator
associated with (V , l, r ), then T is a super O-operator of (A, [ , ]) associated with
(V , l − r, r − l).
Now, we construct pre-Lie superalgebras using super O-operators on Lie
superalgebras.
Proposition 1.3 ([54]) Let (A, [ , ]) be a Lie superalgebra and (V , ρ) be a
representation of A. Suppose that T : V −→ A is a super O-operator of weight zero
associated with (V , ρ). Then, the even bilinear map
u ◦ v = ρ(T (u))v, ∀ u, v ∈ H(V )
defines a pre-Lie superalgebra structure on A.
Remark 1.1 ([54]) Let (A, [ , ]) be a Lie superalgebra and R be the super
Ooperator (of weight zero) associated with the adjoint representation (A, ad). Then the
even binary operation given by x ◦ y = [R(x ), y], for all x , y ∈ H(A), defines a
pre-Lie superalgebra structure on A.
As a direct consequence, since a Rota–Baxter operator on a pre-Lie superalgebra is
also a Rota–Baxter operator of its sub-adjacent Lie superalgebra, we have the following
observation.
Proposition 1.4 Let A1 = (A, ◦, R) be a Rota–Baxter pre-Lie superalgebra of weight
zero. Then A2 = (A, ∗, R) is a Rota–Baxter pre-Lie superalgebra of weight zero,
where the even binary operation is defined by
x ∗ y = R(x ) ◦ y − (−1)|x||y| y ◦ R(x ).
Example 1.1 In this example, we calculate Rota–Baxter operators of weight zero
on the Lie superalgebra osp(1, 2) and give the corresponding pre-Lie superalgebras.
Starting from the orthosymplectic Lie superalgebra, we consider in the sequel the
matrix realization of this superalgebra.
Let osp(1, 2) = A0 ⊕ A1 be the Lie superalgebra where A0 is spanned by
The defining relations (we give only the ones with nonzero values in the right-hand
side) are
[e1, e2] = 2e2, [e1, e3] = −2e3, [e2, e3] = e1,
[e3, e5] = e4, [e2, e4] = e5,
[e1, e4] = −e4, [e1, e5] = e5,
[e5, e4] = e1, [e5, e5] = −2e2, [e4, e4] = 2e3.
The Rota–Baxter operators of weight zero on the Lie superalgebra osp(1, 2) with
respect to the homogeneous basis {e1, e2, e3, e4, e5} are:
R1(e1) = a1e1 + a2e2 − (2a83a+12aa32)2 e3, R1(e2) = − (2a23a+2aa122)2 e1 + (22a(23−a33+aa2)2a)1 e2
+ (2a32+a13a2)2 e3, R1(e3)=a3e1 + (2a38+a1a2)2 e2 + a21((2aa23−+6aa23)) e3, R1(e4)=0, R1(e5)=0,
a1 = 0, a2 = −2a3.
R2(e1) = a1e1 + a2e2, R2(e2) = − 2aa212 e1 − 32a1 e2 + a
2a213 e3, R2(e3) = 8aa21 e2
2
2
+ a21 e3, R2(e4) = 0, R2(e5) = 0, a1 = 0, a2 = 0.
R3(e1) = a1e1 − 2aa312 e3, R3(e2) = a21 e2 + 2a132 e3, R3(e3) = a3e1 + 2aa31 e2
a3 2
− 32a1 e3, R3(e4) = 0, R3(e5) = 0, a1 = 0, a3 = 0.
R4(e1) = 0, R4(e2) = 0, R4(e3) = a3e1 + a4e2, R4(e4) = 0, R4(e5) = 0.
− a1e3, R17(e4) = 0, R17(e5) = 0, a3 = 0.
R5(e1) = a1e1 − 4a3e2 − 2aa312 e3, R5(e2) = − 4a13 e1 + a1e2 + 2a132 e3, R5(e3) =
a2 a3
a13(41a63a−33a2) e3, R16(e3) = a3e1 + aa2a13 e2 + a1(a42a−34a3) e3, R16(e4) = 0, R16(e5) =
0, a1 = 0, a3 = 0.
2 a2 a3
R17(e1) = a1e1 − aa13 e3, R17(e2) = − 4a13 e1 + 4a132 e3, R17(e3) = a3e1
R18(e1) = a1e1 − 4a3e2 − 2aa312 e3, R18(e2) = 2aa312 e1 − 72a1 e2 + 2a132 e3, R18(e3)
a3
2
= a3e1 + 2aa31 e2 + 52a1 e3, R18(e4) = 0, R18(e5) = 0, a1 = 0, a3 = 0.
a2
R19(e1) = a1e1 + 4a3e2, R4(19) = − 4a13 e1 − a1e2, R19(e3) = a3e1 +
R22(e1) = 4aa652 e2 + a6e3, R22(e2) = −a5e2 − 4aa65 e3, R22(e3) = a
4a253 e2 +
2
6
aR52e33(,e1R) 23=(e4a)2=e1,0,RR2323(e(e25)) == 00,, aR52=3(e03,) a=6 =−0a22. e2 + a4e2, R23(e4) = 0,
The constants ai are parameters.
Now, we define Rota–Baxter operators on an A-module, where A is a Rota–Baxter
Lie superalgebra.
Definition 1.7 Let (A, [ , ], R) be a Rota–Baxter Lie superalgebra of weight zero.
A Rota–Baxter operator on an A-module V (relative to R) is a map RV : V −→ V
such that, for all x ∈ H(A) and v ∈ H(V ),
[R(x ), RV (v)] = RV [R(x ), v] + [x , RV (v)] ,
[RV (v), R(x )] = RV [RV (v), x ] + [v, R(x )] ,
where the action ρ(x )(v) is denoted by [x , v].
Proposition 1.5 Let (A, [ , ], R) be a Rota–Baxter Lie superalgebra of weight zero,
V an A-module and RV a Rota–Baxter operator on V . Define new actions of A on V
by
x ◦ v = [R(x ), v],
v ◦ x = [RV (v), x ].
Equipped with these actions, V is a bimodule over the pre-Lie superalgebra
(Remark 1.1).
Proof Let x , y be a homogeneous elements in A and v in V . We have
l(x )r (y)(v) − (−1)|x||y|r (y)l(x ) − r (x ◦ y)(v) + (−1)|x||y|r (y)r (x )(v)
= (−1)|y||v|[R(x ), [RV (v), y]] − (−1)|y||v|[RV ([R(x ), v]), y]
− (−1)|v|(|x|+|y|)[RV (v), [R(x ), y]] + (−1)|v|(|x|+|y|)[RV ([RV (v), x ]), y]
= (−1)|y||v|[R(x ), [RV (v), y]] − (−1)|y||v| [RV ([R(x ), v] + [x , RV (v)]), y]
− (−1)|v|(|x|+|y|)[RV (v), [R(x ), y]]
= (−1)|y||v|[R(x ), [RV (v), y]] − (−1)|y||v|[[R(x ), RV (v)], y]
− (−1)|v|(|x|+|y|)[RV (v), [R(x ), y]]
= 0.
Then
l(x )r (y)(v) − (−1)|x||y|r (y)l(x ) = r (x ◦ y)(v) − (−1)|x||y|r (y)r (x )(v).
Similarly, we show that l(x ) ◦ l(y)v − l(x ◦ y)v = (−1)|x||y| l(y) ◦ l(x )v − l(y ◦ x )v .
Now, we construct a functor from a full sub-category of the category of Rota–Baxter
Lie-admissible (or associative) superalgebras to the category of pre-Lie superalgebras.
The Lie-admissible algebras were studied by Albert in 1948 and Goze and Remm
in 2004, they introduced the notion of G-associative algebras where G is a subgroup
of the permutation group S3 (see [
35
]). The graded case was studied by Ammar and
Makhlouf in 2010 (see [
4
] for more details).
Definition 1.8 (1) A Lie-admissible superalgebra is a superalgebra (A, μ) in which
the supercommutator bracket, defined for all homogeneous x , y in A by
[x , y] = μ(x , y) − (−1)|x||y|μ(y, x ),
satisfies the super-Jacobi identity (1.5).
(2) Let G be a subgroup of the permutation group S3. A Rota–Baxter G-associative
superalgebra of weight λ ∈ K is a G-associative superalgebra (A, ·) together with
an even linear self-map R : A −→ A that satisfies the identity
R(x ) · R(y) = R(R(x ) · y + x · R(y) − λx · y),
(1.13)
for all homogeneous elements x , y, z in A.
Theorem 1.1 Let (A, ·, R) be a Rota–Baxter Lie-admissible superalgebra of weight
zero. Define an even binary operation “∗” on any homogeneous element x , y ∈ A by
x ∗ y = R(x ) · y − (−1)|x||y| y · R(x ) = [R(x ), y].
(1.14)
Then AL = (A, ∗) is a pre-Lie superalgebra.
Proof A direct consequence of Remark 1.1, since a Rota–Baxter operator on a
Lieadmissible superalgebra is also a Rota–Baxter operator of its supercommutator Lie
superalgebra.
Theorem 1.2 Let (A, μ, R) be a Rota–Baxter associative superalgebra of weight
λ = −1. Define the even binary operation “◦” on any homogeneous element x , y ∈ A
by
= R(x )y − (−1)|x||y| y R(x ) − x y.
x ◦ y = μ(R(x ), y) − (−1)|x||y|μ(y, R(x )) − μ(x , y)
Then AL = (A, ◦) is a pre-Lie superalgebra.
(1.15)
Proof For all homogeneous elements x, y, z in A, we have
x ◦ (y ◦ z) = R(x)(R(y)z) − (−1)|y||z| R(x)(z R(y)) − R(x)(yz)
− (−1)|x|(|y|+|z|)(R(y)z)R(x) + (−1)|x|(|y|+|z|)+|y||z|(z R(y))R(x)
+ (−1)|x|(|y|+|z|)(yz)R(x) − x(R(y)z) + (−1)|y||z|(z R(y))x + x(yz),
and
(x ◦ y) ◦ z = R(R(x)y)z − (−1)|x||y| R(y R(x))z
−R(x y)z − (−1)|z|(|x|+|y|)(R(x)y)R(z)
− (R(x)y)z + (−1)|x||y|(y R(x))z + (x y)z.
+ (−1)|z|(|x|+|y|)+|x||y|(y R(x))R(z) + (−1)|z|(|x|+|y|)(x y)z
Then, we obtain
asAL (x, y, z) − (−1)|x||y|asAL (y, x, z)
= x ◦ (y ◦ z) − (x ◦ y) ◦ z − (−1)|x||y| y ◦ (x ◦ z) + (−1)|x||y|(y ◦ x) ◦ z
= R(x)(R(y)z)−(−1)|y||z| R(x)(z R(y))−R(x)(yz)−(−1)|x|(|y|+|z|)(R(y)z)R(x)
+ (−1)|x|(|y|+|z|)+|y||z|(z R(y))R(x) + (−1)|x|(|y|+|z|)(yz)R(x) − x(R(y)z)
+ (−1)|x|(|y|+|z|)+|y||z|(z R(y))x + x(yz)
− R(R(x)y)z + (−1)|x||y| R(y R(x))z + R(x y)z
− (−1)|z|(|x|+|y|)(x y)z
+ (−1)|z|(|x|+|y|)(R(x)y)R(z) − (−1)|z|(|x|+|y|)+|x||y|(y R(x))R(z)
+ (R(x)y)z + (−1)|x||y|(y R(x))z + (x y)z − (−1)|x||y| R(y)(R(x)z)
+ (−1)|x|(|y|+|z|) R(y)(z R(x))
+ (−1)|x||y| R(y)(x z) + (−1)|y||z|(R(x)z)R(y)
− (−1)|z|(|x|+|y|)(z R(x))R(y) − (−1)|y||z|(x z)R(y)
+ (−1)|x||y| y(R(x)z) − (−1)|z|(|x|+|y|)(z R(x))y
+(−1)|z|(|x|+|y|)(x R(y))R(z)
+(−1)|x||y| y(x z) + (−1)|x||y| R(R(y)x)z
−R(x R(y))z − (−1)|x||y| R(yx)z − (−1)|z|(|x|+|y|)+|x||y|(R(y)x)R(z)
+(−1)|z|(|x|+|y|)+|x||y|(yx)z−(−1)|x||y|(R(y)x)z + (x R(y))z + (−1)|x||y|(yx)z.
The above sum vanishes by associativity and the Rota–Baxter identity (1.13) with
λ = −1.
Corollary 1.2 Let (A, μ, R) be a Rota–Baxter associative superalgebra of weight
λ = −1. Then R is still a Rota–Baxter operator of weight λ = −1 on the pre-Lie
superalgebra (A, ◦) defined in (1.15).
As a consequence of Theorem 1.2 and Corollary 1.2, we have:
Proposition 1.6 Let (A, μ, R) be a Rota–Baxter associative superalgebra of weight
λ = −1. Then the binary operation defined, for any homogeneous elements x , y in A,
by
[x , y] = R(x )y − (−1)|x||y| y R(x ) − x y + x R(y) − (−1)|x||y| R(y)x + (−1)|x||y| yx ,
defines a Rota–Baxter Lie superalgebra (A, [ , ], R) of weight λ = −1.
2 L-dendriform Superalgebras
The notion of L-dendriform algebra was introduced by Bai, Liu and Ni in 2010 (see
[
14
]). In this section, we extend this notion to the graded case, and define L-dendriform
superalgebra. Then we study relationships between associative superalgebras,
Ldendriform superalgebras and pre-Lie superalgebras. Moreover, we introduce the
notion of Rota–Baxter operator (of weight zero) on the A-bimodule and we provide
a construction of associative bimodules from bimodules over L-dendriform
superalgebras and a construction of L-dendriform bimodules from bimodules over pre-Lie
superalgebras.
2.1 L-dendriform Superalgebras and Associative Superalgebras
2.1.1 Definition and Some Basic Properties
Definition 2.1 A L-dendriform superalgebra is a triple (A, , ) consisting of a
Z2graded vector space A and two even bilinear maps , : A ⊗ A −→ A satisfying,
for all homogeneous elements x , y, z in A,
x
x
(y
(y
z) = (x
z) = (x
y)
y)
−(−1)|x||y|(y
+(−1)|x||y| y
z + (x
y)
x )
(x
z + (−1)|x||y| y
z − (−1)|x||y|(y
z) − (−1)|x||y|(y
x )
x )
(x
z)
z,
The associated bracket to a L-dendriform superalgebra is defined as [x , y] = x
(−1)|x||y| y x .
Definition 2.2 (1) Let (A, , ) be a L-dendriform superalgebra, V be a Z2-graded
vector space, and l , r , l , r : A −→ E nd(V ) be four even linear maps.
The tuple (V , l , r , l , r ) is an A-bimodule if for any homogeneous elements
x , y ∈ A and u, v ∈ V, the following identities are satisfied
(a) [l (x ), l (y)] = l ([x , y]),
(b) [l (x ), l (y)] = l (x ◦ y) + (−1)|x||y|l (y)l (x ),
(c) r (x y)(v) = r (y)r (x )(v)+r (y)r (x )(v)+(−1)|x||v|l (x )r (y)(v)−
(−1)|x||v|r (y)l (x )(v) − (−1)|x||v|r (y)l (x )(v),
(2.1)
(2.2)
y −
(d) r (x
y)(v) = r (y)r (x )(v)−(−1)|x||v| l (x )r (y)(v)−l (x )r (y)(v)
+r (y)l (x )(v) ,
(e) l (x )r (y)(v) − r (y)l (x )(v) = (−1)|x||v|r (x • y)(v) − (−1)|x||v|r (y)
r (x )(v).
where x ◦ y = x y − (−1)|x||y| y x , and x • y = x y + x y.
Moreover, The tuple (V , V , V , l , r , l , r ) is an A-bimodule
Ksuperalgebra if the following identities are satisfied
(a) l (x )(u V v) − (−1)|x||u|u V l (x )(v) = l (x )(u) V v + l (x )(u) V
v − (−1)|x||u|r (x )(u) V v − (−1)|x||u|r (x )(u) V v,
(b) l (x )(u V v)−(−1)|x||u|u V l (x )(v)=l (x )(u) V v−(−1)|x||u|
r (x )(u) V v + (−1)|x||u|u V l (x )v,
(c) u V l (x )(v) = r (x )(u) V v − (−1)|x||u|l (x )(u V v) +
(−1)|x||u|l (x )(u V v) − (−1)|x||u|l (x )(u) V v.
(2) Let (A, , ) be a L-dendriform superalgebra and (V , V , V , l , r , l , r )
be an A-bimodule K-superalgebra. An even linear map T : V −→ A is called a
super O-operator of weight λ ∈ K associated with (V , V , V , l , r , l , r )
if T satisfies for any homogeneous elements u, v in V
T (u)
T (u)
T (v) = T l (T (u))v + (−1)|u||v|r (T (v))u + λu
T (v) = T l (T (u))v + (−1)|u||v|r (T (v))u + λu
V v ,
V v .
In particular, a super O-operator of weight λ ∈ K of the L-dendriform
superalgebra (A, , ) associated with the bimodule (A, L , R , L , R ) is called a
Rota–Baxter operator (of weight λ) on (A, , ), that is, R satisfies for any
homogeneous elements x , y in A
R(x )
R(x )
R(y) = R R(x )
R(y) = R R(x )
y + R(x )
y + R(x )
u + λx
y + λx
y ,
y .
The following theorem provides a construction of L-dendriform superalgebras using
super O-operators of associative superalgebras.
Theorem 2.1 Let (A, μ) be an associative superalgebra and (V , l, r ) be a
Abimodule. If T is a super O-operator of weight zero associated with (V , l, r ), then
there exists a L-dendriform superalgebra structure on V defined by
u
v = (−1)|u||v|r (T (v))u, ∀ u, v ∈ H(V ).
(2.3)
Proof For any homogeneous elements u, v and w in V , we have
u (v
w) = l(T (u))l(T (v))w,
(u v) w = (−1)|u||v|l T (r(T (v))u) w,
(−1)|u||v|v (u
w) = (−1)|u||v|l(T (v))l(T (u))w,
(−1)|u||v|(v u) w = l T (r(T (u))v) w,
(−1)|u||v|(v u) w = (−1)|u||v|l T (l(T (v))u) w.
Therefore, (V , , ) is a L-dendriform superalgebra.
A direct consequence of Theorem 2.1 is the following construction of a
Ldendriform superalgebra from a Rota–Baxter operator (of weight zero) of an
associative superalgebra.
Corollary 2.1 Let (A, μ, R) be a Rota–Baxter associative superalgebra of weight
zero. Then, the even binary operations given by
x
y = μ(R(x ), y),
x
y = μ(x , R(y)),
∀ x , y ∈ H(A)
defines a L-dendriform superalgebra structure on A.
Definition 2.3 Let (A, , ) be a L-dendriform superalgebra and R : A −→ A be
a Rota–Baxter operator of weight zero. A Rota–Baxter operator on A-bimodule V
(relative to R) is a map RV : V −→ V such that for all homogeneous elements x in
A and v in V
R(x )
RV (v)
R(x )
RV (v)
RV (v) = RV R(x )
v + x
R(x ) = RV RV (v)
x + v
RV (v) = RV R(x )
v + x
R(x ) = RV RV (v)
x + v
RV (v) ,
R(x ) ,
RV (v) ,
R(x ) .
Proposition 2.1 Let (A, μ) be an associative superalgebra, R : A −→ A a Rota–
Baxter operator on A, V an A-bimodule and RV a Rota–Baxter operator on V . Define
a new actions of A on V by
x
x
v = μ(R(x ), v), v
v = μ(x , RV (v)), v
x = μ(RV (v), x ),
x = μ(v, R(x )).
Equipped with these actions, V becomes an A-bimodule over the associated
Ldendriform superalgebra.
Corollary 2.2 Let (V , l , r , l , r ) be an A-bimodule of a dendriform
superalgebra (A, , ). Let (A, μ) be the associated associative superalgebra. If T is a
super O-operator associated with (V , l , r , l , r ), then T is a super O-operator
of (A, μ) associated with (V , l + l , r + r ).
2.2 L-dendriform Superalgebras and Pre-Lie Superalgebras
We have the following observation.
Proposition 2.2 Let (A, , ) be a L-dendriform superalgebra
(1) The even binary operation ◦ : A ⊗ A −→ A given by
x ◦ y = x
y − (−1)|x||y| y
x , ∀ x , y ∈ H(A)
defines a pre-Lie superalgebra (A, ◦) which is called the associated vertical
preLie superalgebra of (A, , ) and (A, , ) is called a compatible L-dendriform
superalgebra structure on the pre-Lie superalgebra (A, ◦).
(2) The even binary operation • : A ⊗ A −→ A given by
x • y = x
y + x
y, ∀ x , y ∈ H(A)
defines a pre-Lie superalgebra (A, •) which is called the associated horizontal
preLie superalgebra of (A, , ) and (A, , ) is called a compatible L-dendriform
superalgebra structure on the pre-Lie superalgebra (A, •).
(3) Both (A, ◦) and (A, •) have the same sub-adjacent Lie superalgebra g(A) defined
by
Corollary 2.3 Let (V , l , r , l , r ) be a bimodule of a L-dendriform superalgebra
(A, , ). Let (A, ◦) be the associated pre-Lie superalgebra. If T is a super
Ooperator associated with (V , l , r , l , r ), then T is a super O-operator of (A, ◦)
associated with (V , l , r ), where l = l + (−1)|u||v|r and r = l + r .
u
Therefore, there is a pre-Lie superalgebra structure on V defined by
u ◦ v = u
v − (−1)|u||v|v
u, ∀ u, v ∈ H(V )
as the associated vertical pre-Lie superalgebra of (V , , ) and T is a homomorphism
of pre-Lie superalgebra.
Furthermore, T (V ) = {T (v) / v ∈ V } ⊂ A is a pre-Lie sub-superalgebra of
(A, ◦) and there is a L-dendriform superalgebra structure on T (V ) given by
T (u)
T (v) = T (u
v), T (u)
T (v) = T (u
v), ∀ u, v ∈ H(V ).
(2.6)
Moreover, the corresponding associated vertical pre-Lie superalgebra structure on
T (V ) is a pre-Lie sub-superalgebra of (A, ◦) and T is a homomorphism of
Ldendriform superalgebra.
Proof For any homogeneous elements u, v and w in V , we have
u (v w) = l(T (u))l(T (v))w,
(u v) w = −l T (r(T (u))v) w,
(−1)|u||v|(v u) w = −(−1)|u||v|l T (r(T (v))u) w,
(−1)|u||v|v (u w) = −(−1)|u||v|r(T (v))l(T (u))w,
(−1)|u||v|(v u) w = (−1)|u||v|r T (r(T (v))u) w.
Hence,
Conversely, we can construct L-dendriform superalgebras from O-operators of
pre-Lie superalgebras.
Theorem 2.2 Let (A, ◦) be a pre-Lie superalgebra and (V , l, r ) be an A-bimodule.
If T is a super O-operator of weight zero associated with (V , l, r ), then there exists a
L-dendriform superalgebra structure on V defined by
u
(v
+ (−1)|u||v|(v
v)
u)
v)
w + (−1)|u||v|(v
u)
w
(u
w)
= l(T (u))l(T (v))w − (−1)|u||v|l(T (v))l(T (u))w − l(T (l(T (u)))v)w
+ l(T (r (T (u))v))w
− (−1)|u||v|l(T (r (T (v))u))w + (−1)|u||v|l(T (l(T (v))u))w
= l(T (u))l(T (v))w − (−1)|u||v|l(T (v))l(T (u))w − l T (u) ◦ T (v) w
(2.4)
(2.5)
+ (−1)|u||v|l T (v) ◦ T (u) w
and
= 0,
u
(v
= −l(T (u))r (T (v))w + (−1)|u||v|r (T (v))l(T (u))w
+ r (T (u) ◦ T (v))w − (−1)|u||v|r (T (v))r (T (u))
= 0.
Therefore, (V , , ) is a L-dendriform superalgebra. The other conditions follow
easily.
A direct consequence of Theorem 2.2, is the following construction of a
Ldendriform superalgebra from a Rota–Baxter operator (of weight zero) of a pre-Lie
superalgebra.
Corollary 2.4 Let (A, ◦) be a pre-Lie superalgebra and R be a Rota–Baxter operator
on A (of weight zero). Then even binary operations given by
x
y = R(x ) ◦ y,
x
y = −(−1)|x||y| y ◦ R(x )
(2.7)
defines a L-dendriform superalgebra structure on A.
Lemma 2.1 Let {R1, R2} be a pair of commuting Rota–Baxter operators (of weight
zero) on a pre-Lie superalgebra (A, ◦). Then R2 is a Rota–Baxter operator (of weight
zero) on the L-dendriform superalgebra (A, , ) defined in (2.7) with R = R1.
Theorem 2.3 Let (A, ◦) be a pre-Lie superalgebra. Then there exists a compatible
L-dendriform superalgebra structure on (A, ◦) such that (A, ◦) is the associated
vertical pre-Lie superalgebra if and only if there exists an invertible super O-operator
(of weight zero) of (A, ◦).
Next, we provide a construction of a L-dendriform bimodule from a bimodule over
a pre-Lie superalgebra.
Proposition 2.3 Let (A, ◦, R) be a Rota–Baxter pre-Lie superalgebra of weight zero,
V an A-bimodule and RV a Rota–Baxter operator on V . Define new actions of A on
V by
x
v = R(x ) ◦ v,
v
v = −(−1)|x||v|v ◦ R(x ), v
x = RV (v) ◦ x ,
x
x = −(−1)|x||v|x ◦ RV (v).
Equipped with actions, V is a bimodule over the L-dendriform superalgebra of
Corollary 2.4.
Proof Let x , y be homogeneous elements in A and v in V . Then, we have
= (−1)|v|(|x|+|y|)v
y) − (−1)|v|(|x|+|y|)(v
y + x
x )
y
− (−1)|y||v|x
(v
y) + (−1)|v|(|x|+|y|)(x
v)
y
= −(R(x ) ◦ y) ◦ RV (v) + (−1)|x||y|(y ◦ R(x )) ◦ RV (v)
− (−1)|x||y| y ◦ RV (x ◦ RV (v))
− (−1)|x||y| y ◦ RV (R(x ) ◦ v) + R(x ) ◦ (y ◦ RV (v))
= −(R(x ) ◦ y) ◦ RV (v) + R(x ) ◦ (y ◦ RV (v)) − (−1)|x||y| y ◦ (R(x ) ◦ RV (v))
+ (−1)|x||y|(y ◦ R(x )) ◦ RV (v)
= 0.
Therefore,
Similarly, we have
[l (x ), r (y)](v) = r (x • y)(v) − (−1)|x||y|r (y)r (x )(v).
r (x
= (−1)|v|(|x|+|y|)v
= −(−1)|v|(|x|+|y|)+|x||y| RV (v) ◦ (y ◦ R(x )) + (−1)|x|(|y|+|v|) y ◦ RV (RV (v) ◦ x )
+ (−1)|v|(|x|+|y|)+|x||y|(RV (v) ◦ y) ◦ R(x ) − (−1)|x|(|y|+|v|)(y ◦ RV (v)) ◦ R(x )
+ (−1)|x|(|y|+|v|) y ◦ RV (v ◦ R(x ))
= (−1)|x|(|y|+|v|) y ◦ (RV (v) ◦ R(x )) − (y ◦ RV (v)) ◦ R(x )
+ (−1)|v|(|x|+|y|)+|x||y| RV (v) ◦ y) ◦ R(x ) − RV (v) ◦ (y ◦ R(x ))
= 0,
then
r (x
y)(v) = (−1)|x||y|r (y)r (x )(v) + l (x )r (y)(v) + [l (x ), r (y)](v).
The others axioms are similar. Therefore, (V , l , r , l , r ) is a bimodule over the
L-dendriform superalgebra (A, , ).
3 Rota–Baxter Operators on Two-dimensional Pre-Lie Superalgebras
The purpose of this section is to compute all Rota–Baxter operators (of weight zero)
on the two-dimensional complex pre-Lie superalgebras given by Zhang and Bai (see
[
15
]). In the following, let C be the ground field of complex numbers and {e1, e2} be a
homogeneous basis of a pre-Lie superalgebra (A, ◦), where {e1} is a basis of A0 and
{e2} is a basis of A1.
By direct computation and by help of a computer algebra system, we obtain the
following results.
Proposition 3.1 The Rota–Baxter operators (of weight zero) on two-dimensional
preLie superalgebras (associative or non-associative) of type B1, B2 and B3 are given
as follows:
3.1 Rota–Baxter Operators on Pre-Lie Superalgebras of Type B1
The pre-Lie superalgebra (B1,1, ◦) : e2 ◦ e1 = e2 has the Rota–Baxter operator
defined as
R1(e1) = a1e1, R1(e2) = 0.
⎧
⎪⎪⎪⎪⎨
The pre-Lie superalgebras of type
(B1,2, ◦) : e1 ◦ e1 = e1, e2 ◦ e1 = e2 (associative).
(B1,3)k , ◦ :
(B1,4, ◦) :
e1 ◦ e1 = ke1,
e1 ◦ e1 = e1,
e2 ◦ e1 = e2, k = 0, 1.
e1 ◦ e2 = e2 (associative).
(B1,5)k , ◦ : e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e2 ◦ e1 = (k + 1)e2, k = 0, −1.
have only the trivial Rota–Baxter operator, that is,
R1(e1) = 0, R1(e2) = 0.
3.2 Rota–Baxter Operators on Pre-Lie Superalgebras of Type B2
The pre-Lie superalgebra (B2,1, ◦) (associative).
R1(e1) = a1e1, R1(e2) = 0.
R2(e1) = a1e1, R2(e2) = 2a1e2.
The pre-Lie superalgebra (B2,2, ◦) (associative)
Rota–Baxter operators RB(B2,2) is:
1
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e2 ◦ e2 = 2 e1.
R1(e1) = 0, R1(e2) = 0.
3.3 Rota–Baxter Operators on Pre-Lie Superalgebras of Type B3
The pre-Lie superalgebra (B3,1, ◦) (associative).
The pre-Lie superalgebra (B3,2, ◦) (associative)
Rota–Baxter operators RB(B3,2) are:
R1(e1) = 0, R1(e2) = a1e2.
The pre-Lie superalgebra (B3,3, ◦) (associative)
R1(e1) = 0, R1(e2) = 0.
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2.
4 Rota–Baxter Operators on Three-dimensional Pre-Lie Superalgebras
4.1 Rota–Baxter Operators on Three-dimensional Pre-Lie Superalgebras with
Two-dimensional Odd Part
We still work over the ground field C of complex numbers. Using the classification
of the three-dimensional pre-Lie superalgebras with one-dimensional even part was
given by Zhang and Bai (see [
15
]). The purpose of this section is to provide, using
Definition 1.6, all Rota–Baxter operators (of weight zero) on these pre-Lie
superalgebras by direct computation. In the following, let {e1, e2, e3} be a homogeneous basis
of a pre-Lie superalgebra (A, ◦), where {e1} is a basis of A0 and {e2, e3} is a basis of
A1.
Proposition 4.1 The Rota–Baxter operators (of weight zero) on three-dimensional
pre-Lie superalgebras (associative or non-associative) with two-dimensional odd part
of type C1, C2h , C3, C4, C5 and C6 are given as follows:
4.1.1 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C1
The pre-Lie superalgebra (C1,1, ◦)
e2 ◦ e3 = −e1, e3 ◦ e1 = e2, e3 ◦ e2 = e1.
Rota–Baxter operators RB(C1,1) are:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + e3.
R2(e1) = 0, R2(e2) = a2e2, R2(e3) = a1e2, a2 = 0.
R3(e1) = a3e1, R3(e2) = 0, R3(e3) = a1e2.
The pre-Lie superalgebra (C1,2)k, ◦ (associative)
e1 ◦ e3 = ke2, e3 ◦ e1 = (k + 1)e2.
Rota–Baxter operators RB((C1,2)k) are:
R1(e1) = 0, R1(e2) = a1e2, R1(e3) = a2e2, k = −1.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e2 + a3e3, k = −1.
R3(e1) = a4e1, R3(e2) = a1e2, R3(e3) = a2e2 + aa41−aa41 e3, a1 = a4, k = −1.
The pre-Lie superalgebra (C1,3, ◦):
e1 ◦ e1 = e1, e3 ◦ e1 = e2.
Rota–Baxter operators RB(C1,3) are:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3.
R2(e1) = 0, R2(e2) = a3e2, R2(e3) = a1e2.
The pre-Lie superalgebra (C1,4, ◦):
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e2 + e3.
Rota–Baxter operators RB(C1,4) are:
R1(e1) = 0, R1(e3) = 0, R1(e2) = a1e2.
R2(e1) = 0, R2(e2) = 0, R2(e3) = 0.
4.1.2 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C2h
The pre-Lie superalgebra (C2h,1, ◦):
e1 ◦ e1 = (h + 1)e1, e2 ◦ e1 = e2, e2 ◦
e3 = −e1, e3 ◦ e1 = he3, e3 ◦ e2 = e1, h ∈ C.
Rota–Baxter operators RB(C2h,1) are:
Case 1: If h = 0, we have
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3.
R2(e1) = 0, R2(e2) = a3e3, R2(e3) = a2e3.
R3(e1) = 0, R3(e2) = 0, R3(e3) = a2e3.
R4(e1) = 0, R4(e2) = a3e3, R4(e3) = a2e3, a3 = 0.
Case 2: If h ∈ C∗, we have
R5(e1) = 0, R5(e2) = 0, R5(e3) = a1e2.
R6(e1) = 0, R6(e2) = a3e3, R6(e3) = 0, a3 = 02.
a
R7(e1) = 0, R7(e2) = a5e2 + a3e3, R7(e3) = − ha53 e2 − ah5 e3, a3 = 0, a5 = 0.
R8(e1) = 0, R8(e2) = 0, R8(e3) = 0.
Case 3: If h = −1, we have
R9(e1) = 0, R9(e2) = a3e3, R9(e3) = 0.
The pre-Lie superalgebra (C2h,2, ◦):
e1 ◦ e1 = (1 − h)e1, e1 ◦ e3 = e2, e2 ◦ e1 = e2, e3 ◦ e1 = e2 + he3, h ∈ C.
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a1e2, a1 = 0.
R3(e1) = 0, R3(e2) = e2, R3(e3) = 0.
Case 3: If h = 1, we have
R4(e1) = a3e1, R4(e2) = 0, R4(e3) = a1e2, a3 = 0.
R5(e1) = 0, R5(e2) = 0, R5(e3) = a1e2, a1 = 0.
R6(e1) = 0, R6(e2) = e2, R6(e3) = 0.
The pre-Lie superalgebra (C2h,3, ◦):
e1 ◦ e1 = (1 − h)e1, e1 ◦ e2 = (1 − h)e2 + e3, e1 ◦
e3 = (1 − h)e3, e2 ◦ e1 = (2 − h)e2 + e3, e3 ◦ e1 = e3, h = 1.
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e3.
R2(e1) = 0, R2(e2) = a2e2 + a3e3, R2(e3) = a2(h − 1)e2 + a2(h − 1)e3.
Case 2: If h = 1, we have
R3(e1) = 0, R3(e2) = 0, R3(e3) = 0.
R4(e1) = 0, R4(e2) = 0, R4(e3) = a1e3, a1 = 0.
R5(e1) = 0, R5(e2) = 0, R5(e3) = a1e3, h3 − 2h2 + 2h − 1 = 0.
The pre-Lie superalgebra (C2h,4, ◦):
e1 ◦ e1 = (h − 1)e1, e1 ◦ e2 = e3, e2 ◦ e1 = e2 + e3, e3 ◦ e1 = he3, h = ±1.
Rota–Baxter operators RB(C2h,4) are:
R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3, h = 0.
R2(e1) = 0, R2(e2) = a1e3, R2(e3) = 0, a1 = 0, h = 0.
R3(e1) = 0, R3(e2) = 0, R3(e3) = 0, h = 0.
The pre-Lie superalgebra (C2h,5, ◦):
e1 ◦ e1 = (1 − h)e1, e1 ◦ e2 = (1 − h)e2 + e3, e1 ◦
e3 = (1 − h)e3, e2 ◦ e1 = (2 − h)e2 + e3, e3 ◦ e1 = e3, h = ±1.
Rota–Baxter operators RB(C2h,5) are:
R1(e1) = 0, R1(e2) = e2, R1(e3) = a1e2 − a1e3, h = 0.
R2(e1) = 0, R2(e2) = 0, R2(e3) = 0, h = 0.
R3(e1) = 0, R3(e2) = a2e2 − (h −a31)2 e3, R3(e3) = a2(hh−1)2 e2 − ah2 e3, a2 =
0, h = 0.
R4(e1) = 0, R4(e2) = a2e2 − a2e3, R4(e3) = a22 e2 − a22 e3, a2 = 0, h = 2.
R5(e1) = 0, R5(e2) = a3e3, R5(e3) = 0, a3 = 0.
The pre-Lie superalgebra (C2h,6)k, ◦ : h = 0 or 1, k = 1 associative other
cases non-associative.
e1 ◦ e1 = ke1, e2 ◦ e1 = e2, e3 ◦ e1 = he3, h, k ∈ C.
Rota–Baxter operators RB((C2h,6)k) are:
Case 1: If h = 0, we have
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3.
R2(e1) = 0, R2(e2) = a3e3, R2(e3) = a2e3.
R3(e1) = 0, R3(e2) = 0, R3(e3) = a2e3.
Case 2: If h ∈ C∗, we have
R4(e1) = 0, R4(e2) = 0, R4(e3) = a1e2.
R5(e1) = 0, R5(e2) = a3e3, R5(e3) = 0.
a2
R6(e1) = 0, R6(e2) = a4e2 + a3e3, R6(e3) = − ha43 e2 − ah4 e3, a3 = 0.
R7(e1) = 0, R7(e2) = 0, R7(e3) = 0.
Case 3: If h = k = 0, we have
R8(e1) = 0, R8(e2) = 0, R8(e3) = a1e2 + a2e3.
R9(e1) = 0, R9(e2) = a3e3, R9(e3) = a2e3.
R10(e1) = a5e1, R10(e2) = 0, R10(e3) = a2e3.
R11(e1) = a5e1, R11(e2) = 0, R11(e3) = 0.
R12(e1) = 0, R12(e2) = a3e3, R12(e3) = a2e3, a3 = 0.
Case 4: If h = 1 and k = 0, we have
R13(e1) = a5e1, R13(e2) = 0, R13(e3) = a1e2.
R14(e1) = a5e1, R14(e2) = a3e3, R14(e3) = 0.
2
R15(e1) = a5e1, R15(e2) = a4e2 + a3e3, R15(e3) = − aa43 e2 − a4e3, a3 = 0.
Case 5: If h = 0, 1 and k = 0, we have
R16(e1) = 0, R16(e2) = 0, R16(e3) = a1e2, a1 = 0.
R17(e1) = 0, R17(e2) = a3e3, R17(e3) = 0, r2,2 = 02.
a
R18(e1) = 0, R18(e2) = a4e2 + a3e3, R18(e3) = − ha43 e2 + ah4 e3, a3 = 0.
The pre-Lie superalgebra (C2h,7)k, ◦ : (C20,7)1 associative.
e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e2 ◦ e1 = (k + 1)e2, e3 ◦ e1 = he3, , k = 0, h = ±1.
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3, a1 = 0.
R2(e1) = 0, R2(e2) = a3e3, R2(e3) = a2e3, a3 = 0.
R3(e1) = 0, R3(e2) = 0, R3(e3) = a2e3.
R4(e1) = 0, R4(e2) = 0, R4(e3) = a1e2, a1 = 0.
R5(e1) = 0, R5(e2) = a3e3, R5(e3) = 0, a3 = 0.
R6(e1) = 0, R6(e2) = 0, R6(e3) = 0.
The pre-Lie superalgebra (C2h,8)k, ◦ : h = −1 or 0, k = 1 associative other
cases are non-associative
e1 ◦ e1 = ke1, e1 ◦ e3 = ke3, e2 ◦ e1 = e2, e3 ◦ e1 = (h + k)e3, h ∈ C, k = 0.
Rota–Baxter operators RB((C2h,8)k) are:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2.
R2(e1) = 0, R2(e2) = 0, R2(e3) = 0.
The pre-Lie superalgebra (C2h)k,9, ◦ : h = 0 or 1, k = −1 associative other
cases are non-associative
e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e1 ◦ e3 = ke3, e2 ◦
e1 = (k + 1)e2, e3 ◦ e1 = (h + k)e3, h ∈ C, k = 0.
Rota–Baxter operators RB((C2h,9)k) are:
2
R1(e1) = 0, R1(e2) = a1e2 + a2e3, R1(e3) = − aa12 e2 − a1e3, a2 = 0, h =
1, k = −1.
R2(e1) = 0, R2(e2) = a2e3, R2(e3) = 0.
R3(e1) = 0, R3(e2) = 0, R3(e3) = a3e2.
R4(e1) = 0, R4(e2) = 0, R4(e3) = 0.
4.1.3 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C3
The pre-Lie superalgebra (C3,1, ◦):
e1 ◦ e1 = 2e1, e2 ◦ e1 = e2, e2 ◦ e3 = −e1, e3 ◦ e1 = e2 + e3, e3 ◦ e2 = e1.
Rota–Baxter operators RB(C3,1) are:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2.
R2(e1) = 0, R2(e2) = a2e1+a3e2, R2(e3) = − a22+a2a3 e2−(a2+a3)e3, a3 = 0.
2
The pre-Lie superalgebra (C3,2)k , ◦ :
e1 ◦ e1 = ke1, e2 ◦ e1 = e2, e3 ◦ e1 = e2 + e3.
Rota–Baxter operators RB((C3,2)k ) are:
Case 1: If k = 0, we have
R1(e1) = a1e1, R1(e2) = 0, R1(e3) = a2e2, a1 = 0.
R2(e1) = a1e1, R2(e2) = 0, R2(e3) = a1e2, a1 = 0.
R3(e1) = a1e1, R3(e2) = 0, R3(e3) = a2e2, a1 = 0.
Case 2: If k ∈ C∗, we have
R4(e1) = 0, R4(e2) = 0, R4(e3) = a2e2.
R5(e1) = 0, R5(e2) = a3e2+a4e3, R5(e3) = − a32+a3a4 e2−(a3+a4)e3, a4 = 0.
a4
R6(e1) = 0, R6(e2) = 0, R6(e3) = 0.
The pre-Lie superalgebra (C3,3)k , ◦ :
e1 ◦ e3 = ke2, e2 ◦ e1 = e2, e3 ◦ e1 = (k + 1)e2 + e3, k = 0.
Rota–Baxter operators RB((C3,3)k ) are:
R1(e1) = a1e1, R1(e2) = 0, R1(e3) = a2e2.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e2.
R3(e1) = 0, R3(e2) = 0, R3(e3) = 0.
The pre-Lie superalgebra (C3,4)k , ◦ :
Rota–Baxter operator RB((C3,4)k ) is:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2.
e1 ◦ e1 = ke1, e1 ◦ e2 = ke2, e1 ◦ e3 = ke3, e2 ◦
e1 = (k + 1)e2, e3 ◦ e1 = e2 + (k + 1)e3, k = 0.
4.1.4 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C4
The pre-Lie superalgebra (C4,1, ◦): (associative)
e2 ◦ e3 = −e1, e3 ◦ e2 = e1.
Rota–Baxter operators RB(C4,1) are:
R1(e1) = a1e1, R1(e2) = a2e2 + a3e3, R1(e3) = a4e2 − a1aa21−+aa23a4 e3, a1 = a2.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a4e2 + a5e3.
2
R3(e1) = a1e1, R3(e2) = a1e2 + a3e3, R3(e3) = − aa13 e2 + a5e3, a3 = 0.
The pre-Lie superalgebra (C4,2, ◦): (associative).
R1(e1) = a1e1, R1(e2) = a2e2 + a3e3, R1(e3) = a4e2 + a5e3.
The pre-Lie superalgebra (C4,3, ◦): (associative)
R1(e1) = 0, R1(e2) = a1e2 + a2e3, R1(e3) = a3e2 + a4e3.
The pre-Lie superalgebra (C4,4, ◦): (associative)
e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = e3.
Rota–Baxter operators RB(C4,4) are:
R1(e1) = 0, R1(e2) = a1e2, R1(e3) = a2e2.
R2(e1) = 0, R2(e2) = a1e2 + a3e3, R2(e3) = 0.
R3(e1) = 0, R3(e2) = a1e2, R3(e3) = 0.
The pre-Lie superalgebra (C4,5, ◦): (associative)
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e3.
Rota–Baxter operators RB(C4,5) are:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2.
2
R2(e1) = 0, R2(e2) = a2e2 + a3e3, R2(e3) = − aa23 e2 − a2e3, a3 = 0.
R3(e1) = 0, R3(e2) = 0, R3(e3) = 0.
The pre-Lie superalgebra (C4,6, ◦): (associative)
e1 ◦ e3 = e2, e3 ◦ e1 = e2.
Rota–Baxter operators RB(C4,6) are:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e2 + a2e3.
R2(e1) = a3e1, R2(e2) = a4e2, R2(e3) = a1e2 + aa33−aa44 e3, a3 = a4.
4.1.5 Rota–Baxter Operators on Pre-Lie Superalgebras of Type C5
The pre-Lie superalgebra (C5,1)k , ◦ : ((C5,1)0 is associative).
e1 ◦ e2 = ke3, e2 ◦ e1 = ke3, e3 ◦ e3 = e1, k = 0 or 1.
Rota–Baxter operators RB((C5,1)k ) are:
Case 1: If k = 0, we have
R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3.
R2(e1) = 0, R2(e2) = a1e3, R2(e3) = a3e2.
R3(e1) = 0, R3(e2) = a4e2, R3(e3) = a3e2.
R4(e1) = 0, R4(e2) = 0, R4(e3) = a2e3.
R5(e1) = a5e1, R5(e2) = a4e2, R5(e3) = aa55−aa44 e3, a4 = a5.
Case 2: If k = 1, we have
R6(e1) = 0, R6(e2) = a1e3, R6(e3) = a2e3.
R7(e1) = 0, R7(e2) = 0, R7(e3) = a2e3.
R8(e1) = 0, R8(e2) = a1e3, R8(e3) = e3.
R9(e1) = a5e1, R9(e2) = a5e2, R9(e3) = a25 e3, a5 = 0.
R10(e1) = a5e1, R10(e2) = a5a−51 e2, R10(e3) = e3, a5 = 1.
The pre-Lie superalgebra (C5,2, ◦):
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e2 ◦ e3 = e1.
Rota–Baxter operators RB(C5,2) are:
R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3.
R3(e1) = 0, R3(e2) = 0, R3(e3) = a3e2.
The pre-Lie superalgebra (C5,3, ◦):
e1 ◦ e1 = e1, e1 ◦ e2 = e2 + e3, e2 ◦ e1 = e2 + e3, e2 ◦ e3 = e1.
Rota–Baxter operators RB(C5,3) are:
R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3.
R3(e1) = 0, R3(e2) = a3e2, R3(e3) = −a3e2.
The pre-Lie superalgebra (C5,4)k , ◦ : (associative)
e2 ◦ e3 =
1
2 + k e1, e3 ◦ e2 =
1 1
2 − k e1, k ≥ 0, k = 2 .
Rota–Baxter operators RB((C5,4)k) are:
2
R8(e1) = a5e1, R8(e2) = a5e2 + a1e3, R8(e3) = aa51 e2 + a5e3, a1 = 0, a5 =
0, k = 0.
R9(e1) = a5e1, R9(e2) = a3e2, R9(e3) = − aa55−aa33 e3, a3 = a5.
4.1.6 Rota–Baxter Operators on Pre-Lie Superalgebras of TypeC6
The pre-Lie superalgebra (C6,1, ◦) (associative)
1
e2 ◦ e2 = 2 e1, e2 ◦ e3 = −e1, e3 ◦ e2 = e1.
Rota–Baxter operators RB(C6,1) are:
R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3, a1 = 0.
R2(e1) = a3e1, R2(e2) = a1e3, R2(e3) = 0.
The pre-Lie superalgebra (C6,2, ◦): (associative)
R1(e1) = a1e1, R1(e2) = a2e3, R1(e3) = a3e3.
R2(e1) = a1e1, R2(e2) = 2a1e2 + a2e3, R2(e3) = a3e3.
The pre-Lie superalgebra (C6,3, ◦) (associative)
1
e1 ◦ e2 = e3, e2 ◦ e1 = e3, e2 ◦ e2 = 2 e1.
Rota–Baxter operators RB(C6,3) are:
R1(e1) = 0, R1(e2) = a1e2, R1(e3) = a2e3.
RR23((ee11)) == aa33ee11,, RR32((ee22)) == 2aa1e33e,2 +R2a(1ee33),=R03,(e3a)3== 203a.3 e3.
The pre-Lie superalgebra (C6,4, ◦): (associative)
1
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e2 ◦ e2 = 2 e1.
Rota–Baxter operators RB(C6,4) are:
R1(e1) = 0, R1(e2) = a1e3, R1(e3) = a2e3.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3.
4.2 Classification of Rota–Baxter Operator on Three-dimensional Pre-Lie
Superalgebras with Two-Dimensional Even Part
In this section, we describe all Rota–Baxter operators of weight zero on the
threedimensional complex pre-Lie superalgebras with two-dimensional even part which
were classified in [
15
] by Zhang and Bai. In the following, let {e1, e2, e3} be a
homogeneous basis of a pre-Lie superalgebra (A, ◦), where {e1, e2} is a basis of A0 and
{e3} is a basis of A1. The computation is obtained using computer algebra system, and
the operators are described with respect to the basis.
Proposition 4.2 The Rota–Baxter operators (of weight zero) on three-dimensional
pre-Lie superalgebras (associative or non-associative) with two-dimensional even
part of type A1, A2, A3, A4, A5, A6, A7h , A8, A9, A10h and A11 are given
as follows:
4.2.1 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A1
The pre-Lie superalgebra ( A1,1, ◦)
e1 ◦ e1 = e1, e2 ◦ e2 = e2.
Rota–Baxter operators RB( A1,1) are:
R1(e1) = 0, R1(e2) = 01, R1(e3) = a1e3.
R2(e1) = 0, R2(e2) = 2 e2, R2(e3) = a1e3.
1
R3(e1) = 0, R3(e2) = e1 + 2 e2, R3(e3) = a1e3.
The pre-Lie superalgebra ( A1,2)k , ◦
D1, ( A1,2)1 associative.
e1 ◦ e1 = e1, e2 ◦ e2 = e2, e3 ◦ e1 = ke3, k ∈ C∗.
Rota–Baxter operators RB(( A1,2)k ) are:
R1(e1) = 0, R1(e2) = 01, R1(e3) = 0.
R2(e1) = 0, R2(e2) = 2 e2, R2(e3) = 0.
1
R3(e1) = 0, R3(e2) = e1 + 2 e2, R3(e3) = 0.
The pre-Lie superalgebra ( A1,3)k1,k2 , ◦
D1.
e1 ◦ e1 = e1, e2 ◦ e2 = e2, e3 ◦ e1 = k1e3, e3 ◦
e2 = k2e3, k1, k2 = 0, k1 ≤ k2, k1 = −k2.
Rota–Baxter operators RB(( A1,3)k1,k2 ) are:
If k2 < 0,
k1 ≤ k2 or (k2 > 0,
k1 < −k2) or (−k2 < k1 < 0) or (0 <
R1(e1) = 0, R1(e2) = 0, R1(e3) = 0.
The pre-Lie superalgebra ( A1,4, ◦)
e1 ◦ e1 = e1, e2 ◦ e2 = e2, e2 ◦ e3 = e3, e3 ◦ e2 = e3.
Rota–Baxter operators RB( A1,4) are:
D1, k1 = k2 = 0 or k1 = 1, k2 =
e1 ◦ e1 = e1, e2 ◦ e2 = e2, e2 ◦ e3 = e3, e3 ◦
e1 = k1e3, e3 ◦ e2 = k2e3, k1 = 0 or k2 = 1.
Rota–Baxter operator RB(( A1,5)k1,k2 ) is:
R1(e1) = 0, R1(e2) = 0, R1(e3) = 0.
The pre-Lie superalgebra ( A1,6, ◦)
e1 ◦ e1 = e1, e2 ◦ e2 = e2, , e2 ◦ e3 = e3, e3 ◦ e2 = e3, e3 ◦ e3 = e2.
Rota–Baxter operators RB( A1,6) are:
R1(e1) = 0, R1(e2) = 01, R1(e3) = 0.
R2(e1) = 0, R2(e2) = 2 e2, R2(e3) = 0.
1
R3(e1) = 0, R3(e2) = e1 + 2 e2, R3(e3) = 0.
4.2.2 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A2
The pre-Lie superalgebra ( A2,1, ◦)
Rota–Baxter operators RB( A2,1) are:
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = a2e3.
1
R2(e1) = 0, R2(e2) = 2 e1, R2(e3) = a2e3.
The pre-Lie superalgebra ( A2,2)k , ◦
D1, ( A2,2)1 is associative.
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e1, e3 ◦ e1 = ke3, k = 0.
Rota–Baxter operators RB(( A2,2)k ) are:
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e2 ◦ e1 = e2, e3 ◦ e1 = ke3, e3 ◦ e2 = e3.
Rota–Baxter operators RB(( A2,3)k ) are:
The pre-Lie superalgebra ( A2,4, ◦)
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e3.
Rota–Baxter operator RB( A2,4) is:
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0.
The pre-Lie superalgebra ( A2,5)k , ◦
D1, ( A2,5)0 is associative.
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = ke3, k = 1.
Rota–Baxter operators RB(( A2,5)k ) are:
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = a2e3, k = 0.
R2(e1) = a1e2, R2(e2) = 0, R2(e3) = 0, k = 0.
The pre-Lie superalgebra ( A2,6)k , ◦
D1:
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = ke3, e3 ◦ e2 = e3.
Rota–Baxter operators RB(( A2,6)k ) are:
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0.
The pre-Lie superalgebra ( A2,7, ◦)
e1 ◦ e1 = e1, e1 ◦ e2 = e2, e1 ◦ e3 = e3, e2 ◦ e1 = e2, e3 ◦ e1 = e3, e3 ◦ e3 = e2.
Rota–Baxter operators RB( A2,7) are:
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0.
4.2.3 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A3
The pre-Lie superalgebra ( A3,1, ◦)
Rota–Baxter operators RB( A3,1) are:
R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = a3e3.
The pre-Lie superalgebra ( A3,4, ◦)
D2: (associative).
e1 ◦ e1 = e1, e3 ◦ e3 = e2.
Rota–Baxter operators RB( A3,4) are
R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = 0.
R2(e1) = a1e2, R2(e2) = a2e2, R2(e3) = 2a2e3.
The pre-Lie superalgebras of type
( A3,2)k , ◦
D1 : e1◦e1 = e1, e3◦e1 = ke3, k = 0, (( A3,2)1 is associative).
D3
D1 :
:
D1
e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = ke3, k = 1,
D1 :
:
e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = ke3, e3 ◦ e2 = e3.
e1 ◦ e1 = e1, e1 ◦ e3 = e3, e3 ◦ e1 = e3, e3 ◦ e3 = e1,
They have the same Rota–Baxter operators:
R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = 0.
4.2.4 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A4
The pre-Lie superalgebra ( A4,1, ◦)
ei ◦ e j = 0, ∀ i, j = 1, 2, 3.
Rota–Baxter operators RB( A4,1) are:
R1(e1) = a1e1 + a2e2, R1(e2) = a3e1 + a4e2, R1(e3) = a5e3.
The pre-Lie superalgebra ( A4,2, ◦)
D1:
e3 ◦ e1 = e3.
Rota–Baxter operators RB( A4,2) are:
R1(e1) = a1e1 + a2e2, R1(e2) = a3e1 + a4e2, R1(e3) = 0.
The pre-Lie superalgebra ( A4,3, ◦)
D2: (associative)
e3 ◦ e3 = e2.
Rota–Baxter operators RB( A4,3) are:
R1(e1) = a1e1 + a2e2, R1(e2) = a3e2, R1(e3) = 2a3e3.
R2(e1) = a1e1 + a2e2, R2(e2) = a4e1 + a3e2, R2(e3) = 0.
4.2.5 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A5
The pre-Lie superalgebra ( A5,1, ◦)
D2 (associative)
e1 ◦ e1 = e2, e3 ◦ e3 = e2.
Rota–Baxter operators RB( A5,1) are:
R1(e1) = a1e2, R1(e2) = a2e2, R1(e3) = 0.
R2(e1) = a1e2, R2(e2) = a2e2, R2(e3) = 2a2e3.
a3 e2, R3(e3) = 0.
RR34((ee11)) == aa33ee11 ++ aa11ee22,, RR34((ee22)) == a223 e2, R4(e3) = a3e3.
The pre-Lie superalgebra ( A5,2, ◦)
Rota–Baxter operators RB( A5,2) are:
a1 e2, R1(e3) = a3e3.
R1(e1) = a1e1 + a2e2, R1(e2) = 2
R2(e1) = a2e2, R2(e2) = a4e2, R2(e3) = 0.
The pre-Lie superalgebra ( A5,3, ◦)
D1:
e1 ◦ e1 = e2, e3 ◦ e1 = e3.
Rota–Baxter operators RB( A5,3) are:
a1 e2, R1(e3) = 0.
R1(e1) = a1e1 + a2e2, R1(e2) = 2
R2(e1) = a2e2, R2(e2) = a3e2, R2(e3) = 0.
The pre-Lie superalgebra ( A5,4, ◦)
D1:
Rota–Baxter operators RB( A5,4) are:
a1 e2, R1(e3) = 0.
R1(e1) = a1e1 + a2e2, R1(e2) = 2
R2(e1) = a2e2, R2(e2) = a3e2, R2(e3) = 0.
4.2.6 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A6
The pre-Lie superalgebra ( A6,1)k , ◦
other cases non-associative
(D4)μ : k = 0 or
− 1 associative
e1 ◦ e2 = −e1, e2 ◦ e2 = −e2, e3 ◦ e2 = ke3.
Rota–Baxter operators RB(( A6,1)k ) are:
Case 1: If k = 0, we have
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = a2e3.
R2(e1) = 0, R2(e2) = a3e1, R2(e3) = a2e3.
Case 2: If k ∈ C∗, we have
R4(e1) = 0, R4(e2) = a3e1, R4(e3) = 0.
R5(e1) = a1e2, R5(e2) = 0, R5(e3) = 0.
R5(e1) = a1e2, R5(e2) = 0, R5(e3) = 0, a1 = 0.
The pre-Lie superalgebra ( A6,2, ◦)
D5:
1
e1 ◦ e2 = −e1, e2 ◦ e2 = −e2, e3 ◦ e2 = − 2 e3, e3 ◦ e3 = e1.
Rota–Baxter operators RB( A6,2) are:
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0.
R2(e1) = 0, R2(e2) = a2e1, R2(e3) = 0.
The pre-Lie superalgebra ( A6,3)k , ◦
other cases are non-associative
e1 ◦ e2 = −e1, e2 ◦ e2 = −e2, e2 ◦ e3 = −e3, e3 ◦ e2 = ke3.
Rota–Baxter operators RB(( A6,3)k ) are:
R1(e1) = a1e2, R1(e2) = 0, R1(e3) = 0, a1 = 0.
R2(e1) = 0, R2(e2) = a2e1, R2(e3) = 0.
4.2.7 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A7h
The pre-Lie superalgebra ( A7h,1, ◦)
D5:
1
e1 ◦ e2 = −e1, e2 ◦ e2 = he2, e3 ◦ e2 = − 2 e3, e3 ◦ e3 = e1, h = −1.
Rota–Baxter operators RB( A7h,1) are:
R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0.
R2(e1) = 0, R2(e2) = a2e2, R2(e3) = 0, h = 0.
(D4)μ : k = 0 or
− 1 associative
R3(e1) = 0, R3(e2) = 0, R3(e3) = 0, h = 0.
The pre-Lie superalgebra ( A7h,2)k, ◦
R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0.
R2(e1) = 0, R2(e2) = 0, R2(e3) = 0.
The pre-Lie superalgebra ( A7h,3)k, ◦
e1 ◦ e2 = −e1, e2 ◦ e2 = he2, e3 ◦ e2 = ke3, h = −1.
Rota–Baxter operators RB(( A7h,3)k) are:
4.2.8 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A8
The pre-Lie superalgebra ( A8,1)k, ◦
(D4)μ:
e1 ◦ e1 = 2e1, e2 ◦ e1 = e2, e2 ◦ e2 = e1, e3 ◦ e1 = ke3.
Rota–Baxter operators RB(( A8,1)k) are:
R1(e1) = 0, R1(e2) = 0, R1(e3) = a1e3, k = 0.
R2(e1) = 0, R2(e2) = 0, R2(e3) = 0.
4.2.9 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A9
The pre-Lie superalgebra ( A9,1)k , ◦
others cases are non-associative.
(D4)μ: k = 0 or 1 associative
e2 ◦ e1 = e1, e2 ◦ e2 = e2, e3 ◦ e2 = ke3.
Rota–Baxter operators RB(( A9,1)k) are:
Case 1: If k = 0, we have
2
R1(e1) = a1e1 + a2e2, R1(e2) = − aa12 e1 − a1e2, R1(e3) = a3e3, a2 = 0.
R2(e1) = 0, R2(e2) = a4e1, R2(e3) = a3e3.
R3(e1) = 0, R3(e2) = 0, R3(e3) = a3e3.
R4(e1) = 0, R4(e2) = a4e1, R4(e3) =2 0.
a
R5(e1) = a1e1 + a2e2, R5(e2) = − a12 e1 − a1e2, R5(e3) = 0, a2 = 0.
R6(e1) = 0, R6(e2) = 0, R6(e3) = 0.
The pre-Lie superalgebras ( A9,2)k, ◦
(D4)μ and ( A9,3, ◦)
D5:
( A9,2)k , ◦
⎧ e2 ◦ e1 = e1
⎪⎪⎨ e2 ◦ e2 = e2
e2 ◦ e3 = e3
⎪⎪⎩ e3 ◦ e1 = ke3 .
⎧ e2 ◦ e2 = e1
⎪⎪⎪⎪ e2 ◦ e2 = e2
( A9,3, ◦) ⎨ e2 ◦ e3 = e3
1
⎪⎪ e3 ◦ e3 = 2 e1
⎪⎪⎩ e3 ◦ e3 = e1 .
4.2.10 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A10h
The pre-Lie superalgebra ( A10h,1)k, ◦
(D4)μ:
e1 ◦ e2 = (h − 1)e1, e2 ◦ e1 = he1, e2 ◦ e2 = e1 + he2, e3 ◦ e2 = ke3, h = 0.
Rota–Baxter operators RB(( A10h,1)k) are:
Case 1: If k = 0, we have
R1(e1) = 0, R1(e2) = a1e1, R1(e3) = a2e3.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3.
Case 2: If k ∈ C∗, we have
R3(e1) = 0, R3(e2) = a1e1, R3(e3) = 0, a1 = 0.
R4(e1) = 0, R4(e2) = 0, R4(e3) = 0.
The pre-Lie superalgebras ( A10h,2)k, ◦
(D4)μ and ( A10h,3, ◦)
D5, where
⎧ e1 ◦ e2 = (h − 1)e1
(A10h,2)k, ◦ ⎪⎪⎪⎨⎪⎧ eee221 ◦◦◦ eee221 === (ehh1e1+− 1h)ee21, h = 0 (A10h,3, ◦) ⎨⎪⎪⎪⎪⎪⎪⎪ eee222 ◦◦◦ eee231 === ehh1ee31+ he2, h = 0
⎪⎪ e2 ◦ e3 = he3
⎪⎪⎩ e3 ◦ e2 = ke3
⎪⎪⎪⎪⎪ e3 ◦ e2 = h − 21 e3
⎪⎩⎪ e3 ◦ e3 = e1
They have the same Rota–Baxter operators, that is,
R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0, a1 = 0.
R2(e1) = 0, R2(e2) = 0, R2(e3) = 0.
4.2.11 Rota–Baxter Operators on Pre-Lie Superalgebras of Type A11
The pre-Lie superalgebra ( A11,1)k , ◦
( D4)μ:
e1 ◦ e2 = −e1, e2 ◦ e2 = e1 − e2, e3 ◦ e2 = ke3.
Rota–Baxter operators RB(( A11,1)k ) are:
Case 1: If k = 0, we have
R1(e1) = 0, R1(e2) = a1e1, R1(e3) = a2e3, a1 = 0.
R2(e1) = 0, R2(e2) = 0, R2(e3) = a2e3.
Case 2: If k ∈ C∗, we have
R3(e1) = 0, R3(e2) = a1e1, R3(e3) = 0, a1 = 0.
R4(e1) = 0, R4(e2) = 0, R4(e3) = 0.
The pre-Lie superalgebras ( A11,2, ◦)
D5 and ( A11,3)k , ◦
( D4)μ, where
( A11,2, ◦) :
( A11,3)k , ◦ :
1
e1 ◦ e2 = −e1, e2 ◦ e2 = e1 − e2, e3 ◦ e2 = − 2 e3,
e3 ◦ e3 = e1.
e1 ◦ e2 = −e1, e2 ◦ e2 = e1 − e2, e2 ◦ e3 = −e3, e3 ◦ e2 = ke3.
They have the same Rota–Baxter operators, that is,
R1(e1) = 0, R1(e2) = a1e1, R1(e3) = 0, a1 = 0.
R2(e1) = 0, R2(e2) = 0, R2(e3) = 0.
Remark 4.1 Using the above classification and Corollary 2.4, one may construct the
two- and three-dimensional L -dendriform superalgebras associated with the Rota–
Baxter pre-Lie superalgebras of dimension 2 and 3 (of weight zero) described above.
Acknowledgements We would like to thank Chengming Bai for his valuable remarks and suggestions.
The study was funded by Laboratoire de Mathématiques Appliquées et Analyse Harmonique with Grant
No. LR11ES52.
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