aa r X i v : . [ m a t h . R A ] A ug Post-Lie algebra structures on the Witt algebra
Xiaomin Tang ∗ Department of Mathematics, Heilongjiang University, Harbin, 150080, P. R. China
Abstract.
In this paper, we characterize the graded post-Lie algebra structures and a class of shiftingpost-Lie algebra structures on the Witt algebra. We obtain some new Lie algebras and give a class of theirmodules. As an application, the homogeneous Rota-Baxter operators and a class of non-homogeneousRota-Baxter operators of weight 1 on the Witt algebra are studied.
Keywords:
Lie algebra, post-Lie algebra, Witt algebra, Rota-Baxter operator
MSC : 17A30, 17B68, 17D25. Introduction and preliminaries
In recent years, post-Lie algebras have aroused the interest of a great many authors, see [1, 3, 4, 5, 8, 9,14, 15, 20]. Post-Lie algebras were introduced around 2007 by B. Vallette [21], who found the structure ina purely operadic manner as the Koszul dual of a commutative trialgebra. Moreover, Vallette [21] provedthat post-Lie algebras have the important algebraic property of being Koszul. As pointed out by Hans Z.Munthe-Kaas [14], post-Lie algebras also arise naturally from the differential geometry of homogeneousspaces and Klein geometries, topics that are closely related to Cartans method of moving frames. Inaddition, post-Lie algebras also turned up in relations with Lie groups [5, 14], classical Yang-Baxterequation [1], Hopf algebra and classical r-matrices [10] and Rota-Baxter operators[11].One of the most important problems in the study of post-Lie algebras is to find the post-Lie algebrastructures on the (given) Lie algebras. In [15], the authors determined all post-Lie algebra structures on sl (2 , C ) of special linear Lie algebra of order 2, and in [20], by using the Gr¨obner basis the package incomputer algebra software Maple, they studied the post-Lie algebra structures on the solvable Lie algebra t (2 , C ) (the Lie algebra of 2 × ∗ Corresponding author:
X. Tang. Email: [email protected] Recall that the Witt algebra W is an important infinite-dimensional Lie algebra with the C -basis { L m | m ∈ Z } and the Lie brackets are defined by[ L m , L n ] = ( m − n ) L m + n . The Witt algebra occurs in the study of conformal field theory and plays an important role in many areasof mathematics and physics. Based on this background, we will study the post-Lie algebra structures on W in this paper. Below we will denote by C and Z the complex number field and the set of integer numbers,respectively. For a fixed integer k , let Z >k = { t ∈ Z | t > k } , Z
A post-Lie algebra ( V, ◦ , [ , ]) is a vector space V over a field k equipped with two k -bilinearoperations x ◦ y and [ x, y ] , such that ( V, [ , ]) is a Lie algebra and [ x, y ] ◦ z = x ◦ ( y ◦ z ) − y ◦ ( x ◦ z ) − < x, y > ◦ z, (1.1) x ◦ [ y, z ] = [ x ◦ y, z ] + [ y, x ◦ z ] (1.2) for all x, y, z ∈ V , where < x, y > = x ◦ y − y ◦ x . We also say that ( V, ◦ , [ , ]) is a post-Lie algebra structureon the Lie algebra ( V, [ , ]) . If a post-Lie algebra ( V, ◦ , [ , ]) such that x ◦ y = y ◦ x for all x, y ∈ V , then itis called a commutative post-Lie algebra. Definition 1.2.
Suppose that ( L, [ , ]) is a Lie algebra. Two post-Lie algebras ( L, [ , ] , ◦ ) and ( L, [ , ] , ◦ ) on the Lie algebra L are said to be isomorphic if there is an automorphism τ of the Lie algebra ( L, [ , ]) such that τ ( x ◦ y ) = τ ( x ) ◦ τ ( y ) for all x, y ∈ L . It is not difficult to verify the following proposition.
Proposition 1.3.
Let ( V, ◦ , [ , ]) be a post-Lie algebra defined by Definition 1.1. Then the followingoperation { x, y } , < x, y > +[ x, y ] , (1.3) induces an another Lie algebra structure on V , where < x, y > = x ◦ y − y ◦ x . Furthermore, if two post-Liealgebras ( V, ◦ , [ , ]) and ( V, ◦ , [ , ]) on the same Lie algebra ( V, [ , ]) are isomorphic, then the two inducedLie algebras ( V, { , } ) and ( V, { , } ) are isomorphic. Remark 1.4.
The left multiplications of the post-Lie algebra ( V, [ , ] , ◦ ) are denoted by L ( x ) , i.e., we have L ( x )( y ) = x ◦ y for all x, y ∈ V . By (1.2), we see that all operators L ( x ) are Lie algebra derivations ofLie algebra ( V, [ , ]) . Our results can be briefly summarized as follows: In Section 2, we classify the graded post-Lie algebrastructures on the Witt algebra W , and then we obtain the induced graded Lie algebras. In Section 3,we classify a class of shifting post-Lie algebra structures on the Witt algebra W , and give the inducednon-graded Lie algebras. In Section 4, we first recall the other definition of post-Lie algebra and give anew definition, and then the modules over some Lie algebras are given. In Section 5, we give the inducedRota-Baxter operators of weight 1 from the post-Lie algebras on W .2. The graded post-Lie algebra structure On the Witt algebra
Recently [17] proved that any commutative post-Lie algebra structure on the Witt algebra W is trivial(namely, x ◦ y = 0 for all x, y ∈ W ). We now will dedicate to the study for the non-commutative case.Since the Witt algebra is graded, it is also natural to suppose first that the algebras should be graded.Hence, in this section, we mainly consider the graded post-Lie algebra structure on the Witt algebra W .Namely, we assume that it satisfies L m ◦ L n = φ ( m, n ) L m + n , ∀ m, n ∈ Z , (2.1)where φ is a complex-value function on Z × Z . Lemma 2.1.
There exists a graded post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying (2.1)if and only if there is a complex-value function f on Z such that φ ( m, n ) = ( m − n ) f ( m ) , (2.2)( m − n )( f ( m + n ) + f ( m ) f ( m + n ) + f ( n ) f ( m + n ) − f ( m ) f ( n )) = 0 , ∀ m, n ∈ Z . (2.3) Proof.
The “if” part is easy to check. Next we prove the “only if” part. By Remark 1.4, L ( x ) is aderivation of W . It is well known that every derivation of Witt algebra is inner [22]. So we have L m ◦ L n = L ( L m )( L n ) = ad( ψ ( L m )) L n = [ ψ ( L m ) , L n ] (2.4)for some linear map ψ from W into itself. Denote by ψ ( L m ) = P i ∈ Z k ( m ) i L i , where k ( m ) i ∈ C for any i ∈ Z . Then we have by (2.4) that L m ◦ L n = P i ∈ Z ( i − n ) k ( m ) i L n + i . This, together with (2.1), yieldsthat ( i − n ) k ( m ) i = 0 for any i ∈ Z \ { m } and φ ( m, n ) = ( m − n ) k ( m ) m . This means that k ( m ) i = 0 for any i = m . Let f ( m ) = k ( m ) m be a complex-valued function on Z , which implies (2.2). Next, by (1.1) with asimple computation we obtain (2.3). (cid:3) Let ( P ( φ i ) , ◦ i ) , i = 1 , W = P ( φ i ) equipped with k -bilinearoperations x ◦ i y such that L m ◦ i L n = φ i ( m, n ) L m + n for all m, n ∈ Z , where φ i , i = 1 , Z × Z . Furthermore, let τ : P ( φ ) → P ( φ ) be a map given by τ ( L m ) = − L − m forall m ∈ Z . Clearly, τ is a Lie automorphism of the Witt algebra ( W, [ , ]). Furthermore, we have Proposition 2.2.
Let ( P ( φ i ) , ◦ i ) , i = 1 , be two algebras and τ : P ( φ ) → P ( φ ) be a map defined asabove. Suppose that P ( φ ) is a post-Lie algebra. Then P ( φ ) is a post-Lie algebra and τ is an isomorphismfrom P ( φ ) to P ( φ ) if and only if φ ( m, n ) = − φ ( − m, − n ) .Proof. Suppose that P ( φ ) is a post-Lie algebra and τ is an isomorphism from P ( φ ) to P ( φ ). Thenwe have τ ( L m ◦ L n ) = − φ ( m, n ) L − ( m + n ) and τ ( L m ) ◦ τ ( L n ) = L − m ◦ L − n = φ ( − m, − n ). By τ ( L m ◦ L n ) = τ ( L m ) ◦ τ ( L n ), it follows that φ ( m, n ) = − φ ( − m, − n ).Conversely, suppose that φ ( m, n ) = − φ ( − m, − n ) for all m, n ∈ Z . Notice that P ( φ ) is a post-Liealgebra, by Lemma 2.1 we know there is a complex-valued function f on Z such that φ ( m, n ) = ( m − n ) f ( m ) , (2.5)( m − n )( f ( m + n ) + f ( m ) f ( m + n ) + f ( n ) f ( m + n ) − f ( m ) f ( n )) = 0 , (2.6)for all m, n ∈ Z . By (2.5), we have φ ( m, n ) = − φ ( − m, − n ) = − ( n − m ) f ( − m ). Let f ( m ) = f ( − m )where f is a complex-valued function on Z , then it follows that φ ( m, n ) = ( m − n ) f ( m ) . (2.7)Furthermore, by (2.6) and f ( m ) = f ( − m ) we have that( m − n )( f ( m + n ) + f ( m ) f ( m + n ) + f ( n ) f ( m + n ) − f ( m ) f ( n )) = 0 . (2.8)Lemma 2.1 with (2.7) and (2.8) tells us that P ( φ ) is a post-Lie algebra. The remainder is to prove that τ is an isomorphism. But one has τ ( L m ◦ L n ) = − φ ( m, n ) L − ( m + n ) = φ ( − m, − n ) L − ( m + n ) = τ ( L m ) ◦ τ ( L m ) , which completes the proof. (cid:3) For a complex-valued function f on Z , we denote I, J by I = { m ∈ Z | f ( m ) = 0 } , J = { m ∈ Z | f ( m ) = − } . Lemma 2.3.
There exists a graded post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying (2.1)if and only if there is a complex-valued function f on Z such that (2.2) holds and it satisfies (i) m ∈ I ∪ J for all m = 0 , and (ii) m, n ∈ I ⇒ m + n ∈ I and m + n ∈ J ⇒ m, n ∈ J for m = n . Proof.
The “if” part is easy to check. Next we prove the “only if” part. By Lemma 2.1, there is a complex-valued function f on Z satisfying (2.2) and (2.3). Let n = 0 in (2.3), we have m ( f ( m ) + f ( m ) ) = 0 . Thus, for m = 0, one has f ( m ) = 0 or f ( m ) = −
1. This implies the conclusion of (i) holds. Now, wechose a pair of m, n ∈ Z with m = n , then it can be obtained by (2.3) that f ( m + n ) + f ( m ) f ( m + n ) + f ( n ) f ( m + n ) − f ( m ) f ( n ) = 0 . It easy to see by the above equation that the conclusion of (ii) holds. (cid:3)
Our main result of this section is the following.
Theorem 2.4.
A graded post-Lie algebra structure satisfying (2.1) on the Witt algebra W must be oneof the following types. ( P ) L m ◦ L n = 0 for all m, n ∈ Z ; ( P ) L m ◦ L n = ( n − m ) L m + n for all m, n ∈ Z ; ( P a ) L m ◦ L n = ( n − m ) L m + n , m > ,naL n , m = 0 , , m < P a ) L m ◦ L n = ( n − m ) L m + n , m < ,naL n , m = 0 , , m > P ) L m ◦ L n = ( ( n − m ) L m + n , m > , , m P ) L m ◦ L n = ( ( n − m ) L m + n , m > , , m P ) L m ◦ L n = ( ( n − m ) L m + n , m − , , m > − P ) L m ◦ L n = ( ( n − m ) L m + n , m > − , , m − where a ∈ C . Conversely, the above types are all the graded post-Lie algebra structure satisfying (2.1) onthe Witt algebra W . Furthermore, the post-Lie algebras P a , P and P are isomorphic to the post-Liealgebras P a , P and P respectively, and other post-Lie algebras are not mutually isomorphic.Proof. Suppose that ( W, [ , ] , ◦ ) is a post-Lie algebra structure satisfying (2.1) on the Witt algebra W .By Lemma 2.3, there is a complex-valued function f on Z such that (2.2) holds and it satisfies (i) and(ii) in Lemma 2.3. We discuss the cases of f (1) , f ( − , f (2) and f ( − f (1) , f ( − , f (2) , f ( − ∈ {− , } , and so that 2 = 16 cases can happen. By a simple discussion, it canbe seen that the 8 cases listed in Tabular 1 are true. We here should point out that Theorem 2.22 of [11]given another method to prove it since they have the same condition as (2.3). Thus, by Lemma 2.1, the graded post-Lie algebra structure on the Witt algebra W must be one of the above 8 types. Conversely,every type of the 8 cases means that there is a complex-valued function f on Z such that (2.2) holdsand, the conclusions (i) and (ii) of Lemma 2.3 are easily verified. Thus, they are all the graded post-Liealgebra structure on the Witt algebra W by Lemma 2.3.cases f ( n ) f (1) f ( − f (2) f ( − f (0) f ( n ) for all n P f ( Z ) = 0 P − − − − − f ( Z ) = − P a − − a ∈ C f ( Z > ) = − f ( Z < ) = 0 and f (0) = a . P a − − a ∈ C f ( Z > ) = 0, f ( Z < ) = − f (0) = a P − f ( Z > ) = − f ( Z ) = 0 P − − − − f ( Z > ) = 0 and f ( Z ) = − P − f ( Z > − ) = 0 and f ( Z − ) = − P − − − − f ( Z > − ) = − f ( Z − ) = 0 Table 1. volume of f ( n )Finally, by Proposition 2.2 we know that the post-Lie algebras P a , P and P are isomorphic to thepost-Lie algebras P a , P and P respectively. We claim that P a is not isomorphic to P a when a = a .If not, then there a linear bijective map τ : P a → P a as the isomorphism of post-Lie algebras. Accordingto Definition 1.2, τ first is an automorphism of the Lie algebra ( W, [ , ]). By the automorphisms of theclassical Witt algebra [7], τ ( L m ) = ǫc m L ǫm for all m ∈ Z , where c ∈ C with c = 0 and ǫ ∈ {± } . This,together with the definitions of P a , yields that a = a . This proves the claim above. Similarly, wehave P a is not isomorphic to P a when a = a . Clearly, the other post-Lie algebras are not mutuallyisomorphic. The proof is completed. (cid:3) Due to Theorem 2.4 and Proposition 1.3, we now are able to give some Lie algebras on the space with C -basis { L i | i ∈ Z } , in which many cases are new and interesting. Proposition 2.5.
Up to isomorphism, the post-Lie algebras in Theorem 2.4 give rise to the followingLie algebras on the space with C -basis { L i | i ∈ Z } , and with the bracket { , } defined by of Proposition 1.3: ( LP ): { L m , L n } = ( m − n ) L m + n for all m, n ∈ Z ; ( LP a ): { L m , L n } = ( n − m ) L m + n , m, n > , ( m − n ) L m + n , m, n < ,naL n , m = 0 , n > ,n ( a − L n m = 0 , n < , , otherwise (unless some cases for n = 0) ( LP ): { L m , L n } = ( n − m ) L m + n , m, n > , ( m − n ) L m + n , m, n , , otherwisewhere a ∈ C .Proof. Theorem 2.4 tells us that, up to isomorphism, there are 5 types of graded post-Lie algebra struc-tures satisfying (2.1) on the Witt algebra, that is P , P , P a , P and P . By Proposition 1.3 and a simplecomputation, we can obtain the 5 types of Lie algebras denoted by LP , LP , LP a , LP and LP , respec-tively. It is easy to verify that the Lie algebras LP , LP are isomorphic to the Lie algebras LP , LP respectively through the linear transformation L m → − L m . The conclusions are easily deducible. (cid:3) Remark 2.6.
Proposition 2.5 tells us that, up to isomorphism, there are types of Lie algebras inducedby graded post-Lie algebras from Proposition 1.3. We note that this fact does not agree with Proposition6.4 in [11] . In fact, it is easy to check that every Lie algebra of Proposition 6.4 in [11] has missed thesame item [ L m , L n ] in the compuation { L m , L n } = L m ◦ L n − L n ◦ L m + [ L m , L n ] . After correcting thesmall error, one can get the same results. A class of shifting post-Lie algebra structures On the Witt algebra
In this section, we mainly consider a class of non-graded post-Lie algebra structures on the Wittalgebra ( W, [ , ]). Namely, we assume that it satisfies L m ◦ L n = φ ( m, n ) L m + n + ̺ ( m, n ) L m + n + ν , ∀ m, n ∈ Z , (3.1)where φ and ̺ are complex-valued functions on Z × Z with ̺ = 0, and ν is a fixed nonzero integer. Themotivation of studying this class of non-graded post-Lie algebra structures is inspired by [19], in which aclass of non-graded left-symmetric algebraic structures on the Witt algebra has been considered. As in thetheory of groups and in the theory of graded Lie algebras, we shall call such non-graded post-Lie algebrato be shifting post-Lie algebra since it with a shifting item. Our results show that the characterizationof all non-graded post-Lie algebra structures on Witt algebra seems difficult. Lemma 3.1.
There exists a shifting post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying(3.1) if and only if there are complex-valued functions f and g on Z such that φ ( m, n ) = ( m − n ) f ( m ) , (3.2) ̺ ( m, n ) = ( m − n + ν ) g ( m ) , (3.3)( m − n )( f ( m + n ) + f ( m ) f ( m + n ) + f ( n ) f ( m + n ) − f ( m ) f ( n )) = 0 , (3.4)( m − n ) g ( m ) g ( n ) = (( m − n + ν ) g ( m ) + ( m − n − ν ) g ( n )) g ( m + n + ν ) , (3.5)( m − n )( f ( m ) + f ( n ) + 1) g ( m + n )= ( n − m + ν )( f ( m + n + ν ) − f ( m )) g ( n ) + ( n − m − ν )( f ( m + n + ν ) − f ( n )) g ( m ) . (3.6) Proof.
We can finish the proof by a similar method to Lemma 2.1. (cid:3)
Let ( P ( φ i , ̺ i , ν i ) , ◦ i ) , i = 1 , W = P ( φ i , ̺ i , ν i ) equipped with k -bilinear operations x ◦ i y such that L m ◦ i L n = φ i ( m, n ) L m + n + ̺ i ( m, n ) L m + n + ν i for all m, n ∈ Z ,where φ i , ̺ i , i = 1 , Z × Z and ν i are nonzero integers. Furthermore,let τ : P ( φ , ̺ i , ν ) → P ( φ , ̺ , ν ) be a linear map given by τ ( L m ) = − L − m for all m ∈ Z . Proposition 3.2.
Let P ( φ i , ̺ i , ν i ) , i = 1 , be two algebras and τ : P ( φ , ̺ i , ν ) → P ( φ , ̺ , ν ) be a mapdefined as above. Suppose that P ( φ , ̺ , ν ) is a post-Lie algebra. Then P ( φ , ̺ , ν ) is a post-Lie algebraand τ is an isomorphism from P ( φ , ̺ , ν ) to P ( φ , ̺ , ν ) if and only if φ ( m, n ) = − φ ( − m, − n ) , ̺ ( m, n ) = − ̺ ( − m, − n ) , ν = − ν . (3.7) Proof.
The proof is similar to Proposition 2.2. (cid:3)
Since a shifting post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) are entirelydetermined by an integer number ν and two complex-valued functions φ and ̺ on Z × Z . According to(3.2) and (3.3) in Lemma 3.1, the classicization of such post-Lie algebras is dependent on the integer ν and two complex-valued functions f and g on Z . It will be proved that the following Table 2 gives allcases of ν, f and g , where b is any nonzero complex number.The above conclusion will be proved by some propositions as follows. First, notice that (3.2) and (3.5)in Lemma 3.1, from Lemma 2.1 and Theorem 2.4 we have the following lemma. Lemma 3.3.
Suppose that ( W, [ , ] , ◦ ) is a shifting post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1). Then φ satisfies (2.2) and f must be determined by one of the cases P , P , P a , P a , P - P in Tabel 1. Taking n = − ν in (3.5), the following equation is often used in our proof. νg ( m ) g ( − ν ) = ( m + 2 ν ) g ( m ) g ( m ) . (3.8) Proposition 3.4. If f takes the form determined by P or P in Tabel 1, then g ( Z ) = 0 . In this case,there is no any shifting post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1).Proof. If f takes the form determined by P , then f ( m ) = 0 for all m ∈ Z . Thus, by (3.6) we have( m − n ) g ( m + n ) = 0. From this, we deduce that g ( Z ) = 0. The case in which f takes the formdetermined by P is similar. (cid:3) Proposition 3.5.
Suppose that f takes the form determined by P a in Table 1, i.e., f ( Z > ) = − , f ( Z < ) =0 and f (0) = a for some a ∈ C . Then the shifting post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) is determined by NP b,ν or NP b,ν in Table 2.Proof. The proof is divided into the following: Assertions 3.5.1, 3.5.2, 3.5.3.
Assertion 3.5.1. (i)
When ν > , we have g ( Z > ) = 0 and g ( Z min {− , − − ν } ) = 0 ; (ii) When ν < , we have g ( Z > max { , − ν } ) = g ( Z − ) = 0 . For all m, n ∈ Z such that { m, n, m + n + ν } ⊂ Z > or { m, n, m + n + ν } ⊂ Z < , by (3.6), it followsthat ( m − n ) g ( m + n ) = 0. The results are easy to check. f ( · ) f ( · ) g ( · ) g ( · ) ν NP b,ν f ( Z > ) = − f ( Z < ) = 0 g ( − ν ) = − b g ( Z \ {− ν } ) = 0 1 , NP b,ν f ( Z > ) = − f ( Z ) = 0 g ( − ν ) = − b g ( Z \ {− ν } ) = 0 − , − NP b,ν f ( Z > ) = − f ( Z ) = 0 g ( − ν ) = − b g ( Z \ {− ν } ) = 0 − , − , − NP b,ν f ( Z > ) = − f ( Z ) = 0 g (2) = − b g ( Z \ { } ) = 0 − NP b,ν f ( Z > ) = − f ( Z ) = 0 g (3) = 2 g (2) = − b g ( Z \ { , } ) = 0 − NP b,ν f ( Z > ) = 0 f ( Z ) = − g ( − ν ) = − b g ( Z \ {− ν } ) = 0 − , − , − NP b,ν f ( Z > ) = 0 f ( Z ) = − g (2) = − b g ( Z \ { } ) = 0 − NP b,ν f ( Z > ) = 0 f ( Z ) = − g (3) = 2 g (2) = − b g ( Z \ { , } ) = 0 − MP b,ν f ( Z ) = − f ( Z > ) = 0 g ( − ν ) = − b g ( Z \ {− ν } ) = 0 − , − MP b,ν f ( Z < ) = − f ( Z > ) = 0 g ( − ν ) = − b g ( Z \ {− ν } ) = 0 1 , MP b,ν f ( Z − ) = − f ( Z > − ) = 0 g ( − ν ) = − b g ( Z \ {− ν } ) = 0 2 , , MP b,ν f ( Z − ) = − f ( Z > − ) = 0 g ( −
2) = − b g ( Z \ {− } ) = 0 1 MP b,ν f ( Z − ) = − f ( Z > − ) = 0 g ( −
3) = 2 g ( −
2) = − b g ( Z \ {− , − } ) = 0 2 MP b,ν f ( Z − ) = 0 f ( Z > − ) = − g ( − ν ) = b g ( Z \ {− ν } ) = 0 2 , , MP b,ν f ( Z − ) = 0 f ( Z > − ) = − g ( −
2) = − b g ( Z \ {− } ) = 0 1 MP b,ν f ( Z − ) = 0 f ( Z > − ) = − g ( −
3) = 2 g ( −
2) = − b g ( Z \ {− , − } ) = 0 2 Table 2. volumes of f ( n ) , g ( n ) and ν Assertion 3.5.2. If ν > or ν − , then g ( Z ) = 0 . Case I. ν >
3. By Assertion 3.5.1 (i), g ( Z > ) = g ( Z − − ν ) = 0. Let m = − n = 1 and m = − n = 2 in(3.6), respectively, one can deduce g (1) = g (2) = 0. Note that − ν < − − ν , one has g ( − ν ) = 0. This,together with m = − ν and n = 0 in (3.5), gives 0 = − νg (0) g ( − ν ), so that g (0) g ( − ν ) = 0. From this,by letting m = − ν and n = 0 in (3.5), we obtain − νg (0) g (0) = 0. In other words, g (0) = 0. In orderto prove that g ( Z ) = 0, it is enough to show that g ( m ) = 0 for all m ∈ {− , − , · · · , − ν } . Because of − ν + 1 < − ν − − ν + 2 < − ν −
1, one has g ( − ν + 1) = g ( − ν + 2) = 0. This, together with m = − ν + 1 , n = − m = − ν + 2 , n = − − ν + 2) g ( − g ( − ν ) = ( − ν + 4) g ( − g ( − ν ) = 0 . As ν >
3, we see that − ν + 2 = 0 and − ν + 4 = 0. This, together with the above equation, yields that g ( − g ( − ν ) = g ( − g ( − ν ) = 0. Next, by applying (3.8) to m = − m = − − ν ) g ( −
1) = ( − ν ) g ( −
2) = 0. Since − ν = 0 and − ν = 0, one has g ( −
1) = g ( −
2) = 0.If we let m = − n = − k in (3.6) where k >
2, it follows that( k − g ( − − k ) = (1 − k + ν ) f ( − − k + ν ) g ( − k ) . (3.9)By letting k = 2 in (3.9), we have g ( −
3) = 0. Again, by letting k = 3 in (3.9), one has g ( −
4) = 0. Inturn, we obtain by (3.9) that g ( Z < ) = 0, therefore the conclusion is proved.Case II. ν −
3. Applying the similar method to Case I, one also can obtain that g ( Z ) = 0. Assertion 3.5.3. f , g and τ must be determined by NP b,ν or NP b,ν in Table 2. By Assertion 3.5.2, g ( Z ) = 0 if ν / ∈ { , , − , − } . Since g = 0, then ν ∈ { , , − , − } .Case 1. ν = 1. From Assertion 3.5.1 (i), g ( Z > ) = g ( Z − ) = 0. Next, we discuss the images f ( k ) for k ∈ {− , − , , , } . Taking m = − n = 1 and m = − n = 2 in (3.6), respectively, one has g (1) = g (2) = 0.If we let n = 0 in (3.5), one has mg ( m ) g (0) = (( m + 1) g ( m ) + ( m − g (0)) g ( m + 1) . (3.10)Taking m = − , − − g ( −
3) = 0 we have that g ( − g (0) = 2 g (0) g (0) , g ( − g (0) = 3 g ( − g (0) , g (0) g ( −
2) = 0 , which implies that g (0) = 0. Thus, g (0) = 0. If we let m = − , n = 1 and m = − , n = 0 in (3.6),respectively, then one has that ( f (0) + 1) g ( −
2) = 0 and 2( f (0) + 1) g ( −
2) = f (0) g ( − g ( −
2) = 0. Let m = − n = 0 in (3.6), then we have ( f (0) + 1) g ( −
1) = 0. The fact of g = 0 means that g ( −
1) = − b = 0 for some b ∈ C . At the same time, we must have that f (0) + 1 = 0, namely, f (0) = a = −
1. By the above part of the analysis and discussion, we see that f ( Z > ) = − , f ( Z < ) = 0 , g ( −
1) = − b, g ( Z \ {− } ) = 0 , for some nonzero b ∈ C . This is of the type NP b, in Table 2.Case 2. ν = 2. By Assertion 3.5.1 (i), g ( Z > ) = g ( Z − ) = 0. Thus, we only discuss the images f ( k ) for k ∈ {− , − , , , } . Taking m = − n = 1 and m = − n = 2 in (3.6), respectively, it followsthat g (1) = g (2) = 0. Let m = − , n = − m = − , n = − g ( − g ( −
2) = 3 g ( − g ( −
1) and g ( − g ( −
2) = 0, which yields that g ( −
1) = 0. Hence g ( −
1) = 0. Weagain obtain g (0) = 0 by taking m = − n = 0 in (3.6). Finally, it follows that ( f (0) + 1) g ( −
2) = 0by taking m = − n = 0 in (3.6). Since g = 0, hence f (0) = a = − g ( −
2) = − b = 0 for somenonzero b ∈ C . It proves that f ( Z > ) = − , f ( Z < ) = 0 , g ( −
2) = − b, g ( Z \ {− } ) = 0 , which is of the type NP b, in Table 2.Case 3. ν = −
1. Similar to the discussion of Case 1, we obtain the type NP b, − in Table 2.Case 4. ν = −
2. Similar to the discussion of Case 2, we obtain the type NP b, − in Table 2. (cid:3) Proposition 3.6.
Suppose that f takes the form determined by P in Table 1, i.e., f ( Z > ) = − , f ( Z ) =0 . Then the shifting post-Lie algebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) is determinedby NP b,ν , NP b,ν or NP b,ν in Table 2.Proof. The proof is divided into the following: Assertions 3.6.1, 3.6.2 and 3.6.3.
Assertion 3.6.1. (i)
When ν > , we have g ( Z > ) = g ( Z − ν ) = 0 ; (ii) When ν < , we have g ( Z > max { , − ν ) = g ( Z ) = 0 . For any m, n ∈ Z with { m, n, m + n + ν } ⊂ Z > or { m, n, m + n + ν } ⊂ Z , one has by (3.6) that( m − n ) g ( m + n ) = 0. The assertion are easy to check. Assertion 3.6.2. If ν > or ν − , then g ( Z ) = 0 . Case I. ν >
1. By Assertion 3.6.1 (i), g ( Z > ) = g ( Z − ν ) = 0. Taking n = 0 in (3.6), then( m ( f ( m ) + 1) − ( m + ν ) f ( m + ν )) g ( m ) = ( ν − m )( f ( m + ν ) − f ( m )) g (0) (3.11)for all m ∈ Z . If we let m ∈ { , , } , then f ( m ) = f ( m + ν ) = − m + ν ) g ( m ) = 0.This implies g (2) = g (3) = g (4) = 0. Take m = 1 and n = − − ν in (3.5), then by g ( − − ν ) = 0,(2 + 2 ν ) g (1) g (0) = 0 . (3.12) If we let m = 1 in (3.11), it follows that − νg (1) = (1 − ν ) g (0). This, together with (3.12), gives g (1) = g (0) = 0 for the case ν >
1. But for the case ν = 1, we also have g (1) = 0 since − νg (1) = (1 − ν ) g (0)and g (0) = 0 since g ( Z − ν ) = 0. In order to prove the conclusion, we only need show that g ( k ) = 0for all k ∈ { − ν, − ν, · · · , − , − } . For such k , we let k = s − ν , where 2 s < ν . Note that k <
0, so f ( k ) = 0 and f ( k + ν ) = f ( s ) = −
1. Applying (3.11) to m = k , we have by g (0) = 0 that( k − ( k + ν )) g ( k ) = 0. In other words, g ( k ) = 0, as desired.Case II. ν −
5. By Assertion 3.6.1 (ii), g ( Z > − ν ) = g ( Z ) = 0. From this, we only need toprove that g ( m ) = 0 for every m ∈ { , , · · · , − ν } . For m ∈ Z with 1 < m < − ν , it follows that m + 2 ν < ν <
0. Taking m = 2 and n = − − ν in (3.5) and (3.6), respectively, we obtain that(4 + ν ) g (2) g ( − − ν ) = 0 , (3.13)(4 + ν ) g ( − ν ) = (4 + 2 ν ) g (2) + 4 g ( − − ν ) . (3.14)Note that 4 + ν = 0, so (3.13) tells us that g (2) g ( − − ν ) = 0. This, together with (3.14), gives that(4 + ν ) g (2) g ( − ν ) = (4 + 2 ν ) g (2) , (3.15)(4 + ν ) g ( − − ν ) g ( − ν ) = 4 g ( − − ν ) . (3.16)If we let m = 2 and m = − − ν in (3.8), respectively, we have that νg (2) g ( − ν ) = (2 + 2 ν ) g (2) , (3.17) νg ( − − ν ) g ( − ν ) = ( ν − g ( − − ν ) . (3.18)It follows by (3.15) and (3.17) that (cid:18) ν νν ν (cid:19) (cid:18) g (2) g ( − ν ) − g (2) (cid:19) = 0 , and by (3.16) and (3.18) that (cid:18) ν ν ν − (cid:19) (cid:18) g ( − − ν ) g ( − ν ) − g ( − − ν )) (cid:19) = 0 . Since ν −
5, we deducedet (cid:18) ν νν ν (cid:19) = 8 + 6 ν = 0 , det (cid:18) ν ν ν − (cid:19) = ( ν − ν + 2) = 0 , which yields that − g (2) = − g ( − − ν ) = 0, and then g (2) = g ( − − ν ) = 0. Thus, from (3.14) we have(4 + ν ) g ( − ν ) = (4 + 2 ν ) g (2) + 4 g ( − − ν ) = 0. Note that 4 + ν = 0, so g ( − ν ) = 0. This, together with(3.8), implies that ( m + 2 ν ) g ( m ) = 0. For m ∈ Z with 1 < m < − ν , we have m + 2 ν < ν < g ( m ) = 0 and thereby g ( m ) = 0 for every m ∈ { , , · · · , − ν } . The proof is completed. Assertion 3.6.3. f , g and τ must be determined by NP b,ν , NP b,ν or NP b,ν in Table 2. By Assertion 3.6.2, g ( Z ) = 0 if ν / ∈ {− , − , − , − } . Since g = 0, then ν ∈ {− , − , − , − } .Case 1. ν = −
1. From Clam 3.6.1 (ii), g ( Z > ) = g ( Z ) = 0. Next, we discuss the images f ( k ) for k ∈ { , , } . Taking m = 0 , n = 3 and m = 0 , n = 4 in (3.6), respectively, one has that g (3) = g (4) = 0.In this case, g (2) = − b = 0 for some b ∈ C . Thus, it summarizes as f ( Z > ) = − , f ( Z ) = 0 , g (2) = − b, g ( Z \ { } ) = 0 . It gives the type NP b, − in Table 2.Case 2. ν = −
2. By Assertion 3.6.1 (ii) we have g ( Z > ) = g ( Z ) = 0. Thus, we only discuss theimages f ( k ) for k ∈ { , , } . Taking m = 0 and n = 4 in (3.6), it follows that g (4) = 0. If we let m = 3and ν = − g (3) g (3) = 2 g (2) g (3). This tells us that if g (2) = 0 then g (3) = 0. Since g = 0, it must be g (2) = − b = 0. In this case, either g (3) = 0 which gives the type NP b, − in Table 2; or g (3) = 2 g (2) = − b = 0 which gives the type NP b, − in Table 2.Case 3. ν = −
3. By Assertion 3.6.1 (ii), g ( Z > ) = g ( Z ) = 0. Thus, we only discuss the images f ( k )for k ∈ { , , } . Taking m = 2 , n = 5 and m = 2 , n = 4 in (3.6), respectively, one has that g (2) g (4) = 0 , g (3) g (4) = 5 g (2) g (3) . (3.19)If we let ν = − m = 2 and m = 4 in (3.8), respectively, we get3 g (2) g (3) = 4 g (2) g (2) , g (2) g (3) = 2 g (4) g (4) . (3.20)Combing (3.19) with (3.20), it implies that g (4) = 10 g (2) and g (2) g (4) = 0. This means that g (2) = g (4) = 0. Since g = 0, we have g (3) = − b = 0 for some b ∈ C \ { } . It is easy to check that this is justthe type NP b, − in Table 2.Case 4. ν = −
4. By Assertion 3.6.1 (ii) we have g ( Z > ) = g ( Z ) = 0. Thus, we only discuss theimages f ( k ) for k ∈ { , , , } . If we let ν = − m = 2 , , g (2) g (4) = 2 g (2) g (2) , g (3) g (4) = 5 g (3) g (3) , g (4) g (5) = 3 g (5) g (5) . (3.21)Taking m = 2 and n = 6 in (3.5), it follows that g (2) g (4) = 0. This, together with (3.21), implies that g (2) = 0. So we have g (2) = 0. If we let m = 3 , n = 6 and m = 3 , n = 5 in (3.5), respectively, we obtainthat g (3) g (5) = 0 and g (4) g (5) = 3 g (3) g (4). This yields that g (4) g (5) = g (3) g (4) = 0. This, togetherwith (3.21), gives that g (3) = g (5) = 0. Thus, g (3) = g (5) = 0. Since g = 0, we have g (4) = − b = 0 forsome b ∈ C \ { } . It is easy to check that this is just the type NP b, − in Table 2. (cid:3) Proposition 3.7.
Suppose that f takes the form determined by P in Table 1. Then the shifting post-Liealgebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) is determined by NP b,ν , NP b,ν or NP b,ν inTable 2.4
Suppose that f takes the form determined by P in Table 1. Then the shifting post-Liealgebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) is determined by NP b,ν , NP b,ν or NP b,ν inTable 2.4 Proof.
The proof is similar to Proposition 3.6. (cid:3)
Next, by using Proposition 3.2, similar to Propositions 3.5, 3.6 and 3.7, one has the following threepropositions.
Proposition 3.8.
Suppose that f takes the form determined by P a in Table 1. Then the shifting post-Liealgebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) is determined by MP b,ν or MP b,ν in Table2. Proposition 3.9.
Suppose that f takes the form determined by P in Table 1. Then the shifting post-Liealgebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) is determined by MP b,ν , MP b,ν or MP b,ν in Table 2. Proposition 3.10.
Suppose that f takes the form determined by P in Table 1. Then the shifting post-Liealgebra structure on the Witt algebra ( W, [ , ]) satisfying (3.1) is determined by MP b,ν , MP b,ν or MP b,ν in Table 2. Our main result in this section is the following.
Theorem 3.11.
A shifting post-Lie algebra structure satisfying (3.1) on the Witt algebra W must be oneof the following types. ( NP b,ν ): ν = 1 or , L m ◦ L n = ( n − m ) L m + n , m > ,nbL n , m = − ν, , m < , m = − ν ;( NP b,ν ): ν = − or − , L m ◦ L n = ( n − m ) L m + n , m > , m = − ν, ( n + ν ) L n − ν + nbL n , m = − ν, , m NP b,ν ): ν = − , − or − , L m ◦ L n = ( n − m ) L m + n , m > , m = − ν ( n + ν ) L n − ν + nbL n , m = − ν, , m NP b,ν ): L m ◦ L n = ( n − m ) L m + n , m > , ( n − L n +2 + b ( n − L n +1 , m = 2 , , m NP b,ν ): L m ◦ L n = ( n − m ) L m + n , m > , ( n − L n +2 + nbL n , m = 2 , ( n − L n +3 + 2( n − bL n +1 , m = 3 , , m NP b,ν ): ν = − , − or − , L m ◦ L n = ( n − m ) L m + n , m ,nbL n , m = − ν, , m > , m = − ν ;( NP b,ν ): L m ◦ L n = ( n − m ) L m + n , m , ( n − bL n +1 , m = 2 , , m > NP b,ν ): L m ◦ L n = ( n − m ) L m + n , m ,nbL n , m = 2 , n − bL n +1 , m = 3 , , m > MP b,ν ): ν = − or − , L m ◦ L n = ( n − m ) L m + n , m ,nbL n , m = − ν, , m > , m = − ν ;( MP b,ν ): ν = 1 or , L m ◦ L n = ( n − m ) L m + n , m < , m = − ν, ( n + ν ) L n − ν + nbL n , m = − ν, , m > MP b,ν ): ν = 2 , or , L m ◦ L n = ( n − m ) L m + n , m − , m = − ν ( n + ν ) L n − ν + nbL n , m = − ν, , m > − MP b,ν ): L m ◦ L n = ( n − m ) L m + n , m − , ( n + 2) L n − + b ( n + 1) L n − , m = − , , m > − MP b,ν ): L m ◦ L n = ( n − m ) L m + n , m − , ( n + 2) L n − + nbL n , m = − , ( n + 3) L n − + 2( n + 1) bL n − , m = − , , m > − MP b,ν ): ν = 2 , or , L m ◦ L n = ( n − m ) L m + n , m > − ,nbL n , m = − ν, , m − , m = − ν ;( MP b,ν ): L m ◦ L n = ( n − m ) L m + n , m > − , ( n + 1) bL n − , m = − , , m − MP b,ν ): L m ◦ L n = ( n − m ) L m + n , m > − ,nbL n , m = − , n + 1) bL n − , m = − , , m − where b is a non-zero number. Conversely, the above types are all shifting post-Lie algebra structuressatisfying (3.1) on the Witt algebra W . Furthermore, the post-Lie algebras NP b,νi are isomorphic to the post-Lie algebras MP b, − νi , i = 1 , , · · · , respectively, and other post-Lie algebras are not mutuallyisomorphic.Proof. Suppose that ( W, [ , ] , ◦ ) is a class of shifting post-Lie algebra structures satisfying (3.1) on the Wittalgebra W . By Propositions 3.4-3.10, there are complex-valued function f and g on Z such that one of 16cases in Table 2 holds. Thus, by Lemma 3.1 we know that the shifting post-Lie algebra structure mustbe one of the above 16 types. Conversely, every type of the 16 cases means that there are complex-valuedfunction f and g on Z such that (3.2) and (3.3) hold and, the Equations (3.4)-(3.6) are easily verified.Thus, by Lemma 3.1 we see that all they are the shifting post-Lie algebra structure satisfying (3.1) on theWitt algebra W . Finally, by Proposition 3.2 we know that the post-Lie algebras NP b,νi are isomorphic tothe post-Lie algebras MP b, − νi , i = 1 , , · · · , (cid:3) Proposition 3.12.
Up to isomorphism, the post-Lie algebras in Theorem 3.11 give rise to the followingLie algebras under the bracket { , } defined in (1.3) of Proposition 1.3: ( LNP b,ν ): ν = 1 or ν = 2 , { L m , L n } = ( n − m ) L m + n , m, n > , ( m − n ) L m + n , m, n < , m, n = − νnbL n , m = − ν, n > ,nbL n − ( n + ν ) L n − ν m = − ν, n < , n = − ν, , otherwise (unless some cases for n = − ν );( LNP b,ν ): ν = − , − or − , { L m , L n } = ( n − m ) L m + n m, n > , m, n = − ν, ( m − n ) L m + n , m, n ,nbL n , m = − ν, n ,nbL n + ( n + ν ) L n − ν m = − ν, n > , n = − ν, , otherwise (unless some cases for n = − ν );( LNP b,ν ): { L m , L n } = ( n − m ) L m + n m, n > , ( m − n ) L m + n , m, n , ( n − bL n +1 , m = 2 , n , ( n − bL n +1 + ( n − L n +2 m = 2 , n > , , otherwise (unless some cases for n = 2);( LNP b,ν ): { L m , L n } = ( n − m ) L m + n m, n > , ( m − n ) L m + n , m, n ,L + bL , m = 2 , n = 3 ,nbL n , m = 2 , n , n − bL n +1 , m = 3 , n ,nbL n + ( n − L n +2 m = 2 , n > , n − bL n +1 + ( n − L n +3 m = 3 , n > , , otherwise (unless some cases for n = 2 , where b is a nonzero number.Proof. The proof is similar to Proposition 2.5. (cid:3) Another understanding of the Post-Lie algebra structures
We should see that there were two different definitions of post-Lie algebra structure, that is post-Liealgebra structure on a Lie algebra and post-Lie algebra structure on pairs of Lie algebras . The former isstudied as above. Now we recall the latter as follows.
Definition 4.1. [3, 4]
Let ( V , [ , ]) and ( V , { , } ) be a pair of Lie algebras on the same linear space V = V = V over a field k . A post-Lie algebra structure on the pair ( V , V ) is a k -bilinear product x ◦ y on V satisfying the following identities: < x, y > = { x, y } − [ x, y ] , { x, y } ◦ z = x ◦ ( y ◦ z ) − y ◦ ( x ◦ z ) ,x ◦ [ y, z ] = [ x ◦ y, z ] + [ y, x ◦ z ] for all x, y, z ∈ V , where < x, y > = x ◦ y − y ◦ x . We also say that ( V , V , ◦ , [ , ] , { , } ) is a post-Lie algebra. Inspired by the above definitions, here we would like to give another definition of a post-Lie algebraas follows. Below we will see that the three definitions of post-Lie algebra are equivalent.
Definition 4.2.
A post-Lie algebra ( V, ◦ , { , } ) is a vector space V over a field k equipped with two k -bilinear operations x ◦ y and { x, y } , such that ( V, { , } ) is a Lie algebra and { x, y } ◦ z = x ◦ ( y ◦ z ) − y ◦ ( x ◦ z ) , (4.1) x ◦ { y, z } − { x ◦ y, z } − { y, x ◦ z } = x ◦ < y, z > − < x ◦ y, z > − < y, x ◦ z > (4.2) for all x, y, z ∈ V , where < x, y > = x ◦ y − y ◦ x . We also say that ( V, ◦ , { , } ) is a post-Lie algebra structureon Lie algebra ( V, { , } ) . Proposition 4.3.
A post-Lie algebra ( V, ◦ , { , } ) defined by Definition 4.2 with the following operation [ x, y ] , { x, y }− < x, y > . defines an another Lie algebra structure on V , where < x, y > = x ◦ y − y ◦ x . Proposition 4.4.
Definitions 1.1, 4.1 and 4.2 of the notion of post-Lie algebra are equivalent. Proof.
If ( V, ◦ , [ , ]) is a post-Lie algebra defined by Definition 1.1, then by Proposition 1.3 we know thatunder the Lie bracket { x, y } = < x, y > +[ x, y ], V admits a new Lie algebra structure. Obviously,( V, ◦ , { , } ) satisfies (4.1) and (4.2). Conversely, when ( V, ◦ , { , } ) is a post-Lie algebra defined by Definition4.2, then by Proposition 4.3 we know that under the Lie bracket [ x, y ] = { x, y }− < x, y > , V also admitsa new Lie algebra structure. We are able to verily that ( V, ◦ , [ , ]) satisfies (1.1) and (1.2). This tell usthat whether ( V, ◦ , [ , ]) or ( V, ◦ , { , } ) all are connotations of two Lie algebras structures, which satisfy theconditions of Definitions 4.1. On the other hand, a post-Lie algebra structure on the pair ( V , V ) definedby Definitions 4.1 imply ( V , ◦ , [ , ]) satisfying (1.1) and (1.2) or ( V , ⊲ , { , } ) satisfying (4.1) and (4.2). (cid:3) Remark 4.5.
Recall that the left multiplications of the algebra A = ( V, ◦ ) are denoted by L ( x ) , i.e., wehave L ( x )( y ) = x ◦ y for all x, y ∈ V . Clearly, by (4.1) we see that the map L : V → End ( V ) givenby x L ( x ) is a linear representation of the Lie algebra ( V, { , } ) when ( V, ◦ , { , } ) is the post-Lie algebradefined by Definition 4.2. It can be seen that the study of post-Lie algebra structures on pairs of Lie algebras given by Definition4.1 is divided into two directions: either when ( V , [ , ]) is a given Lie algebra, to determine the product ◦ ; or when ( V , { , } ) is a given Lie algebra, to determine the product ◦ . By Proposition 4.4, the firstdirection is characterizing of the post-Lie algebra structures ( V , [ , ] , ◦ ) on the Lie algebra ( V , [ , ]) givenby Definition 1.1, and another direction is characterizing of the post-Lie algebra structure ( V , { , } , ◦ ) onthe Lie algebra ( V , { , } ) given by Definition 4.2. For the first case in which V = W is the Witt algebra,the graded or some shifting post-Lie algebra structures are studied in Sections 2 and 3. The problem ofanother case in which V = W is the Witt algebra, should be interesting. We are not going to discussthis problem here. But, inspired by [13, 19], we may give two non-trivial examples as follows. Example 4.1.
The following cases give two class of graded post-Lie algebra structures on W satisfying(2.1) given by Definition 4.2. φ ( m, n ) = − ( α + n + αǫm )(1 + ǫn )1 + ǫ ( m + n ) , ∀ m, n ∈ Z , where α, ǫ ∈ C satisfy ǫ = 0 or ǫ − Z , or φ ( m, n ) = ( − n − t, if m + n + t = 0 , ( n + t )( n + t − β ) β − t , if m + n + t = 0 , ∀ m, n ∈ Z , where β ∈ C and t ∈ Z satisfy β = t . Example 4.2.
Let φ ( m, n ) = − ( n + α ) and ̺ ( m, n ) = µ in (3.1), where α, µ ∈ C . This is a shiftingpost-Lie algebra structure on the Witt algebra given by Definition 4.2 as follows. L m ◦ L n = − ( n + α ) L m + n + µL m + n + ν . By Propositions 2.5 and 3.12, we find 7 classes of Lie algebras up to isomorphism. They are LP , LP a , LP , LNP b,ν , LNP b,ν , LNP b,ν and LNP b,ν . Note that Remark 4.5 tells us that the post-Lie ( L, ◦ , { , } )admits a module structure of Lie algebra ( L, { , } ). Thus, by Theorems 2.4 and 3.11, we have the followingresults on module structures of some Lie algebras. Proposition 4.6.
Let V = Span C { v i | i ∈ Z } be a linear vector space over the complex number field. Then V becomes a module over some Lie algebras under the following acts. (1) For the Lie algebra LP : L m .v n = 0 , ∀ m, n ∈ Z . (2) For the Lie algebra LP a : L m .v n = ( n − m ) v m + n , m > ,nav n , m = 0 , , m < For the Lie algebra LP : L m .v n = ( ( n − m ) v m + n , m > , , m For the Lie algebra
LNP b,ν : ν = 1 or , L m .v n = ( n − m ) v m + n , m > ,nbv n , m = − ν, , m < , m = − ν ;(5) For the Lie algebra
LNP b,ν : ν = − , − or − , L m .v n = ( n − m ) v m + n , m > , m = − ν ( n + ν ) L n − ν + nbL n , m = − ν, , m For the Lie algebra NP b,ν : L m .v n = ( n − m ) v m + n , m > , ( n − v n +2 + b ( n − v n +1 , m = 2 , , m For the Lie algebra NP b,ν : L m .v n = ( n − m ) v m + n , m > , ( n − v n +2 + nbv n , m = 2 , ( n − v n +3 + 2( n − bv n +1 , m = 3 , , m , where a, b ∈ C with b = 0 . Application to Rota-Baxter operators
Now let us recall the definition of Rota-Baxter operator.
Definition 5.1.
Let L be a complex Lie algebra. A Rota-Baxter operator of weight λ ∈ C is a linearmap R : L → L satisfying [ R ( x ) , R ( y )] = R ([ R ( x ) , y ] + [ x, R ( y )]) + λR ([ x, y ]) , ∀ x, y ∈ L. (5.1) Note that if R is a Rota-Baxter operator of weight λ = 0, then λ − R is a Rota-Baxter operator ofweight 1. Therefore, one only needs to consider Rota-Baxter operators of weight 0 and 1. Lemma 5.2. [1]
Let L be a complex Lie algebra and R : L → L a Rota-Baxter operator of weight .Define a new operation x ◦ y = [ R ( x ) , y ] on L . Then ( L, [ , ] , ◦ ) is a post-Lie algebra given by Definition1.1. In this section, by use of Lemma 5.2, Theorems 2.4 and 3.11, we mainly consider the homogeneousRota-Baxter operator of weight 1 on the Witt algebra W such that R : W → W given by R ( L m ) = f ( m ) L m , ∀ m ∈ Z (5.2)and the non-homogeneous Rota-Baxter operator of weight 1 on the Witt algebra W such that R ( L m ) = f ( m ) L m + g ( m ) L m + ν , ∀ m ∈ Z (5.3)where ν is a nonzero integer number and f, g are complex-valued functions on Z with g = 0.Up till now, the authors [11] have presented the homogeneous Rota-Baxter operators on the Wittalgebras. Inspired by this, we will prove the following. Theorem 5.3.
A homogeneous Rota-Baxter operator of weight satisfying (5.2) on the Witt algebra W must be one of the following types ( R ): R ( L m ) = 0 , ∀ m ∈ Z ;( R ): R ( L m ) = − L m , ∀ m ∈ Z ;( R a ): R ( L m ) = − L m , m > ,aL , m = 0 , , m < , ( R a ): R ( L m ) = − L m , m < ,aL , m = 0 , , m > , ( R ): R ( L m ) = ( − L m , m > , , m R ): R ( L m ) = ( − L m , m , , m > R ): R ( L m ) = ( − L m , m > − , , m − R ): R ( L m ) = ( − L m , m − , , m > − , where a ∈ C . Proof.
Due to Lemma 5.2, if we define a new operation x ◦ y = [ R ( x ) , y ] on W , then ( W, [ , ] , ◦ ) is apost-Lie algebra. By (5.2) we have L m ⊲ L n = [ R ( L m ) , L n ] = ( m − n ) f ( m ) L m + n , ∀ m, n ∈ Z . This means that ( W, [ , ] , ⊲ ) is a graded post-Lie algebra structure satisfying (2.1) on W with φ ( m, n ) =( m − n ) f ( m ). By Theorem 2.4, we see that f must be one of the eight cases listed in Table 1, which canbe get the eight forms of R one by one. On the other hand, it is easy to verify that every form of R listedin the above is a Rota-Baxter operator of weight 1 satisfying (5.2). The proof is completed. (cid:3) Theorem 5.4.
A non-homogeneous Rota-Baxter operator of weight satisfying (5.3) on the Witt algebra W must be one of the following types ( NR b ): ν = 1 or , R ( L m ) = − L m , m > , − bL , m = − ν, , m < , m = − ν ;( NR b ): ν = − or − , R ( L m ) = − L m , m > , m = − ν, − L − ν − bL , m = − ν, , m NR b ): ν = − , − or − , R ( L m ) = − L m , m > , m = − ν − L − ν − bL , m = − ν, , m NR b ): R ( L m ) = − L m , m > , − L − bL , m = 2 , , m NR b ): R ( L m ) = − L m , m > , − L − bL , m = 2 , − L − bL , m = 3 , , m NR b ): ν = − , − or − , R ( L m ) = − L m , m , − bL , m = − ν, , m > , m = − ν ;( NR b ): R ( L m ) = − L m , m , − bL , m = 2 , , m > NR b ): R ( L m ) = − L m , m , − bL , m = 2 , − bL , m = 3 , , m > MR b ): ν = − or − , R ( L m ) = − L m , m , − bL , m = − ν, , m > , m = − ν ;( MR b ): ν = 1 or , R ( L m ) = − L m , m < , m = − ν, − L − ν − bL , m = − ν, , m > MR b ): ν = 2 , or , R ( L m ) = − L m , m − , m = − ν − L − ν − bL , m = − ν, , m > − MR b ): R ( L m ) = − L m , m − , − L − − bL − , m = − , , m > − MR b ): R ( L m ) = − L m , m − , − L − − bL , m = − , − L − − bL − , m = − , , m > − MR b ): ν = 2 , or , R ( L m ) = − L m , m > − , − bL , m = − ν, , m − , m = − ν ;( MR b ): R ( L m ) = − L m , m > − , − bL − , m = − , , m − MR b ): R ( L m ) = − L m , m > − , − bL , m = − , − bL − , m = − , , m − , where b is a non-zero number. Conversely,the above operators are all the non-homogeneous Rota-Baxteroperators of weight satisfying (5.3) on the Witt algebra W .Proof. Due to Lemma 5.2, if we define a new operation x ◦ y = [ R ( x ) , y ] on W , then ( W, [ , ] , ◦ ) is apost-Lie algebra. By (5.3) we have L m ◦ L n = [ R ( L m ) , L n ] = ( m − n ) f ( m ) L m + n + ( m − n + ν ) g ( m ) L m + n + ν , ∀ m, n ∈ Z . This means that ( W, [ , ] , ◦ ) is a shifting post-Lie algebra structure satisfying (2.4) on W with φ ( m, n ) =( m − n ) f ( m ) and ̺ ( m, n ) = ( m − n + ν ) g ( m ). By Theorem 3.11, we see that f, g and ν must be one ofthe 16 cases listed in Table 2, which can get the 16 forms of R one by one. On the other hand, it is easyto verify that every form of R listed in the above is a Rota-Baxter operator of weight 1 satisfying (5.3).The proof is completed. (cid:3) Remark 5.5.
The Rota-Baxter operators given by Theorem 5.3 just are the all homogeneous Rota-Baxteroperators of weight described in [11] . But the Rota-Baxter operators given by Theorem 5.4 are new andnon-homogeneous. Remark 5.6.
The Rota-Baxter operators on the Witt algebra W can be given a class of solutions of theclassical Yang-Baxter equation (CYBE) on W ⋉ ad ∗ W ∗ . The details can be found in [11] , which discussthe homogeneous case. Along the same lines, we can also consider the non-homogeneous case by use ofTheorem 5.4. It is not discussed here. Acknowledgments
This work is supported in part by NSFC (Grant No. 11771069), NSF of Heilongjiang Province (GrantNo. A2015007), the Fund of Heilongjiang Education Committee (Grant No. 12531483 and No. HDJCCX-2016211).
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