Biderivations of the higher rank Witt algebra without anti-symmetric condition

Xiaomin Tang; Yu Yang

doi:10.1515/math-2018-0042

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Biderivations of the higher rank Witt algebra without anti-symmetric condition

Xiaomin Tang und Yu Yang

Veröffentlicht/Copyright: 30. April 2018

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Aus der Zeitschrift Open Mathematics Band 16 Heft 1

Abstract

The Witt algebra 𝔚_d of rank d(≥ 1) is the derivation algebra of Laurent polynomial algebras in d commuting variables. In this paper, all biderivations of 𝔚_d without anti-symmetric condition are determined. As an applications, commutative post-Lie algebra structures on 𝔚_d are obtained. Our conclusions recover and generalize results in the related papers on low rank or anti-symmetric cases.

Keywords: Biderivation; Higher rank Witt algebra; Anti-symmetric; Post-Lie algebra

MSC 2010: 17B05; 17B40

1 Introduction

Let 𝔽 be a field of characteristic zero. We denote by ℤ the sets of all integers. We fix a positive integer d ≥ 1 and denote by 𝔚_d the derivation Lie algebra of the Laurent polynomial algebra A=Fz1±1,⋯,zd±1 in d commuting variables z₁, …, z_d over 𝔽. It is well known that the infinite-dimensional Lie algebra 𝔚_d is called the Witt algebra of rank d. Its representations have attracted a lot of attention from many mathematicians [1,2,3,4,5]. According to their notations, the Witt algebra can be described as follows.

For i ∈ {1, 2, …, d}, set ∂i=zi∂zi . For any n ∈ ℤ^d (considered as row vectors, i.e., n = (n₁, …, n_d) where n_i ∈ ℤ), set zⁿ = z1n1z2n2⋯zdnd. We fix the vector space 𝔽^d of 1 × d matrices. Denote the standard basis by e₁, e₂, …, e_d, which are the row vectors of the identity matrix I_d. Let (·, ·) be the standard symmetric bilinear form such that (u, v) = uv^T ∈ 𝔽, where v^T is the matrix transpose. For u ∈ 𝔽^d and r ∈ ℤ^d, we denote D(u, r) = zr∑i=1dui∂i. Then we have

Du,r,Dv,s=Dw,r+s,u,v∈Fd,r,s∈Zd,(1)

where w = (u, s)v − (v, r)u. Therefore, the Witt algebra of rank d is the 𝔽-linear space

Wd=spanF{D(u,r)|u∈Fd,r∈Zd}

with brackets determined by (1). Note that D(u, r) is linear only with respect to the first component u. It is clear that 𝔚_d has a basis as

{D(e1,r),D(e2,r),⋯,D(ed,r)|r∈Zd}.

Recall that

h=Du,0|u∈Fd=spanF{D(e1,0),D(e2,0),⋯,D(ed,0)}

is the Cartan subalgebra of 𝔚_d.

It is well-known that derivations and generalized derivations are very important subjects in the research of both algebras and their generalizations. In recent years, biderivations have interested a great number of authors, see [6,7,8,9,10,11,12,13,14,15,16], [7], Brešar et al. showed that all biderivations on commutative prime rings are inner biderivations, and determined the biderivations of semiprime rings. The notion of biderivations of Lie algebras was introduced in [15]. Since then, biderivations of Lie algebras have been studied by many authors. It may be useful and interesting for computing the biderivations of some important Lie algebras. In particular, the authors in [11] determined anti-symmetric biderivations for all 𝔚_d. All biderivations of 𝔚₁ without anti-symmetric condition were later obtained in [14]. In the present paper, we shall use the methods of [14] to determine all biderivations of 𝔚_d for all d ≥ 1.

Next, let us introduce the definition of biderivation. For an arbitrary Lie algebra L, a bilinear map f : L × L → L is called a biderivation of L if it is a derivation with respect to both components. Namely, for each x ∈ L, both linear maps ϕ_x and ψ_x form L into itself given by ϕ_x = f(x, ·) and ψ_x = f(·, x) are derivations of L, i.e.,

f([x,y],z)=[x,f(y,z)]+[f(x,z),y],f(x,[y,z])=[f(x,y),z]+[y,f(x,z)](2)

for all x, y, z ∈ L. Denote by B(L) the set of all biderivations of L. For λ ∈ ℂ, it is easy to verify that the bilinear map f : L × L → L given by f(x, y) = λ [x, y] for all y ∈ L is a biderivation of L. Such biderivation is said to be inner. Recall that f is anti-symmetric if f(x, y) = -f(y, x) for all x, y ∈ L.

In this paper, we will prove that every biderivation of 𝔚_d without anti-symmetric condition is inner. As an application, we characterize the commutative post-Lie algebra structures on 𝔚_d.

2 Biderivations of the Witt algebras

We first give some lemmas which will be useful for our proof.

Lemma 2.1

([17]). Every derivation of 𝔚_dis inner.

Lemma 2.2

Suppose thatf ∈ B(𝔚_d). Then there are linear mapsϕandψfrom 𝔚_dinto itself such that

f(x,y)=[ϕ(x),y]=[x,ψ(y)]

for allx, y ∈ 𝔚_d.

Proof

Since f is a biderivation of 𝔚_d, then for a fixed element x ∈ 𝔚_d the map ϕ_x : L → L given by ϕ_x(y) = f(x, y) is a derivation of 𝔚_d by (2). Therefore, from Lemma 2.1 we know that ϕ_x is an inner derivation of 𝔚_d. Therefore, there is a map ϕ : 𝔚_d → 𝔚_d such that ϕ_x = adϕ(x), i.e., f(x, y) = [ϕ(x), y]. Since f is bilinear, it is easy to verify that ϕ is linear. Similarly, if we define a map ψ_z from 𝔚_d into itself given by ψ_z(y) = f(y, z) for all y ∈ 𝔚_d, then one can obtain a linear map ψ from 𝔚_d into itself such that f(x, y) = ad(−ψ (y))(x) = [x, ψ(y)]. The proof is completed. □

Lemma 2.3

Letf ∈ B(𝔚_d) andϕ, ψbe determined by Lemma 2.2. For any i, j ∈ {1, …, d} and r, s ∈ ℤ^d, we assume that

ϕ(D(ei,r))=∑k=1d∑n∈Zdak,n(i,r)D(ek,n),(3)

ψ(D(ej,s))=∑k=1d∑n∈Zdbk,n(j,s)D(ek,n)(4)

whereak,n(i,r),bk,n(j,s) ∈ 𝔽. Then

∑k=1d∑n∈Zdak,n(i,r)skD(ej,n+s)−∑k=1d∑n∈Zdak,n(i,r)njD(ek,n+s)=∑k=1d∑n∈Zdbk,n(j,s)niD(ek,n+r)−∑k=1d∑n∈Zdbk,n(j,s)rkD(ei,n+r).(5)

Proof

Lemma 2.2 tells us that

f(D(ei,r),D(ej,s))=[ϕ(D(ei,r)),D(ej,s)]=[D(ei,r),ψ(D(ej,s))]

for all i, j ∈ {1, …, d} and r, s ∈ ℤ^d. From (3) and (4), the conclusion follows by direct computations. □

Lemma 2.4

Letf ∈ B(𝔚_d) andϕ, ψbe determined by Lemma 2.2. Then the Cartan subalgebra 𝔥 is an invariant subspace of both mapsϕandψ.

Proof

Note that D(e_i, 0), i = 1, … d span the Cartan subalgebra 𝔥, so it is enough to prove that ϕ(D(e_i, 0)), ψ(D(e_i, 0)) ∈ 𝔥 for each i ∈ {1,2, …, d}. For any fixed i ∈ ℤ, applying (3) for r = 0 we have that

ϕ(D(ei,0))=∑k=1d∑n∈Zdak,n(i,0)D(ek,n).(6)

We will prove that ak,n(i,0) = 0 in (6) for all n ∈ ℤ^d ∖ {0}, and so that ϕ(D(e_i, 0)) = ∑k=1dak,0(i,0)D(ek,0)∈h. The proof of ψ(D(e_i, 0)) ∈ 𝔥 is similar.

Now for an arbitrary s ∈ ℤ^d ∖ {0}, we assume that s_j ≠ 0 for some j ∈ {1,2, …, d}. It follows by letting r = 0 in (5) that

∑k=1d∑n∈Zdak,n(i,0)skD(ej,n+s)−∑k=1d∑n∈Zdak,n(i,0)njD(ek,n+s)=∑k=1d∑n∈Zdbk,n(j,s)niD(ek,n).(7)

It is clear that the right-hand side of (7) does not contain any non-zero elements in 𝔥, thereby the left-hand side is so. From this, one has that

∑k=1dak,−s(i,0)skD(ej,0)+∑k=1dak,−s(i,0)sjD(ek,0)=0,

which implies that

(2aj,−s(i,0)sj+∑k=1k≠jdak,−s(i,0)sk)D(ej,0)+∑k=1k≠jdak,−s(i,0)sjD(ek,0)=0.(8)

Thanks to s_j ≠ 0, we have by (8) that ak,−s(i,0) = 0 for every k ≠ j. Once again applying (8), we see that 2aj,−s(i,0)sj = 0, i.e., aj,−s(i,0) = 0. In other words, ak,−s(i,0) = 0 for all k = 1, …, d. Notice the arbitrariness of s, the proof is completed. □

Lemma 2.5

Letf ∈ B(𝔚_d) andϕ, ψbe determined by Lemma 2.2. Then we have

ϕ(D(ei,r))≡ai,r(i,r)D(ei,r)modh,(9)

ψ(D(ej,s))≡bj,s(j,s)D(ej,s)modh(10)

for all i, j ∈ {1, …, d} and r, s ∈ ℤ^d ∖{0}.

Proof

We will only prove (10), the proof for (9) is similar. Continuing the use of the assumptions (3) and (4), we also have that (7) holds. This, together with Lemma 2.4 meaning ak,n(i,0) = 0 for all n ∈ ℤ^d ∖{0}, yields that

(∑k=1dak,0(i,0)sk)D(ej,s)=∑k=1d∑n∈Zdn≠0bk,n(j,s)niD(ek,n).(11)

Therefore, from (11) we see that bk,n(j,s)ni = 0 for all i = 1, … d and (k, n) ≠ (j, s) with n ≠ 0. This implies that bk,n(j,s) = 0 for all (k, n) ≠ (j, s) since n ≠ 0. It has been obtained that

ψ(D(ej,s))=∑k=1dbk,0(j,s)D(ek,0)+bj,s(j,s)D(ej,s),

which proves (10). □

Lemma 2.6

Let f ∈ B(𝔚_d) andϕ, ψ be determined by Lemma 2.2. Then there is λ ∈ 𝔽 such that

ϕ(D(ei,r))=λD(ei,r),ψ(D(ej,s))=λD(ej,s)

for all i, j ∈ {1, …, d} and r, s ∈ ℤ^d ∖ {0}.

Proof

We use the assumptions (3) and (4). With Lemmas 2.4 and 2.5, Equation (5) becomes

(∑k=1dak,0(i,r)sk)D(ej,s)+ai,r(i,r)siD(ej,r+s)−ai,r(i,r)rjD(ei,r+s)=−(∑k=1dbk,0(j,s)rk)D(ei,r)+bj,s(j,s)siD(ej,r+s)−bj,s(j,s)rjD(ei,r+s).

It follows that

∑k=1dak,0(i,r)sk=∑k=1dbk,0(j,s)rk=0,∀s,r∈Zd∖{0},withs≠r,(12)

ai,r(i,r)si=bj,s(j,s)si,ai,r(i,r)rj=bj,s(j,s)rj,∀s,r∈Zd∖{0},i≠j.(13)

Although r ≠ 0, but we still can find a subset {s͠⁽¹⁾, …, s͠^(d)} of ℤ^d such that s͠⁽¹⁾, …, s͠^(d) are 𝔽-linearly independent with s͠^(t) ≠ r, t = 1, …, d. Let s run over the vectors s͠⁽¹⁾, …, s͠^(d) in (12), then we see that

s~(1)⋮s~(d)a1,0(i,r)⋮ad,0(i,r)=0,

which implies that

a1,0(i,r)=⋯=ad,0(i,r)=0.

Similarly, we have

b1,0(j,s)=⋯=bd,0(j,s)=0.

Next, by taking s = (1, …, 1) ≐ e in (13), we have ai,r(i,r)=bj,e(j,e) for all i ≠ j. This tells us that ai,r(i,r)=b1,e(1,e) for any i ≠ 1 and ai,r(i,r)=b2,e(2,e) for any i ≠ 2. It follows that ai,r(i,r) is a constant denoted by λ for all i = 1, …, d and r ∈ ℤ^d ∖ {0}. Similarly, we obtain that bj,s(j,s) is a constant denoted by μ for all j = 1, …, d and s ∈ ℤ^d ∖ {0}. Finally, by a2,e(2,e)=b1,e(1,e) we have λ = μ, which completes the proof. □

Lemma 2.7

Let f ∈ B(𝔚_d), andϕ, ψ be determined by Lemma 2.2, λ ∈ 𝔽 be given by Lemma 2.7. Then

ϕ(D(ei,0))=λD(ei,0),ψ(D(ej,0))=λD(ej,0)

for all i, j ∈ {1, …, d}.

Proof

We use the assumptions (3) and (4). By Lemma 2.4, we have

ϕ(D(ei,0))=∑k=1dak,0(i,0)D(ek,0),ψ(D(ej,0))=∑k=1dbk,0(j,0)D(ek,0)

for all i, j ∈ ℤ. Namely, it follows that, in (3) and (4), ak,n(i,0)=bk,n(j,0)=0 for all n ∈ ℤ^d ∖ {0}. Note that Lemma 2.6 tells us that, in (3) and (4), ak,n(i,r)=δi,kδn,rλ and bk,n(j,s)=δj,kδn,sλ for any i, j ∈ ℤ and r, s ∈ ℤ^d ∖ {0}. All these together with letting r = 0 in (5), deduce that

∑k=1dak,0(i,0)skD(ej,s)=λsiD(ej,s).

Then we have

s1a1,0(i,0)+⋯+si(ai,0(i,0)−λ)+⋯+sdad,0(i,0)=0

for all s ∈ ℤ^d ∖ {0}. Let s run over the vectors e₁, e₂, …, e_d, we have ai,0(i,0)=λ and ak,0(i,0)=0 for every k ≠ i. This proves that ϕ(D(e_i, 0)) = λD(e_i, 0). Similarly, we can obtain that ψ(D(e_j, 0)) = λD(e_j, 0). The proof is completed. □

Our main result is the following.

Theorem 2.8

Every biderivation of 𝔚_dwithout anti-symmetric condition is inner.

Proof

Suppose that f is a biderivation of 𝔚_d. Let ϕ be determined by Lemma 2.2, λ ∈ 𝔽 be given by Lemma 2.7. Note that 𝔚_d is spanned by D(u, 0), D(u, r) for all u ∈ 𝔽^d and r ∈ ℤ^d ∖ {0}. Then by Lemmas 2.6 and 2.7, we see that ϕ(x) = λx for all x ∈ 𝔚_d. Now, it follows by Lemma 2.2 that

f(x,y)=[ϕ(x),y]=[λx,y]=λ[x,y]

for all x, y ∈ 𝔚_d, as desired. □

3 An application

The anti-symmetric biderivation can be applied to linear commuting maps, commuting automorphisms and derivations, see [8]. Another application of biderivation without the anti-symmetric condition is the characterization of post-Lie algebra structures. Post-Lie algebras have been introduced by Valette in connection with the homology of partition posets and the study of Koszul operads [18]. As [19] point out, post-Lie algebras are natural common generalization of pre-Lie algebras and LR-algebras in the geometric context of nil-affine actions of Lie groups. Recently, many authors have studied some post-Lie algebras and post-Lie algebra structures [19, 20, 21, 22, 23]. In particular, the authors of [19] study the commutative post-Lie algebra structure on Lie algebra. Let us recall the following definition of a commutative post-Lie algebra.

Definition 3.1

Let (L,[,]) be a Lie algebra over 𝔽. A commutative post-Lie algebra structure on L is a 𝔽-bilinear product x ∘ y on L and satisfies the following identities:

x∘y=y∘x,[x,y]∘z=x∘(y∘z)−y∘(x∘z),x∘[y,z]=[x∘y,z]+[y,x∘z].

for all x, y, z ∈ L. It is also said that (L,[,],∘) is a commutative post-Lie algebra.

Lemma 3.2

([14]). Let (L,[,],∘) be a commutative post-Lie algebra. If we define a bilinear map f : L × L → L given by f(x, y) = x ∘ y for all x, y ∈ L, thenf is a biderivation of L.

Theorem 3.3

Any commutative post-Lie algebra structure on the generalized Witt algebra 𝔚_dis trivial. Namely, x ∘ y = 0 for all x, y ∈ 𝔚_d.

Proof

Suppose that (𝔚_d, [, ], ∘) is a commutative post-Lie algebra. By Lemma 3.2 and Theorem 2.8, we know that there is λ ∈ 𝔽 such that x ∘ y = λ [x, y] for all x, y ∈ 𝔚_d. Since the post-Lie algebra is commutative, so we have λ[x, y] = λ[y, x]. It implies that λ = 0. The proof is completed. □

Acknowledgement

We would like to thank the referee for invaluable comments and suggestions. This work was supported in part by the NNSFC [grant number 11771069], the NSF of Heilongjiang Province [grant number A2015007] and the Funds of the Heilongjiang Education Committee [grant numbers 12531483 and HDJCCX-2016211).

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Received: 2017-11-08

Accepted: 2018-02-26

Published Online: 2018-04-30

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Biderivation; Higher rank Witt algebra; Anti-symmetric; Post-Lie algebra

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