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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 16, Issue 1


Volume 13 (2015)

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

Xiaomin Tang
  • Corresponding author
  • Department of Mathematics, Heilongjiang University, Harbin 150080, China
  • Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Heilongjiang University, Harbin 150080, China
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/ Yu Yang
Published Online: 2018-04-30 | DOI: https://doi.org/10.1515/math-2018-0042


The Witt algebra 𝔚d of rank d(≥ 1) is the derivation algebra of Laurent polynomial algebras in d commuting variables. In this paper, all biderivations of 𝔚d without anti-symmetric condition are determined. As an applications, commutative post-Lie algebra structures on 𝔚d 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 𝔚d the derivation Lie algebra of the Laurent polynomial algebra A=Fz1±1,,zd±1 in d commuting variables z1, …, zd over 𝔽. It is well known that the infinite-dimensional Lie algebra 𝔚d 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=zizi . For any n ∈ ℤd (considered as row vectors, i.e., n = (n1, …, nd) where ni ∈ ℤ), set zn = z1n1z2n2zdnd. We fix the vector space 𝔽d of 1 × d matrices. Denote the standard basis by e1, e2, …, ed, which are the row vectors of the identity matrix Id. Let (·, ·) be the standard symmetric bilinear form such that (u, v) = uvT ∈ 𝔽, where vT is the matrix transpose. For u ∈ 𝔽d and r ∈ ℤd, we denote D(u, r) = zri=1duii. Then we have


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


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


Recall that


is the Cartan subalgebra of 𝔚d.

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 𝔚d. 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 𝔚d for all d ≥ 1.

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


for all x, y, zL. 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 × LL given by f(x, y) = λ [x, y] for all yL 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, yL.

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

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 𝔚d is inner.

Lemma 2.2

Suppose that fB(𝔚d). Then there are linear maps ϕ and ψ from 𝔚d into itself such that


for all x, y ∈ 𝔚d.


Since f is a biderivation of 𝔚d, then for a fixed element x ∈ 𝔚d the map ϕx : LL given by ϕx(y) = f(x, y) is a derivation of 𝔚d by (2). Therefore, from Lemma 2.1 we know that ϕx is an inner derivation of 𝔚d. Therefore, there is a map ϕ : 𝔚d → 𝔚d such that ϕx = 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 ψz from 𝔚d into itself given by ψz(y) = f(y, z) for all y ∈ 𝔚d, then one can obtain a linear map ψ from 𝔚d into itself such that f(x, y) = ad(−ψ (y))(x) = [x, ψ(y)]. The proof is completed. □

Lemma 2.3

Let fB(𝔚d) and ϕ, ψ be determined by Lemma 2.2. For any i, j ∈ {1, …, d} and r, s ∈ ℤd, we assume that



where ak,n(i,r),bk,n(j,s) ∈ 𝔽. Then



Lemma 2.2 tells us that


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

Lemma 2.4

Let fB(𝔚d) and ϕ, ψ be determined by Lemma 2.2. Then the Cartan subalgebra 𝔥 is an invariant subspace of both maps ϕ and ψ.


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


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

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


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


which implies that


Thanks to sj ≠ 0, we have by (8) that ak,s(i,0) = 0 for every kj. 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

Let fB(𝔚d) and ϕ, ψ be determined by Lemma 2.2. Then we have



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


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 ∈ ℤd ∖{0}, yields that


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


which proves (10). □

Lemma 2.6

Let fB(𝔚d) and ϕ, ψ be determined by Lemma 2.2. Then there is λ ∈ 𝔽 such that


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


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


It follows that



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


which implies that


Similarly, we have


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

Lemma 2.7

Let fB(𝔚d), and ϕ, ψ be determined by Lemma 2.2, λ ∈ 𝔽 be given by Lemma 2.7. Then


for all i, j ∈ {1, …, d}.


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


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 ∈ ℤd ∖ {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 ∈ ℤd ∖ {0}. All these together with letting r = 0 in (5), deduce that


Then we have


for all s ∈ ℤd ∖ {0}. Let s run over the vectors e1, e2, …, ed, we have ai,0(i,0)=λ and ak,0(i,0)=0 for every ki. This proves that ϕ(D(ei, 0)) = λD(ei, 0). Similarly, we can obtain that ψ(D(ej, 0)) = λD(ej, 0). The proof is completed. □

Our main result is the following.

Theorem 2.8

Every biderivation of 𝔚d without anti-symmetric condition is inner.


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


for all x, y ∈ 𝔚d, 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 xy on L and satisfies the following identities:


for all x, y, zL. 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 × LL given by f(x, y) = xy for all x, yL, then f is a biderivation of L.

Theorem 3.3

Any commutative post-Lie algebra structure on the generalized Witt algebra 𝔚d is trivial. Namely, xy = 0 for all x, y ∈ 𝔚d.


Suppose that (𝔚d, [, ], ∘) is a commutative post-Lie algebra. By Lemma 3.2 and Theorem 2.8, we know that there is λ ∈ 𝔽 such that xy = λ [x, y] for all x, y ∈ 𝔚d. 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. □


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).


  • [1]

    Liu G., Zhao K., Irreducible modules over the derivation algebras of rational quantum tori. J. Algebra, 2011, 340(1), 28-34 Web of ScienceCrossrefGoogle Scholar

  • [2]

    Lin W., Tan S., Representations of the Lie algebra of derivations for quantum torus. J. Algebra, 2004, 275(1), 250-274 CrossrefGoogle Scholar

  • [3]

    Mazorchuk V., Zhao K., Supports of weight modules over Witt algebras. P. Roy. Soc. Ed.: Sec. A Math., 2011, 141(1), 155-170 CrossrefGoogle Scholar

  • [4]

    Rao S. E., Irreducible representations of the Lie-algebra of the diffeomorphisms of a d-dimensional torus. J. Algebra, 1996, 182(2), 401-421 CrossrefGoogle Scholar

  • [5]

    Zhao K., Weight modules over generalized Witt algebras with 1-dimensional weight spaces. Forum Math., 2004, 16(5), 725-748 Google Scholar

  • [6]

    Benkovič D., Biderivations of triangular algebras. Linear Algebra Appl., 2009, 431(9), 1587-1602 CrossrefWeb of ScienceGoogle Scholar

  • [7]

    Brešar M., On generalized biderivations and related maps. J. Algebra, 1995, 172(3), 764-786 CrossrefGoogle Scholar

  • [8]

    Chen Z., Biderivations and linear commuting maps on simple generalized Witt algebras over a field. Elec. J. Linear Algebra, 2016, 31(1), 1-12 CrossrefGoogle Scholar

  • [9]

    Cheng X., Wang M., Sun J., Zhang H., Biderivations and linear commuting maps on the Lie algebra gca. Linear Multilinear Algebra, 2017, 65(12), 2483-2493 Web of ScienceCrossrefGoogle Scholar

  • [10]

    Du Y., Wang Y., Biderivations of generalized matrix algebras. Linear Algebra Appl., 2013, 438(11), 4483-4499 Web of ScienceCrossrefGoogle Scholar

  • [11]

    Han X., Wang D., Xia C., Linear commuting maps and biderivations on the Lie algebras W (a, b). J. Lie theory, 2016, 26(3), 777-786 Google Scholar

  • [12]

    Liu X., Guo X., Zhao K., Biderivations of the block Lie algebras, Linear Algebra Appl., 2018, 538(2), 43-55 CrossrefWeb of ScienceGoogle Scholar

  • [13]

    Tang X., Biderivations of finite-dimensional complex simple Lie algebras. Linear Multilinear Algebra, 2018, 66(2), 250-259 Web of ScienceCrossrefGoogle Scholar

  • [14]

    Tang X., Biderivations, linear commuting maps and commutative post-Lie algebra structures on W-algebras. Comm. Algebra, 2017, 45(12), 5252-5261 CrossrefGoogle Scholar

  • [15]

    Wang D., Yu X., Chen Z., Biderivations of the parabolic subalgebras of simple Lie algebras. Comm. Algebra, 2011, 39(11), 4097-4104 CrossrefGoogle Scholar

  • [16]

    Wang D., Yu X., Biderivations and linear commuting maps on the Schrödinger-Virasoro Lie algebra. Comm. Algebra, 2013, 41(6), 2166-2173 CrossrefGoogle Scholar

  • [17]

    Ikeda T., Kawamoto N., On the derivations of generalized Witt algebras over a field of characteristic zero. Hiroshima Math. J., 1990, 20(1), 47-55 CrossrefGoogle Scholar

  • [18]

    Vallette B., Homology of generalized partition posets. J. Pure Appl. Algebra, 2007, 208(2), 699-725 CrossrefWeb of ScienceGoogle Scholar

  • [19]

    Burde D., Dekimpe K., Vercammen K., Affine actions on Lie groups and post-Lie algebra structures. Linear Algebra Appl., 2012, 437(5), 1250-1263 CrossrefWeb of ScienceGoogle Scholar

  • [20]

    Burde D., Moens W. A., Commutative post-Lie algebra structures on Lie algebras. J. Algebra, 2016, 467, 183-201 CrossrefWeb of ScienceGoogle Scholar

  • [21]

    Munthe-Kaas H. Z., Lundervold A., On post-Lie algebras, Lie-Butcher series and moving frames. Found. Comput. Math., 2013, 13(4), 583-613 CrossrefWeb of ScienceGoogle Scholar

  • [22]

    Pan Y., Liu Q., Bai C., Guo L., PostLie algebra structures on the Lie algebra sl(2, ℂ). Elec. J. Linear Algebra, 2012, 23(1), 180-197 Google Scholar

  • [23]

    Tang X., Zhang Y., Post-Lie algebra structures on solvable Lie algebra t (2,ℂ). Linear Algebra Appl., 2014, 462, 59-87 Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-11-08

Accepted: 2018-02-26

Published Online: 2018-04-30

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 447–452, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0042.

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© 2018 Tang and Yang, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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