2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Journal of Algebra and Its ApplicationsVol. 16, No. 2 (2017) 1750190 (21 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219498817501900

Grobner–Shirshov bases for Lie Ω-algebrasand free Rota–Baxter Lie algebras

Jianjun Qiu∗ and Yuqun Chen†

School of Mathematical SciencesSouth China Normal UniversityGuangzhou 510631, P. R. China

∗jianjunqiu@126.com†yqchen@scnu.edu.cn

Received 24 May 2016Accepted 11 October 2016Published 1 December 2016

Communicated by Leonid Bokut

We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov Ω-words on a set Xand prove that the set of all the nonassociative Lyndon–Shirshov Ω-words forms a linearbasis of the free Lie Ω-algebra on the set X. From this, we establish Grobner–Shirshovbases theory for Lie Ω-algebras. As applications, we give Grobner–Shirshov bases of afree λ-Rota–Baxter Lie algebra, of a free modified λ-Rota–Baxter Lie algebra, and of afree Nijenhuis Lie algebra and, then linear bases of these three algebras are obtained.

Keywords: Grobner–Shirshov basis; Lyndon–Shirshov word; Lie Ω-algebra; Rota–BaxterLie algebra; Nijenhuis Lie algebra.

Mathematics Subject Classification: 16S15, 13P10, 17A50, 16T25

1. Introduction

Let (R, ·) be an algebra over a field k, λ ∈ k and P : R → R a linear map satisfying

P (x) · P (y) = P (P (x) · y) + P (x · P (y)) + λP n(x · y), x, y ∈ R,

where n is a nonnegative integer. If n = 1 (respectively, n = 0, n = 2 and λ =−1), then the operator P is called a λ-Rota–Baxter (respectively modified λ-Rota–Baxter, Nijenhuis) operator on the algebra (R, ·) and the triple (R, ·, P ) is called aλ-Rota–Baxter (respectively modified λ-Rota–Baxter, Nijenhuis) algebra.

The Rota–Baxter operator on an associative algebra was introduced by Baxterto solve an analytic problem in probability [6] and was studied by Rota [55] in com-binatorics later. The Rota–Baxter operator of weight λ = 0 on a Lie algebra is also

†Corresponding author.

1750190-1

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

called the operator form of the classical Yang–Baxter equation due to Semenov–Tian–Shansky’s work [57]. The modified Rota–Baxter operator on an associativealgebra was introduced by Ebrahimi-Fard [32] with motivation from modified clas-sical Yang–Baxter equation on a Lie algebra [57]. The Nijenhuis operator on anassociative algebra was introduced by Carinena et al. [24] to study quantum bi-Hamiltonian systems. In [62], the Nijenhuis operators were constructed by analogywith Poisson–Nijenhuis geometry from relative Rota–Baxter operators. The Nijen-huis operator on a Lie algebra plays an important role in the study of integrability ofnonlinear evolution equations [28]. Recently, there are some results on Rota–Baxteroperators on Lie algebras and related topic (see for instance [1, 4, 5, 37, 50, 53]).

The constructions of a free λ-Rota–Baxter associative algebra (respectively afree modified λ-Rota–Baxter associative algebra, a free Nijenhuis associative alge-bra) were given by using different methods, for example, [3, 14, 25, 33–36, 38–42,49, 55]. We will apply Grobner–Shirshov bases method to construct free λ-Rota–Baxter Lie algebras, free modified λ-Rota–Baxter Lie algebras and free NijenhuisLie algebras.

Grobner bases and Grobner–Shirshov bases were invented independently byShirshov for ideals of free (commutative, anti-commutative) nonassociative algebras[60, 61], free Lie algebras [59, 61] and implicitly free associative algebras [59, 61](see also [7, 9]), by Hironaka [46] for ideals of the power series algebras (both for-mal and convergent), and by Buchberger [21] for ideals of the polynomial algebras.Grobner bases and Grobner–Shirshov bases theories have been proved to be veryuseful in different branches of mathematics, including commutative algebra andcombinatorial algebra. It is a powerful tool to solve the following classical prob-lems: normal form; word problem; conjugacy problem; rewriting system; automa-ton; embedding theorem; PBW theorem; extension; homology; growth function;Dehn function; complexity; etc. See, for example, the books [2, 18, 22, 23, 27, 30],the papers [7–9, 13, 29, 36, 52], and the surveys [11, 15–17].

The concept of Ω-algebra was introduced by Kurosh [47] under an influence ofthe concept of multioperator group of Higgins [45]. An Ω-algebra over a field k

is a k-algebra A with a set of multilinear operators Ω, where Ω =⋃∞

m=1 Ωm andeach Ωm is a set of m-ary multilinear operators on A. A λ-Rota–Baxter Lie algebra(respectively modified λ-Rota–Baxter Lie algebra, Nijenhuis Lie algebra) can beseen as a Lie Ω-algebra with a single operator. In [14], Grobner–Shirshov basestheory for associative Ω-algebras is established and it was used to find a linear basisof a free λ-Rota–Baxter associative algebra. For Grobner–Shirshov bases theory ofvarious Ω-algebras and their applications, see for instance [12, 14, 31, 36, 54].

As we know, the free Lie algebras play a great role in the Lie theory, especiallythe linear bases of the free Lie algebras, which are difficult to find in the history.Magnus [51] and Witt [63] showed that the Lie algebra of Lie polynomials is actuallythe free Lie algebra independently in 1937. Shirshov 1953 [58] and Witt 1956 [64]showed the Shirshov–Witt theorem, i.e. any Lie subalgebra of a free Lie algebrais itself a free Lie algebra. Shirshov [58] proved the Shirshov–Lazard elimination

1750190-2

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

theorem for a free Lie algebra, which is some years earlier than the work ofLazard [48] and this is often called the Lazard elimination (see, for instance, [56]).The first explicit basis (called Hall basis) of the free Lie algebra Lie(X) on the set X

was given by Hall in 1950 [43] based on work of Hall [44] on groups. In 1958, Chen,Fox, Lyndon [26] and Shirshov [59] introduced the nonassociative Lyndon–Shirshovwords on X and proved that they form a linear basis of Lie(X), independently.One of the main applications of the Lyndon–Shirshov basis is the Shirshov theoryof Grobner–Shirshov bases for Lie algebras [59].

The paper is organized as follows. In Sec. 2, we review the concept and somerelated properties of the Lyndon–Shirshov words on a set X , generalize the associa-tive (respectively nonassociative) Lyndon–Shirshov words to the associative (respec-tively nonassociative) Lyndon–Shirshov Ω-words on a set X and show that the setof all the nonassociative Lyndon–Shirshov Ω-words forms a linear basis of the freeLie Ω-algebra on the set X . In Sec. 3, we review the Grobner–Shirshov bases theoryfor associative Ω-algebras, give the definition of the Grobner–Shirshov bases for LieΩ-algebras and establish the Composition-Diamond lemma for Lie Ω-algebras. InSec. 4, we give Grobner–Shirshov bases of a free λ-Rota–Baxter Lie algebra, of afree modified λ-Rota–Baxter Lie algebra, and of a free Nijenhuis Lie algebra and,then linear bases of these three algebras are obtained by the Composition-Diamondlemma for Lie Ω-algebras.

2. Free Lie Ω-Algebras

2.1. Lyndon–Shirshov words

In this subsection, we review the concept and some properties of the Lyndon–Shirshov words, which can be found in [10, 59, 61].

For any set X , we define the following notations:S(X): the set of all nonempty associative words on X .X∗: the set of all associative words on X including the empty word 1.X∗∗: the set of all nonassociative words on X .For any u ∈ X∗, let us denote by deg(u) the degree (length) of u. Let > be a

well order on X . Define the lex-order >lex and the deg-lex order >deg−lex on X∗

with respect to > by:

(i) 1 >lex u for any nonempty word u, and if u = xiu′ and v = xjv

′, wherexi, xj ∈ X , then u >lex v if xi > xj , or xi = xj and u′ >lex v′ by induction.

(ii) u >deg−lex v if deg(u) > deg(v), or deg(u) = deg(v) and u >lex v.

A nonempty associative word w is called an associative Lyndon–Shirshov wordon X , if w = uv >lex vu for any decomposition of w = uv, where 1 = u, v ∈ X∗.

A nonassociative word (u) ∈ X∗∗ is said to be a nonassociative Lyndon–Shirshovword on X with respect to the lex-order >lex, denoted by [u], if

(a) u is an associative Lyndon–Shirshov word on X ;

1750190-3

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

(b) if (u) = ((v)(w)), then both (v) and (w) are nonassociative Lyndon–Shirshovwords on X ;

(c) if (v) = ((v1)(v2)), then v2 ≤lex w.

Let ALSW(X) (respectively NLSW(X)) denote the set of all the associative(respectively nonassociative) Lyndon–Shirshov words on X with respect to thelex-order >lex. It is well known that for any u ∈ ALSW(X), there exists a uniqueShirshov standard bracketing way [u] such that [u] ∈ NLSW(X). Then NLSW(X) =[u] |u ∈ ALSW(X).

Let k〈X〉 be the free associative algebra on X over a field k and Lie (X) be theLie subalgebra of k〈X〉 generated by X under the Lie bracket (uv) = uv − vu. Itis well-known that Lie(X) is a free Lie algebra on the set X and NLSW(X) is alinear basis of Lie(X).

For any f ∈ k〈X〉, let f be the leading word of f with respect to the deg-lexorder >deg−lex on X∗.

Lemma 2.1 ([10, 59, 61]). For any nonassociative word (u) ∈ X∗∗, (u) has arepresentation

(u) =∑

αi[ui]

in Lie(X), where each αi ∈ k and ui ∈ ALSW(X).

Lemma 2.2 ([10, 59, 61]). If u ∈ ALSW(X), then [u] = u.

Lemma 2.3 ([10, 59, 61]). For any u ∈ X∗, there exists a unique decomposition

u = u1u2 · · ·um,

where each ui ∈ ALSW(X) and uj ≤lex uj+1, 1 ≤ j ≤ m − 1.

Lemma 2.4 ([10, 59, 61]). Let u = avb, where u, v ∈ ALSW(X) and a, b ∈ X∗.Then

[u] = [a[vc]d],

where b = cd for some c, d ∈ X∗. Let

[u]v = [u][vc] →[[···[[[v][c1]][c2]]···][cm]],

where c = c1c2 · · · cm with each ci ∈ ALSW(X), c1 ≤lex c2 ≤lex · · · ≤lex cm and[[· · · [[[v][c1]][c2]] · · ·][cm]] is the left normed word on [v], [c1], [c2], . . . , [cm]. Then,

[u]v = a[v]b +∑

αiai[v]bi,

where each αi ∈ k, ai, bi ∈ X∗ and aivbi <deg−lex avb = u. It follows that [u]v = u.

1750190-4

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

2.2. Lyndon–Shirshov Ω-words

In this subsection, we define the Lyndon–Shirshov Ω-words on a set X .Let

Ω =∞⋃

m=1

Ωm,

where Ωm is a set of m-ary operators for any m ≥ 1. For any set Y , let

Ω(Y ) :=∞⋃

m=1

ω(m)(y1, y2, . . . , ym) | yi ∈ Y, 1 ≤ i ≤ m, ω(m) ∈ Ωm.

Let X be a set. Define 〈Ω; X〉0 := S(X) and (Ω; X)0 := X∗∗. Assume that wehave defined 〈Ω; X〉n−1 and (Ω; X)n−1. Define

〈Ω; X〉n := S(X ∪ Ω(〈Ω; X〉n−1)),

(Ω; X)n := (X ∪ Ω((Ω; X)n−1))∗∗.

Then it is easy to see that for any n ≥ 0,

〈Ω; X〉n ⊆ 〈Ω; X〉n+1, (Ω; X)n ⊆ (Ω; X)n+1.

Set

〈Ω; X〉 :=∞⋃

n=0

〈Ω; X〉n, (Ω; X) :=∞⋃

n=0

(Ω; X)n.

The elements of 〈Ω; X〉 (respectively (Ω; X)) are called associative (respectivelynonassociative) Ω-words on X .

If u ∈ X ∪ Ω(〈Ω; X〉), then u is called prime. Therefore, for any u ∈ 〈Ω; X〉, u

can be uniquely expressed in the canonical form

u = u1u2 · · ·un, n ≥ 1,

where each ui is prime. The breath of u, denoted by bre(u), is defined to be thenumber n. The degree of u, denoted by deg(u), is defined to be the total number ofall occurrences of all x ∈ X and θ ∈ Ω in u. By the depth of u, denoted by dep(u),we mean dep(u) := minn |u ∈ 〈Ω; X〉n. Let (u) be a nonassociative Ω-word.Define the depth of (u) by dep((u))=dep(u).

For example, if u = ω(3)(x2x1x1, x1, ω(1)(x2x2x1))x2x1, where x1, x2 ∈ X and

ω(3), ω(1) ∈ Ω, then deg(u)=11, bre(u)=3 and dep(u)=2.Let u = u1u2 · · ·un, n ≥ 1, where each ui is prime. Define

wt(u) := (deg(u), bre(u), u1, u2, . . . , un).

Let X and Ω be well-ordered sets with the orders >X and >Ω, respectively. Definethe Deg-lex order >Dl on 〈Ω; X〉 as follows. For any u = u1u2 · · ·un, v = v1v2 · · · vm,where ui, vj are prime, define

u >Dl v if wt(u) > wt(v) lexicographically,

1750190-5

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

where if ui = ω(ui1, ui2, . . . , uit), vi = θ(vi1, vi2, . . . , vil) and deg(ui)=deg(vi), thenui > vi if

(ω, ui1, ui2, . . . , uit) > (θ, vi1, vi2, . . . , vil) lexicographically.

Let be the restriction of >Dl on X ∪ Ω(〈Ω; X〉). We define the Lyndon–Shirshov Ω-words on the set X by induction on the depth of Ω-words.

For n = 0, define

ALSW(Ω; X)0 := ALSW(X),

NLSW(Ω; X)0 := NLSW(X) = [u] |u ∈ ALSW(Ω; X)0with respect to the lex-order lex on X∗.

Assume that we have defined

ALSW(Ω; X)n−1 and NLSW(Ω;X)n−1 = [u] |u ∈ ALSW(Ω; X)n−1.Define

ALSW(Ω; X)n := ALSW(X ∪ Ω(ALSW(Ω; X)n−1))

with respect to the lex-order lex.For any u ∈ X ∪ Ω(ALSW(Ω; X)n−1), define the bracketing way on u by

[u] :=

u if u ∈ X,

ω(m)([u1], [u2], . . . , [um]) if u = ω(m)(u1, u2, . . . , um).

Let

[X ∪ Ω(ALSW(Ω; X)n−1)] := [u] |u ∈ X ∪ Ω(ALSW(Ω; X)n−1).Then, the order on X ∪ Ω(ALSW(Ω; X)n−1) induces an order on [X ∪Ω(ALSW(Ω; X)n−1)] by [u] [v] if u v for any u, v ∈ X∪Ω(ALSW(Ω; X)n−1). Ifu = u1u2 · · ·um ∈ ALSW(Ω; X)n, where each ui ∈ X ∪ Ω(ALSW(Ω; X)n−1), thenwe define

[u] := [[u1][u2] · · · [um]]

is the Shirshov standard bracketing way on [u1], [u2], . . . , [um], which means [u] anonassociative Lyndon–Shirshov word on [u1], [u2], . . . , [um] with respect to thelex-order lex. Define

NLSW(Ω; X)n := [u] |u ∈ ALSW(Ω; X)n.It is easy to see that

NLSW(Ω; X)n = NLSW([X ∪ Ω(ALSW(Ω; X)n−1)]).

Set

ALSW(Ω; X) :=∞⋃

n=0

ALSW(Ω; X)n,

NLSW(Ω; X) :=∞⋃

n=0

NLSW(Ω; X)n.

1750190-6

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

Then

NLSW(Ω; X) = [u] |u ∈ ALSW(Ω; X).

The elements of ALSW(Ω; X) (respectively NLSW(Ω; X)) are called the associative(respectively nonassociative) Lyndon–Shirshov Ω-words. By the above definitions,for any associative Lyndon–Shirshov Ω-word u, u is associated a unique nonasso-ciative Lyndon–Shirshov Ω-word [u]. For example, if

u = ω(3)(x2x1x1, x1, ω(1)(x2x2x1))x2x1,

where x1, x2 ∈ X with x2 > x1, and ω(3), ω(1) ∈ Ω, then u ∈ ALSW(Ω; X) and

[u] = (ω(3)(((x2x1)x1), x1, ω(1)((x2(x2x1))))(x2x1)) ∈ NLSW(Ω; X).

2.3. Free Lie Ω-algebras

In this subsection, we prove that the set NLSW(Ω; X) of all the nonassociativeLyndon–Shirshov Ω-words is a linear basis of the free Lie Ω-algebra on the set X .

An associative (respectively nonassociative, Lie) Ω-algebra over a field k is anassociative (respectively nonassociative, Lie) algebra A with a set of multilinearoperators Ω.

For any associative Ω-algebra (A, ·, Ω), it is easy to see that (A, [, ], Ω) is a LieΩ-algebra under the Lie bracket [a, a′] = a · a′ − a′ · a, a, a′ ∈ A.

Let X be a set. An associative (respectively nonassocitive, Lie) Ω-algebra F (X)together with a injective map i : X → F (X) is called a free associative (respectivelynonassocitive, Lie) Ω-algebra on X , if for any associative (respectively nonassoci-tive, Lie) Ω-algebra A and any map σ : X → A, there exists a unique Ω-algebrahomomorphism σ : F (X) → A such that σi = σ.

Let k〈Ω; X〉 be the k-linear space spanned by 〈Ω; X〉. Then k〈Ω; X〉 is a freeassociative Ω-algebra on the set X , see [14]. Let us denote Lie(Ω; X), the Lie Ω-subalgebra of k〈Ω; X〉 generated by X under the Lie bracket (uv) = uv − vu. Theelements of k〈Ω; X〉 (respectively Lie(Ω; X)) are called associative (respectively Lie)Ω-polynomials on X .

Lemma 2.5. For any (u) ∈ (Ω; X), (u) has a representation

(u) =∑

αi[ui],

in Lie(Ω; X), where each αi ∈ k and [ui] ∈ NLSW(Ω; X).

Proof. The proof is by induction on dep((u)). If dep((u))=0, by Lemma 2.1, wehave (u) =

∑αi[ui], where each αi ∈ k and [ui] ∈ NLSW(X) ⊆ NLSW(Ω; X).

Assume that the result is true for any (u) with dep((u))≤ n − 1.Let dep((u))=n ≥ 1. There are two cases to consider.

1750190-7

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

Case 1. If (u) = θ((u1), (u2), . . . , (um)), then dep((ui))≤ n − 1, 1 ≤ i ≤ m. Thus,by induction, we may assume that (ui) = [ui], where each [ui] ∈ NLSW(Ω; X).Therefore,

(u) = θ([u1], [u2], . . . , [um]) = [θ(u1, u2, . . . , um)] ∈ NLSW(Ω; X).

Case 2. If (u) = (a1a2 · · ·am), m ≥ 2, where each ai is prime, then by Case 1, wemay assume that (u) = ([a1][a2] · · · [am]), where each ai ∈ X ∪ Ω(ALSW(Ω; X)n−1).Thus, (u) is a nonassociative word on [a1], [a2], . . . , [am]. By Lemma 2.1, we have

([a1][a2] · · · [am]) =∑

αi[ui],

where each αi ∈ k and [ui] ∈ NLSW([a1], [a2], . . . , [am]) ⊆ NLSW(Ω; X).

For any f ∈ k〈Ω; X〉, let f be the leading Ω-word of f with respect to the order>Dl on 〈Ω; X〉.

Lemma 2.6. If u ∈ ALSW(Ω; X), then [u] = u with respect to the order >Dl on〈Ω; X〉.

Proof. The proof is by induction on dep(u). If dep(u)= 0, then [u] = u byLemma 2.2. Assume that the result is true for any u ∈ ALSW(Ω; X) withdep(u)≤ n − 1.

Let dep(u)= n ≥ 1. There are two cases to consider.

Case 1. If bre(u)=1 and u = θ(u1, u2, . . . , um), then [u] = θ([u1], [u2], . . . , [um]).By induction, we have [ui] = ui, 1 ≤ i ≤ m. Thus,

[u] = θ([u1], [u2], . . . , [um]) = θ(u1, u2, . . . , um) = u.

Case 2. If bre(u)> 1 and u = a1a2 · · · am, where each ai is prime, then byLemma 2.2, we have

[u] = [[a1][a2] · · · [am]] = [a1][a2] · · · [am] +∑

αi[ai1 ][ai2 ] · · · [aim ],

where each i1, i2, . . . , im = 1, 2, . . . , m and ai1ai2 · · ·aim <Dl a1a2 · · · am. ByCase 1, we have [at] = at and [aij ] = aij . It follows that [u] = a1a2 · · ·am = u.

Lemma 2.7. The set NLSW(Ω; X) is a linear basis of Lie(Ω; X).

Proof. Suppose∑m

i=1 αi[ui] = 0 in Lie(Ω; X), where each αi ∈ k, ui ∈ALSW(Ω; X) and ui >Dl ui+1. If α1 = 0, then by Lemma 2.6, we have

∑mi=1 αi[ui] =

u1, a contradiction. Therefore, NLSW(Ω; X) is linear independent. By Lemma 2.5,NLSW(Ω; X) is a linear basis of Lie(Ω;X).

1750190-8

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

Let k(Ω; X) be the k-linear space spanned by (Ω;X). It is easy to see thatk(Ω; X) is a free nonassociative Ω-algebra on the set X . Let RΩ denote the setconsisting of the following relations in k(Ω; X):

((u)(v)) = −((v)(u)),

(((u)(v))(w)) = (((u)(w))(v)) + ((u)((v)(w))),

where (u), (v), (w) ∈ (Ω; X). Then FLΩ(X) := k(Ω; X)/Id(RΩ) is a free LieΩ-algebra on the set X .

Theorem 2.8. Lie(Ω; X) is a free Lie Ω-algebra on the set X with a linear basisNLSW(Ω; X).

Proof. Let i : X → FLΩ(X), x → x + Id(RΩ) and ϕ : X → k〈Ω; X〉, x →x. Since FLΩ(X) is a free Lie Ω-algebra on X , there is a unique Lie Ω-algebrahomomorphism ϕ : FLΩ(X) → k〈Ω; X〉 such that ϕi = ϕ. It is easy to see that

ϕ(FLΩ(X)) = Lie(Ω; X).

Similar to the proof of Lemma 2.5, we have for any (u) ∈ (Ω; X),

(u) + Id(RΩ) =∑

βj [uj] + Id(RΩ),

where each βj ∈ k and uj ∈ ALSW(Ω; X). If

ϕ

(m∑

i=1

αi[ui] + Id(RΩ)

)=

m∑i=1

αi[ui] = 0

in k〈Ω; X〉, where each αi ∈ k, ui ∈ ALSW(Ω; X), then by Lemma 2.7, we have eachαi = 0. Thus, ϕ is injective. It follows that FLΩ(X) ∼= Lie(Ω; X), i.e. Lie(Ω; X) isa free Lie Ω-algebra on the set X .

3. Grobner–Shirshov Bases for Lie Ω-Algebras

3.1. Composition-diamond lemma for associative Ω-algebras

In this subsection, we review the Grobner–Shirshov bases theory for associativeΩ-algebras, which can be found in [14].

Let k〈Ω; X〉 be the free associative Ω-algebra on X and /∈ X . By a -Ω-wordon X , we mean any expression in 〈Ω; X ∪ 〉 with only one occurrence of . Let〈Ω; X〉 denote the set of all -Ω-words on X . If π is a -Ω-word and s ∈ k〈Ω; X〉,then we call π|s := π|→s an s-Ω-word.

Now, we assume that 〈Ω; X〉 is equipped with a monomial order >. This meansthat > is a well order on 〈Ω; X〉 such that for any v, w ∈ 〈Ω; X〉 and π ∈ 〈Ω; X〉,if w > v, then π|w > π|v.

For every Ω-polynomial f ∈ k〈Ω; X〉, let f be the leading Ω-word of f withrespect to the order >. If the coefficient of f is 1, then we call that f is monic. Wealso call a set S ⊆ k〈Ω; X〉 monic if each s ∈ S is monic.

1750190-9

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

Let f, g ∈ k〈Ω; X〉 be monic. Then we define two kinds of compositions.

(i) If there exists an associative Ω-word w = fa = bg for some a, b ∈ 〈Ω; X〉 suchthat bre(w)<bre(f)+bre(g), then we call (f, g)w := fa − bg the intersectioncomposition of f and g with respect to the ambiguity w.

(ii) If there exists an associative Ω-word w = f = π|g

for some π ∈ 〈Ω; X〉, thenwe call (f, g)w := f − π|g the inclusion composition of f and g with respect tothe ambiguity w.

Let S ⊆ k〈Ω; X〉 be monic. The composition (f, g)w is called trivial modulo(S, w) if

(f, g)w =∑

αiπi|si ,

where each αi ∈ k, πi ∈ 〈Ω; X〉, si ∈ S and πi|si< w. If this is the case, we write

(f, g)w ≡ass 0 mod(S, w).

In general, for any two associative Ω-polynomials p and q, p ≡ass q mod(S, w)means that p − q =

∑αiπi|si , where each αi ∈ k, πi ∈ 〈Ω; X〉, si ∈ S and

πi|si < w.A monic set S is called a Grobner–Shirshov basis in k〈Ω; X〉 if any composition

(f, g)w of f, g ∈ S is trivial modulo (S, w).

Lemma 3.1 ([14], Composition-Diamond lemma for associative Ω-algebras). Let S ⊆ k〈Ω; X〉 be monic, > a monomial order on 〈Ω; X〉 and Idass(S)the ideal of k〈Ω; X〉 generated by S. Then the following statements are equivalent:

(i) S is a Grobner–Shirshov basis in k〈Ω; X〉.(ii) f ∈ Idass(S) ⇒ f = π|s for some π ∈ 〈Ω; X〉 and s ∈ S.(iii) The set Irr(S) = w ∈ 〈Ω; X〉 |w = π|s, π ∈ 〈Ω; X〉, s ∈ S is a linear basis

of the associative Ω-algebra k〈Ω; X |S〉 := k〈Ω; X〉/Idass(S).

3.2. Composition-diamond lemma for Lie Ω-algebras

In this subsection, we establish the Composition-Diamond lemma for Lie Ω-algebras, which is a generalization of Shirshov’s Composition-Diamond lemma forLie algebras.

Let >Dl be the order defined as before. It is easy to see that >Dl is a mono-mial order on 〈Ω; X〉. We always use this order in this subsection, in particular, aGrobner–Shirshov basis in k〈Ω; X〉 is with respect to the order >Dl .

Lemma 3.2. Let π ∈ 〈Ω; X〉 and v, π|v ∈ ALSW(Ω; X). Then there is a π1 ∈〈Ω; X〉 and c ∈ 〈Ω; X〉 such that [π|v] = [π1 |[vc]], where c may be empty. Let

[π|v]v := [π1 |[vc]]|[vc] →[[···[[[v][c1]][c2]]···][cm]],

1750190-10

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

where c = c1c2 · · · cm with each ci ∈ ALSW(Ω; X), c1 lex c2 lex · · · lex cm and[[· · · [[[v][c1]][c2]] · · ·][cm]] is the left normed word on [v], [c1], [c2], . . . , [cm]. Then

[π|v]v = π|[v] +∑

αiπi|[v],

where each αi ∈ k and πi|v <Dl π|v. It follows that [π|v]v = π|v with respect to theorder >Dl .

Proof. The proof is by induction on the depth of π|v. If dep(π|v)=0, then the resultis true by Lemma 2.4. Assume that the result is true for any π|v with dep(π|v)≤n − 1.

Let dep(π|v)=n ≥ 1. There are two cases to consider.

Case 1. If π|v = avb, where a, b ∈ 〈Ω; X〉. Let a = a1a2 · · ·al, l ≥ 0 and b =b1b2 · · · bt, t ≥ 0, where ai and bj are prime. By Lemma 2.4, we can obtain

[π|v] = [[a1][a2] · · · [al][vc][bj+1] · · · [bt]] = [a[vc]d],

where c = b1b2 · · · bj and d = bj+1bj+2 · · · bt. Let c = c1c2 · · · cm, where each ci ∈ALSW(Ω; X) and c1 lex c2 lex · · · lex cm. Then,

[π|v]v = [[a1][a2] · · · [al][· · · [[[v][c1]][c2]] · · · [cm]]][bj+1] · · · [bt]]

and by Lemma 2.4, we have

[π|v]v = [a1][a2] · · · [al][v][b1][b2] · · · [bt] +∑

βi[di1 ] · · · [dip ][v][dip+1 ] · · · [dip+t ].

Therefore, by Lemma 2.6, we have

[π|v]v = a[v]b +∑

αja′j [v]b′j ,

where each a′jvb′j <Dl avb.

Case 2. If π|v = aθ(u1, u2, . . . , π′|v, . . . , uq)b, where a, b ∈ 〈Ω; X〉 and may be

empty, then we have

[π|v] = [aθ(u1, u2, · · · , [π′|v], · · · , uq)b].

By induction,

[π′|v] = [π′′|[vc]], [π′|v]v = π′|[v] +∑

αiπ′i|[v],

where π′i|v <Dl π′|v. Therefore,

[π|v] = [aθ(u1, u2, . . . , [π′′|[vc] ], . . . , uq)b]

1750190-11

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

and

[π|v]v = [aθ(u1, . . . , π′|v, . . . , uq)b]v

= [aθ([u1], . . . , [π′|v]v, . . . , [uq])b]

= aθ([u1], . . . , [π′|v]v, . . . , [uq])b +∑

αiaiθ([u1], . . . , [π′|v]v, . . . , [uq])bi

= aθ(u1, . . . , π′|[v], . . . , uq)b +

∑βja

′jθ(u

′j1 , u

′j2 . . . , π′′

j |[v], . . . , u′jq

)b′j

= π|[v] +∑

βja′jθ(u

′j1 , u

′j2 , . . . , π

′′j |[v], . . . , u

′jq

)b′j ,

where each a′jθ(u

′j1

, . . . , π′′j |v, . . . , u′

jq)b′j <Dl π|v. It follows that [π|v]v = π|v.

Definition 3.3. Let π ∈ 〈Ω; X〉 and f ∈ Lie(Ω; X) ⊆ k〈Ω; X〉 be monic. Ifπ|

f∈ ALSW(Ω; X), then

[π|f]f

:= [π|f]f|[f] →f

, (replace [f ] by f in [π|f]f)

is called a special normal f -word.

Corollary 3.4. Let f ∈ Lie(Ω; X) and π|f∈ ALSW(Ω; X). Then

[π|f]f

= π|f

+∑

αiπi|f ,

where each αi ∈ k and πi|f <Dl π|f.

Let f, g ∈ Lie(Ω; X) be monic. There are two kinds of compositions.

(a) If w = fa = bg, where a, b ∈ 〈Ω; X〉 with bre(w)<bre(f)+bre(g), then〈f, g〉w := [fa]

f− [bg]

gis called the intersection composition of f and g with

respect to the ambiguity w.(b) If w = f = π|

g, then 〈f, g〉w := f − [π|

g]g

is called the inclusion composition off and g with respect to the ambiguity w.

If S is a monic subset of Lie(Ω; X), then the composition 〈f, g〉w is called trivialmodulo (S, w) if

〈f, g〉w =∑

αi[πi|si ]si,

where each αi ∈ k, si ∈ S, [πi|si ]siis a special normal si-word and πi|si

<Dl w. Ifthis is the case, then we write

〈f, g〉w ≡ 0 mod(S, w).

In general, for any two Lie Ω-polynomials p and q, p ≡ q mod(S, w)means thatp− q =

∑αi[πi|si ]si

, where each αi ∈ k, si ∈ S, [πi|si ]siis a special normal si-word

and πi|si<Dl w.

Definition 3.5. A monic set S is called a Grobner–Shirshov basis in Lie(Ω;X) ifany composition 〈f, g〉w of f, g ∈ S is trivial modulo (S, w).

1750190-12

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

Lemma 3.6. Let f, g ∈ Lie(Ω; X) ⊂ k〈Ω; X〉 be monic. Then

〈f, g〉w − (f, g)w ≡ass 0 mod(f, g, w).

Proof. If 〈f, g〉w and (f, g)w are compositions of intersection, where w = fa = bg,then

〈f, g〉w = [fa]f− [bg]

g= fa +

∑αiaifa′

i − bg −∑

βjbjgb′j,

where aifa′i, bj gb′j <Dl w. It follows that

〈f, g〉w − (f, g)w ≡ass 0 mod(f, g, w).

If 〈f, g〉w and (f, g)w are compositions of inclusion, where w = f = π|g, then

〈f, g〉w = f − [π|g ]g = f − π|g −∑

αiπi|g ,

where πi|g <Dl w. It follows that

〈f, g〉w − (f, g)w ≡ass 0 mod(f, g, w).

Lemma 3.7. Let S ⊂ Lie(Ω; X) ⊂ k〈Ω; X〉 be monic. With the order >Dl on〈Ω; X〉, the following two statements are equivalent:

(i) S is a Grobner–Shirshov basis in Lie(Ω; X).(ii) S is a Grobner–Shirshov basis in k〈Ω; X〉.

Proof. (i) ⇒ (ii). Suppose that S is a Grobner–Shirshov basis in Lie(Ω;X). Then,for any composition 〈f, g〉w, we have 〈f, g〉w =

∑αi[πi|si ]si

, where each αi ∈k, πi|si

<Dl w, πi ∈ 〈Ω; X〉, si ∈ S. By Corollary 3.4, we have 〈f, g〉w =∑

βtπt|st ,

where each βt ∈ k, πt|st<Dl w. Therefore, by Lemma 3.6, we can obtain that

(f, g)w ≡ass 0 mod(S, w). Thus, S is a Grobner–Shirshov basis in k〈Ω; X〉.(ii) ⇒ (i). Assume that S is a Grobner–Shirshov basis in k〈Ω; X〉. Then, for

any composition 〈f, g〉w in S, we have 〈f, g〉w ∈ Lie(Ω; X) and 〈f, g〉w ∈ Idass(S).By the Composition-Diamond lemma for associative Ω-algebras, 〈f, g〉w = π1|s1

∈ALSW(Ω; X). Let

h1 = 〈f, g〉w − α1[π1|s1 ]s1,

where α1 is the coefficient of 〈f, g〉w. Then, h1 <Dl 〈f, g〉w, h1 ∈ Idass(S) andh1 ∈ Lie(Ω; X). Since 〈Ω; X〉 is a well-ordered set with the order >Dl , the resultcan be obtained by induction on 〈f, g〉w.

Lemma 3.8. Let S ⊂ Lie(Ω; X) be monic and

Irr(S) := [w]|w ∈ ALSW(Ω; X), w = π|s, s ∈ S, π ∈ 〈Ω; X〉.Then, for any h ∈ Lie(Ω; X), h can be expressed as

h =∑

αi[ui] +∑

βj [πj |sj ]sj,

where each ui ∈ ALSW(Ω; X), ui ≤Dl h and sj ∈ S, πj |sj≤Dl h.

1750190-13

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

Proof. Since f ∈ Lie(Ω; X), h =∑

αi[ui], where each ui ∈ ALSW(Ω; X) andui >Dl ui+1. If [u1] ∈ Irr(S), then let h1 = h − α1[u1]. Otherwise, there existss1 ∈ S such that u1 = π|

s1. Let h1 = h − [π|s1 ]s1

. In both of the above two cases,we have h1 ∈ Lie(Ω; X) and h1 <Dl h. Then, by induction on h, we can obtain theresult.

The following theorem is the Composition-Diamond lemma for Lie Ω-algebras.It is an analogue of Shirshov’s Composition lemma for Lie algebras [59], whichwas specialized to associative algebras by Bokut [9], see also Bergman [7] andBuchberger [19, 21].

Theorem 3.9 (Composition-Diamond lemma for Lie Ω-algebras). Let S ⊆Lie(Ω; X) be a nonempty monic set and IdLie(S) the ideal of Lie(Ω; X) generatedby S. Then the following statements are equivalent:

(i) S is a Grobner–Shirshov basis in Lie(Ω; X).(ii) f ∈ IdLie(S) ⇒ f = π|s ∈ ALSW(Ω; X) for some s ∈ S and π ∈ 〈Ω; X〉.(iii) The set Irr(S) = [w]|w ∈ ALSW(Ω; X), w = π|s, s ∈ S, π ∈ 〈Ω; X〉 is a

linear basis of the Lie Ω-algebra Lie(Ω; X |S) := Lie(Ω; X)/IdLie(S).

Proof. (i) ⇒ (ii) Since f ∈ IdLie(S) ⊆ Idass(S), by Lemmas 3.7 and 3.1, we havef = π|s ∈ ALSW(Ω; X) for some s ∈ S and π ∈ 〈Ω; X〉.

(ii) ⇒ (iii) Suppose that∑

αi[ui] = 0 in Lie(Ω; X |S), where each [ui] ∈ Irr(S)and ui >Dl ui+1. That is,

∑αi[ui] ∈ IdLie(S). Then each αi must be 0. Otherwise,

say α1 = 0, since∑

αi[ui] = u1 and by (ii), we have u1 /∈ Irr(S), a contradiction.Therefore, Irr(S) is linear independent. By Lemma 3.8, Irr(S) is a linear basis ofLie(Ω; X |S).

(iii) ⇒ (i) For any composition 〈f, g〉w of f, g ∈ S, we have 〈f, g〉w ∈ IdLie(S).Then, by (iii) and Lemma 3.8, 〈f, g〉w =

∑βj [πj |sj ]sj

, where each βj ∈ k, πj ∈〈Ω; X〉, sj ∈ S, πj |sj

<Dl w. This proves that S is a Grobner–Shirshov basis inLie(Ω; X).

4. Applications

In this section, we give Grobner–Shirshov bases of a free λ-Rota–Baxter Lie algebra,of a free modified λ-Rota–Baxter Lie algebra, and of a free Nijenhuis Lie algebraand, then linear bases of these three algebras are obtained by Theorem 3.9.

4.1. Free λ-Rota–Baxter Lie algebras

A Lie algebra L equiped with a linear map P : L → L satisfying

[P (x)P (y)] = P ([P (x)y]) + P ([xP (y)]) + λP ([xy]), x, y ∈ L

is called a Rota–Baxter Lie algebra of weight λ or λ-Rota–Baxter Lie algebra. Itis easy to see that any λ-Rota–Baxter Lie algebra (L, P ) is a Lie Ω-algebra, whereΩ = P.

1750190-14

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

Let Lie(P; X) be the free Lie P-algebra on a set X . Let S denote the setconsisting of the following Lie P-polynomials in Lie(P; X):

fu,v = (P ([u])P ([v])) − P ((P ([u])[v])) − P (([u]P ([v]))) − λP (([u][v])),

where u, v ∈ ALSW(P; X) and u >Dl v.It is easy to see

RBL(X) := Lie(P; X |S) = Lie(P; X)/IdLie(S)

is a free λ-Rota–Baxter Lie algebra on the set X .

Theorem 4.1. With the order >Dl on 〈P; X〉, the set S is a Grobner–Shirshovbasis in Lie(P; X). It follows that the set

Irr(S) =

[w] ∈ NLSW(P; X)

∣∣∣∣∣w = π|P (u)P (v), π ∈ 〈P; X〉u, v ∈ ALSW(P; X), u >Dl v

is a linear basis of the free λ-Rota–Baxter Lie algebra RBL(X) = Lie(P; X |S)on X.

Proof. All the possible compositions of Lie P-polynomials in S are listed asbelow:

〈fu,v, fv,w〉w1 , w1 = P (u)P (v)P (w), u >Dl v >Dl w,

〈fπ|P(u)P (v) ,w, fu,v〉w2 , w2 = P (π|

P (u)P (v))P (w), u >Dl v, π|P (u)P (v) >Dl w,

〈fu,π|P(v)P (w)

, fv,w〉w3 , w3 = P (u)P (π|P(v)P (w)), v >Dl w, u >Dl π|

P (v)P (w) .

We check that all the compositions are trivial.

〈fu,v, fv,w〉w1

= [fu,vP (w)]fu,v

− [P (u)fv,w]fv,w

= ((P ([u])P ([w]))P ([v])) − (P ((P ([u])[v]))P ([w])) − (P (([u]P ([v])))P ([w]))

−λ(P (([u][v]))P ([w])) + (P ([u])P ((P ([v])[w]))) + (P ([u])P (([v]P ([w]))))

+ λ(P ([u])P (([v][w]))).

By direct computation, we have mod(S, w1):

((P ([u])P ([w]))P ([v]))

≡ (P ((P ([u])[w])) + P (([u]P ([w]))) + λP (([u][w])))P ([v]))

≡ P ((P ((P ([u])[w]))[v])) + P (((P ([u])[w])P ([v]))) + λP ((P ([u])[w]))[v]))

+ P ((P (([u]P ([w])))[v])) + P ((([u]P ([w]))P ([v]))) + λP ((([u]P ([w]))[v]))

+ λP ((P (([u][w]))[v])) + λP ((([u][w])P ([v]))) + λ2P ((([u][w])[v])),

1750190-15

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

(P (([u]P ([v])))P ([w]))

≡ P ((P (([u]P ([v])))[w])) + P ((([u]P ([v]))P ([w]))) + λP ((([u]P ([v]))[w]))

≡ P ((P (([u]P ([v])))[w])) + P ((([u]P ([w]))P ([v]))) + P (([u](P ([v])P ([w])))

+ λP ((([u]P ([v]))[w]))

≡ P ((P (([u]P ([v])))[w])) + P ((([u]P ([w]))P ([v]))) + P (([u]P ((P ([v])[w]))))

+ P (([u]P (([v]P ([w]))))) + λP (([u]P (([v][w])))) + λP ((([u]P ([v]))[w])),

(P ((P ([u])[v]))P ([w]))

≡ P ((P ((P ([u])[v]))[w])) + P (((P ([u])[v])P ([w]))) + λP (((P ([u])[v])[w]))

≡ P ((P ((P ([u])[v]))[w])) + P (((P ([u])P ([w]))[v])) + P ((P ([u])([v]P ([w]))))

+ λP (((P ([u])[v])[w]))

≡ P ((P ((P ([u])[v]))[w])) + P ((P ((P ([u])[w]))[v])) + P ((P (([u]P ([w])))[v]))

+ λP ((P (([u][w]))[v])) + P ((P ([u])([v]P ([w])))) + λP (((P ([u])[v])[w])),

(P (([u][v]))P ([w])) ≡ P ((P (([u][v]))[w])) + P ((([u][v])P ([w]))) + λP ((([u][v])[w])),

(P ([u])P ((P ([v])[w])))

≡ P ((P ([u])(P ([v])[w]))) + P (([u]P ((P ([v])[w]))) + λP (([u](P ([v])[w])))

≡ P ((P ([u])P ([v]))[w])) + P ((P ([v])(P ([u])[w]))) + P (([u](P ((P ([v])[w])))

+ λP (([u](P ([v])[w])))

≡ P ((P ((P ([u])[v]))[w])) + P ((P (([u]P ([v])))[w])) + λP ((P (([u][v]))[w]))

+ P ((P ([v])(P ([u])[w]))) + P (([u](P ((P ([v])[w]))))) + λP (([u](P ([v])[w]))),

(P ([u])P (([v]P ([w])))

≡ P ((P ([u])([v]P ([w])))) + P (([u]P (([v]P ([w]))))) + λP (([u]([v]P ([w]))),

(P ([u])P (([v][w]))) ≡ P ((P ([u])([v][w]))) + P (([u]P (([v][w])))) + λP (([u]([v][w]))).

Therefore, we have

〈fu,v, fv,w〉w1 = [fu,vP (w)]fu,v

− [P (u)fv,w]fv,w

≡ 0 mod(S, w1).

Let

r(π|fu,v

) = [π|fu,v

]fu,v

− [π|fu,v

].

1750190-16

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

Then,

〈fπ|P (u)P (v) ,w, fu,v〉w2

= fπ|P(u)P (v) ,w − [P (π|fu,v )P (w)]

P (u)P (v)

= fπ|P(u)P (v),w − (P ([π|fu,v

]P (u)P (v))P (w))

= P ((P ([π|P (u)P (v) ])[w])) + P (([π|

P (u)P (v) ]P ([w]))) + λP (([π|P (u)P (v) ][w]))

− (P (r(π|fu,v

))P ([w]))

≡ P ((P (r(π|fu,v

))[w])) + P ((r(π|fu,v

)P ([w]))) + λP ((r(π|fu,v

)[w]))

−P ((P (r(π|fu,v

))[w])) − P ((r(π|fu,v

)P ([w]))) − λP ((r(π|fu,v

)[w]))

≡ 0 mod(S, w2),

Similarly, we have

〈fu,π|P(v)P (w)

, fv,w〉w3 ≡ 0 mod(S, w3).

Therefore, S is a Grobner–Shirshov basis in Lie(P; X). By Theorem 3.9, theset Irr(S) is a linear basis of the free λ-Rota–Baxter Lie algebra RBL(X).

4.2. Free modified λ-Rota–Baxter Lie algebras

A Lie algebra L equipped with a linear map P : L → L satisfying

[P (x)P (y)] = P ([P (x)y]) + P ([xP (y)]) + λ[xy], x, y ∈ L

is called a modified λ-Rota–Baxter Lie algebra. If λ = −1, the modified λ-Rota–Baxter Lie algebra is also called a Baxter Lie algebra in [20]. It is easy to see thatany modified λ-Rota–Baxter Lie algebra (L, P ) is a Lie Ω-algebra, where Ω = P.

Let Lie(P; X) be the free Lie P-algebra on a set X . Let SB denote the setconsisting of the following Lie P-polynomials in Lie(P; X):

fBu,v = (P ([u])P ([v])) − P ((P ([u])[v])) − P (([u]P ([v]))) − λ([u][v]),

where u, v ∈ ALSW(P; X) and u >Dl v.It is easy to see

MRBL(X) := Lie(P; X |SB) = Lie(P; X)/IdLie(SB)

is a free modified λ-Rota–Baxter Lie algebra on X .Similar to the proof of Theorem 4.1, we have the following theorem.

Theorem 4.2. With the order >Dl on 〈P; X〉, the set SB is a Grobner–Shirshovbasis in Lie(P; X). It follows that the set

Irr(SB) =

[w] ∈ NLSW(P; X)

∣∣∣∣∣w = π|P (u)P (v) , π ∈ 〈P; X〉

u, v ∈ ALSW(P; X), u >Dl v

is a linear basis of the free modified λ-Rota–Baxter Lie algebra MRBL(X) =Lie(P; X |SB) on X.

1750190-17

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

4.3. Free Nijenhuis Lie algebras

A Lie algebra L equiped with a linear map P : L → L satisfying

[P (x)P (y)] = P ([P (x)y]) + P ([xP (y)]) − P 2([xy]), x, y ∈ L

is called a Nijenhuis Lie algebra.Let Lie(P; X) be the free Lie P-algebra on a set X . Let SN denote the set

consisting of the following Lie P-polynomials in Lie(P; X):

fNu,v = (P ([u])P ([v])) − P ((P ([u])[v])) − P (([u]P ([v]))) + P 2(([u][v])),

where u, v ∈ ALSW(P; X) and u >Dl v.It is easy to see that

NL(X) := Lie(P; X |SN) = Lie(P; X)/IdLie(SN )

is a free Nijenhuis Lie algebra on X .Similar to the proof of Theorem 4.1, we have the following theorem.

Theorem 4.3. With the order >Dl on 〈P; X〉, the set SN is a Grobner–Shirshovbasis in Lie(P; X). It follows that the set

Irr(SN ) =

[w] ∈ NLSW(P; X)

∣∣∣∣∣w = π|P (u)P (v) , π ∈ 〈P; X〉

u, v ∈ ALSW(P; X), u >Dl v

is a linear basis of the free Nijenhuis Lie algebra NL(X) = Lie(P; X |SN) on X.

Acknowledgment

This paper was supported by the NNSF of China (11171118, 11571121).

References

[1] K. Abdaoui, S. Mabrouk and A. Makhlouf, Rota–Baxter operators on pre-lie super-algebras and beyond, preprint (2015), arXiv:1512.08043.

[2] W. W. Adams and P. Loustaunau, An introduction to Grobner Bases, GraduateStudies in Mathematics, Vol. 3 (American Mathematical Society, 1994).

[3] M. Aguiar and W. Moreira, Combinatorics of free Baxter algebra, Electron. J. Com-bin. 13 (2006) R17.

[4] H. An and C. Bai, From Rota–Baxter algebras to pre-Lie algebras, J. Phys. A: Math.Theory 41 (2008) 015201.

[5] C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang–Baxter equation and PostLie algebras, Commun. Math. Phys. 297 (2010) 553–596.

[6] G. Baxter, An analytic problem whose solution follows from a simple algebraic iden-tity, Pacific J. Math. 10 (1960) 731–742.

[7] G. M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978) 178–218.[8] L. A. Bokut, Unsolvability of the word problem, and subalgebras of finitely presented

Lie algebras, Izv. Akad. Nauk. SSSR Ser. Mat. 36 (1972) 1173–1219.[9] L. A. Bokut, Imbeddings into simple associative algebras, Algebra i Logika 15 (1976)

117–142.

1750190-18

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

[10] L. A. Bokut and Y. Q. Chen, Grobner–Shirshov bases: After A.I. Shirshov, SoutheastAsian Bull. Math. 31 (2007) 1057–1076.

[11] L. A. Bokut and Y. Q. Chen, Grobner–Shirshov bases and their calculation, Bull.Math. Sci. 4 (2014) 325–395.

[12] L. A. Bokut, Y. Q. Chen and J. P. Huang, Grobner–Shirshov bases for L-algebras,Int. J. Algebra Comput. 23(3) (2013) 547–571.

[13] L. A. Bokut, Y. Q. Chen and Q. H. Mo, Grobner–Shirshov bases and embeddings ofalgebras, Int. J. Algebra Comput. 20 (2010) 875–900.

[14] L. A. Bokut, Y. Q. Chen and J. J. Qiu, Grobner–Shirshov bases for associative alge-bras with multiple operations and free Rota–Baxter algebras, J. Pure Appl. Algebra214 (2010) 89–100.

[15] L. A. Bokut, Y. Q. Chen and K. P. Shum, Some new results on Grobner–Shirshovbases, in Proc. Int. Conf. Algebra 2010, Advances in algebraic structures, World Sci.Publ., Hackensack, NJ (2012), pp. 53–102.

[16] L. A. Bokut, Y. Fong, W.-F. Ke and P. S. Kolesnikov, Grobner and Grobner–Shirshovbases in algebra and conformal algebras, Fund. Appl. Math. 6(3) (2000) 669–706.

[17] L. A. Bokut and P. S. Kolesnikov, Grobner–Shirshov bases: From their incipiency tothe present, J. Math. Sci. 116(1) (2003) 2894–2916.

[18] L. A. Bokut and G. Kukin, Algorithmic and Combinatorial Algebra (Kluwer AcademicPublishers, Dordrecht, 1994).

[19] B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, Ph.D. thesis, University of Innsbruck, Austria (1965)(in German).

[20] M. Bordemann, Generalized Lax pairs, themodified classical Yang–Baxter equation,and affine geometry of Lie groups, Commun. Math. Phys. 135 (1990) 201–216.

[21] B. Buchberger, An algorithmical criteria for the solvability of algebraic systems ofequations, Aequationes Math. 4 (1970) 374–383.

[22] B. Buchberger, G. E. Collins, R. Loos and R. Albrecht, Computer Algebra, Sym-bolic and Algebraic Computation, Computing supplementum, Vol. 4 (Springer-Verlag,New York, 1982).

[23] B. Buchberger and F. Winkler, Grobner Bases and Applications, London mathemat-ical society lecture note series, Vol. 251 (Cambridge University Press, Cambridge,1998).

[24] J. Carinena, J. Grabowski and G. Marmo, Quantum bi-Hamiltonian systems, Int. J.Mod. Phys. A 15 (2000) 4797–4810.

[25] P. Cartier, On the structure of free Baxter algebras, Adv. in Math. 9 (1972) 253–265.[26] K.-T. Chen, R. H. Fox and R. C. Lyndon, Free differential calculus. IV. The quotient

groups of the lower central series, Ann. Math. 68(2) (1958) 81–95.[27] D. A. Cox, J. Little and D. O’Shea, Ideals, Varieties and Algorithms: An Introduc-

tion to Computational Algebraic Geometry and Commutative Algebra, UndergraduateTexts in Mathematics (Springer-Verlag, New York, 1992).

[28] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations(Wiley, Chichester, 1993).

[29] V. Dotsenko and A. Khoroshkin, Grobner bases for operads, Duke Math. J. 153(2010) 363–396.

[30] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Grad-uate Texts in Mathematics, Vol. 150 (Springer-Verlag, Berlin and New York, 1995).

[31] V. Drensky and R. Holtkamp, Planar trees, free nonassociative algebras, invariants,and elliptic integrals, Algebra Discrete Math. 2 (2008) 1–41.

1750190-19

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

J. Qiu & Y. Chen

[32] K. Ebrahimi-Fard, Loday-type algebras and the Rota–Baxter relation, Lett. Math.Phys. 61 (2002) 139–147.

[33] K. Ebrahimi-Fard, On the associative nijenhuis relation, Electr. J. Comb. 11 (2004)R38.

[34] K. Ebrahimi-Fard and L. Guo, Rota–Baxter algebras and dendriform dialgebras, J.Pure Appl. Alg. 212 (2008) 320–339.

[35] K. Ebrahimi-Fard and L. Guo, Free Rota–Baxter algebras and rooted trees, J. Alg.Appl. 7 (2008) 167–194.

[36] X. Gao, L. Guo and M. Rosenkranz, Free integro-differential algebras and Grobner–Shirshov bases, J. Algebra 442 (2015) 354–396.

[37] V. Gubarev and P. Kolesnikov, Grobner–Shirshov basis of the universal envelopingRota–Baxter algebra of a Lie algebra, preprint (2016), arXiv:1602.07409 [math.RA].

[38] L. Guo, An Introduction to Rota–Baxter Algebra (Higher education press, Beijing,2012).

[39] L. Guo, Operated semigroups, Motzkin paths and rooted trees, J. Algebra Comb. 29(2009) 35–62.

[40] L. Guo and W. Keigher, Baxter algebras and Shuffle products, Adv. Math. 150 (2000)117–149.

[41] L. Guo and W. Keigher, On free Baxter algebras: Completions and the internalconstruction, Adv. Math. 151 (2000) 101–127.

[42] L. Guo and W. Keigher, On differential Rota–Baxter algebras, J. Pure Appl. Algebra212 (2008) 522–540.

[43] M. Hall, A basis of free Lie rings and higher commutators in free groups, Proc. Amer.Math. Soc. 1 (1950) 575–581.

[44] P. Hall, A contribution to the theory of groups of prime power order, Proc. LondonMath. Soc. 36(2) (1934) 29–95.

[45] P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. 6 (1956)366–416.

[46] H. Hironaka, Resolution of singulatities of an algebtaic variety over a field if charac-teristic zero, I, II, Ann. Math. 79 (1964) 109–203, 205–326.

[47] A. G. Kurosh, Free sums of multiple operator algebras, Siberian Math. J. 1 (1960)62–70.

[48] M. Lazard, Groupes, anneaux de Lie et probleme de Burnside, Istituto Matematicodell ’ Universita di Roma (1960).

[49] P. Lei and L. Guo, Nijenhuis algebras, NS algebras, and N-dendriform algebras,Front. Math. China 7(5) (2012) 827–846.

[50] X. Li, D. Hou and C. Bai, Rota–Baxter operators on pre-lie algebras, J. NonlinearMath. Phys. 14(2) (2007) 269–289.

[51] W. Magnus, Uber Beziehungen zwischen hoheren Kommutatoren, J. Reine Angew.Math. 177 (1937) 105–115.

[52] A. A. Mikhalev, The composition lemma for color Lie superalgebras and for Liep-superalgebras, Contemp. Math. 131(2) (1992) 91–104.

[53] J. Pei, C. Bai and L. Guo, Rota–Baxter operators on sl(2, C) and solutions of theclassical Yang–Baxter equation, J. Math. Phys. 55(2) (2014) 021701.

[54] J. J. Qiu and Y. Q. Chen, Composition-Diamond lemma for λ-differential associativealgebras with multiple operators, J. Algebra Appl. 9 (2010) 223–239.

[55] G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc.5 (1969) 325–329.

[56] C. Reutenauer, Free Lie Algebras (Oxford Science Publications, 1993).

1750190-20

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.

2nd Reading

November 29, 2016 16:17 WSPC/S0219-4988 171-JAA 1750190

Grobner–Shirshov bases for Lie Ω-algebras

[57] M. A. Semenov-Tian-Shansky, What is a classical R-matrix? Funct. Anal. Appl. 17(4)(1983) 259–272.

[58] A. I. Shirshov, Subalgebras of free Lie algebras, Uspekhi Mat. Nauk 8(3) (1953) 173.[59] A. I. Shirshov, Some algorithmic problem for Lie algebras, Sibirsk. Mat. Z. 3 (1962)

292–296 (in Russian); SIGSAM Bull. 33(2) (1999) 3–6.[60] A. I. Shirshov, Some algorithmic problem for ε-algebras, Sibirsk. Mat. Z. 3 (1962)

132–137.[61] Selected works of A.I. Shirshov, eds. L. A. Bokut, V. Latyshev, I. Shestakov and E.

Zelmanov (Birkhauser, Basel, Boston, Berlin, 2009), translated by M. Bremmer andM. Kochetov.

[62] K. Uchino, Twisting on associative algebras and Rota–Baxter type operators, J.Noncommut. Geom. 4 (2010) 349–379.

[63] E. Witt, Treue darstellung Liescher ringe, J. Reine Angew. Math. 177 (1937) 152–160.

[64] E. Witt, Die Unterringe der freien Lienge Ringe, Math. Z. 64 (1956) 195–216.

1750190-21

J. A

lgeb

ra A

ppl.

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

FL

IND

ER

S U

NIV

ER

SIT

Y L

IBR

AR

Y o

n 12

/02/

16. F

or p

erso

nal u

se o

nly.