Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for - - PowerPoint PPT Presentation

Groebner-Shirshov bases for Rota-Baxter algebras and PBW Theorem for dendriform algebras

Yuqun Chen (Joint work with L.A. Bokut) South China Normal University, China The 4th International Workshop on Differential Algebra and Related Topics, Beijing, October 27-30, 2010

Introduction

What is a Gr¨

• bner-Shirshov basis?

Introduction

What is a Gr¨

• bner-Shirshov basis?

Let k(X) be a free algebra (for example, Lie algebra, non-associative algebra, anti-commutative non-associative algebra, associative algebra, polynomial algebra, Ω-algebra, Rota-Baxter algebra, differential algebra, etc) generated by a set X over a field k (or a commutative algebra K), I an ideal of k(X) and S ⊂ k(X).

Introduction

What is a Gr¨

• bner-Shirshov basis?

Let k(X) be a free algebra (for example, Lie algebra, non-associative algebra, anti-commutative non-associative algebra, associative algebra, polynomial algebra, Ω-algebra, Rota-Baxter algebra, differential algebra, etc) generated by a set X over a field k (or a commutative algebra K), I an ideal of k(X) and S ⊂ k(X). Definition The set S is a Gr¨

• bner-Shirshov basis of the ideal I if

(i) S is an ideal generator: Id(S) = I; (ii) f ∈ I ⇒ ¯ f = a¯ sb for some s ∈ S and X-words a, b, where ¯ f is the leading term of the polynomial f .

Introduction

What is a Gr¨

• bner-Shirshov basis?

Let k(X) be a free algebra (for example, Lie algebra, non-associative algebra, anti-commutative non-associative algebra, associative algebra, polynomial algebra, Ω-algebra, Rota-Baxter algebra, differential algebra, etc) generated by a set X over a field k (or a commutative algebra K), I an ideal of k(X) and S ⊂ k(X). Definition The set S is a Gr¨

• bner-Shirshov basis of the ideal I if

(i) S is an ideal generator: Id(S) = I; (ii) f ∈ I ⇒ ¯ f = a¯ sb for some s ∈ S and X-words a, b, where ¯ f is the leading term of the polynomial f . Remark 1: Gr¨

• bner-Shirshov basis is NOT a linear basis but a GOOD

set S ⊂ k(X) of defining relations of the ideal I. k(X|S) := k(X)/Id(S) = k(X)/I.

Introduction

Remark 2: The above definition is not valid for dialgebras and conformal algebras.

Introduction

Remark 2: The above definition is not valid for dialgebras and conformal algebras. Gr¨

• bner bases and Gr¨
• bner-Shirshov bases were invented

independently by

Introduction

Remark 2: The above definition is not valid for dialgebras and conformal algebras. Gr¨

• bner bases and Gr¨
• bner-Shirshov bases were invented

independently by A.I. Shirshov 1962 for non-associative algebras, (commutative, anti-commutative) non-associative algebras, Lie algebras and implicitly associative algebras (L.A. Bokut gave an explicitly approach in 1976);

Introduction

Remark 2: The above definition is not valid for dialgebras and conformal algebras. Gr¨

• bner bases and Gr¨
• bner-Shirshov bases were invented

independently by A.I. Shirshov 1962 for non-associative algebras, (commutative, anti-commutative) non-associative algebras, Lie algebras and implicitly associative algebras (L.A. Bokut gave an explicitly approach in 1976); G.M. Bergman 1978 (A.I. Shirshov 1962, L.A. Bokut 1976) for free associative algebras;

Introduction

Remark 2: The above definition is not valid for dialgebras and conformal algebras. Gr¨

• bner bases and Gr¨
• bner-Shirshov bases were invented

independently by A.I. Shirshov 1962 for non-associative algebras, (commutative, anti-commutative) non-associative algebras, Lie algebras and implicitly associative algebras (L.A. Bokut gave an explicitly approach in 1976); G.M. Bergman 1978 (A.I. Shirshov 1962, L.A. Bokut 1976) for free associative algebras;

• H. Hironaka 1964 for power series algebras (both formal and

convergent);

• B. Buchberger 1970 (1965, Ph.D thesis) for polynomial algebras

–Gr¨

• bner bases.

Introduction

By using Gr¨

• bner-Shirshov bases, Shirshov 1962 proved that the word

problem is solvable for one relator Lie algebra: L = Lie(X|f = 0).

Introduction

By using Gr¨

• bner-Shirshov bases, Shirshov 1962 proved that the word

problem is solvable for one relator Lie algebra: L = Lie(X|f = 0). ∀w ∈ L, ? ⇒ w = 0

Introduction

By using Gr¨

• bner-Shirshov bases, Shirshov 1962 proved that the word

problem is solvable for one relator Lie algebra: L = Lie(X|f = 0). ∀w ∈ L, ? ⇒ w = 0 In general, Gr¨

• bner-Shirshov bases theory is a powerful tool to solve

the following classical problems.

Introduction

By using Gr¨

• bner-Shirshov bases, Shirshov 1962 proved that the word

problem is solvable for one relator Lie algebra: L = Lie(X|f = 0). ∀w ∈ L, ? ⇒ w = 0 In general, Gr¨

• bner-Shirshov bases theory is a powerful tool to solve

the following classical problems. (i) normal form; (ii) word problem; (iii) rewritting system; (iv) embedding theorems; (v) extentions; (vi) growth function; Hilbert series; etc.

Free Rota-Baxter algebras

Definition Let A be an associative algebra over k and λ ∈ k. Let a k-linear

• perator P : A → A satisfy

P(x)P(y) = P(P(x)y) + P(xP(y)) + λP(xy), ∀x, y ∈ A. Then A is called a Rota-Baxter algebra with weight λ.

Free Rota-Baxter algebras

Definition Let A be an associative algebra over k and λ ∈ k. Let a k-linear

• perator P : A → A satisfy

P(x)P(y) = P(P(x)y) + P(xP(y)) + λP(xy), ∀x, y ∈ A. Then A is called a Rota-Baxter algebra with weight λ. Definition A free Rota-Baxter algebra with weight λ on a set X is a Rota-Baxter algebra RB(X) generated by X with a natural mapping i : X → A such that, for any Rota-Baxter algebra A with weight λ and any map f : X → A, there exists a unique homomorphism ˜ f : RB(X) → A such that ˜ f · i = f .

Free Rota-Baxter algebras

Free Rota-Baxter algebra RB(X) is constructed by L. Guo.

Free Rota-Baxter algebras

Free Rota-Baxter algebra RB(X) is constructed by L. Guo. Notation Let X be a nonempty set, S(X) the free semigroup generated by X without identity and P a symbol of a unary operation. For any two nonempty sets Y and Z, denote by ΛP(Y , Z) = (∪r≥0(YP(Z))rY ) ∪ (∪r≥1(YP(Z))r) ∪(∪r≥0(P(Z)Y )rP(Z)) ∪ (∪r≥1(P(Z)Y )r), where for a set T, T 0 means the empty set. Remark In ΛP(Y , Z), there are no words with a subword P(z1)P(z2) where z1, z2 ∈ Z.

Free Rota-Baxter algebras

Define Φ0 = S(X) . . . . . . Φn = ΛP(Φ0, Φn−1) . . . . . . Let Φ(X) = ∪n≥0Φn

Free Rota-Baxter algebras

Define Φ0 = S(X) . . . . . . Φn = ΛP(Φ0, Φn−1) . . . . . . Let Φ(X) = ∪n≥0Φn Remark For any u ∈ Φ(X), u has a unique form u = u1u2 · · · un where ui ∈ X ∪ P(Φ(X)) , i = 1, 2, . . . , n, and ui, ui+1 can not both have forms as p(u′

i) and p(u′ i+1). Such ui is called prime.

Free Rota-Baxter algebras

Let kΦ(X) be a free k-module with k-basis Φ(X) and λ ∈ k . Extend linearly P : kΦ(X) → kΦ(X), u → P(u) where u ∈ Φ(X).

Free Rota-Baxter algebras

Let kΦ(X) be a free k-module with k-basis Φ(X) and λ ∈ k . Extend linearly P : kΦ(X) → kΦ(X), u → P(u) where u ∈ Φ(X). Multiplication Firstly, for u, v ∈ X ∪ P(Φ(X)), define u · v = P(P(u′) · v′) + P(u′ · P(v′)) + λP(u′ · v′), if u = P(u′), v = P(v′); uv,

• therwise.

Secondly, for any u = u1u2 · · · us, v = v1v2 · · · vl ∈ Φ(X) where ui, vj are prime, i = 1, 2, . . . , s, j = 1, 2, . . . , l, define u · v = u1u2 · · · us−1(us · v1)v2 · · · vl.

Free Rota-Baxter algebras

Let kΦ(X) be a free k-module with k-basis Φ(X) and λ ∈ k . Extend linearly P : kΦ(X) → kΦ(X), u → P(u) where u ∈ Φ(X). Multiplication Firstly, for u, v ∈ X ∪ P(Φ(X)), define u · v = P(P(u′) · v′) + P(u′ · P(v′)) + λP(u′ · v′), if u = P(u′), v = P(v′); uv,

• therwise.

Secondly, for any u = u1u2 · · · us, v = v1v2 · · · vl ∈ Φ(X) where ui, vj are prime, i = 1, 2, . . . , s, j = 1, 2, . . . , l, define u · v = u1u2 · · · us−1(us · v1)v2 · · · vl. Then RB(X) := kΦ(X) is the free Rota-Baxter algebra with weight λ generated by X.

Composition-Diamond lemma

From now on, k is always a field of characteristic 0.

Composition-Diamond lemma

From now on, k is always a field of characteristic 0. Order Φ(X)

Composition-Diamond lemma

From now on, k is always a field of characteristic 0. Order Φ(X) For any u = u1u2 · · · un ∈ Φ(X) and for a set T ⊆ X ∪ {P}, denote by degT (u) the number of occurrences of t ∈ T in u. Let wt(u) = (deg{P}∪X (u), deg{P}(u), u1, · · · , un).

Composition-Diamond lemma

From now on, k is always a field of characteristic 0. Order Φ(X) For any u = u1u2 · · · un ∈ Φ(X) and for a set T ⊆ X ∪ {P}, denote by degT (u) the number of occurrences of t ∈ T in u. Let wt(u) = (deg{P}∪X (u), deg{P}(u), u1, · · · , un). For any u, v ∈ Φ(X), define by induction u > v ⇔ wt(u) > wt(v) lexicographically, where P(u) > P(v) ⇔ u > v.

Composition-Diamond lemma

From now on, k is always a field of characteristic 0. Order Φ(X) For any u = u1u2 · · · un ∈ Φ(X) and for a set T ⊆ X ∪ {P}, denote by degT (u) the number of occurrences of t ∈ T in u. Let wt(u) = (deg{P}∪X (u), deg{P}(u), u1, · · · , un). For any u, v ∈ Φ(X), define by induction u > v ⇔ wt(u) > wt(v) lexicographically, where P(u) > P(v) ⇔ u > v. It is clear that > is a well ordering on Φ(X). Throughout this talk, we will use this ordering.

Composition-Diamond lemma

From now on, k is always a field of characteristic 0. Order Φ(X) For any u = u1u2 · · · un ∈ Φ(X) and for a set T ⊆ X ∪ {P}, denote by degT (u) the number of occurrences of t ∈ T in u. Let wt(u) = (deg{P}∪X (u), deg{P}(u), u1, · · · , un). For any u, v ∈ Φ(X), define by induction u > v ⇔ wt(u) > wt(v) lexicographically, where P(u) > P(v) ⇔ u > v. It is clear that > is a well ordering on Φ(X). Throughout this talk, we will use this ordering. For any 0 = f ∈ RB(X), f has the leading term ¯ f . Denote by lc(f ) the coefficient of the leading term ¯ f . If lc(f ) = 1, we call f monic.

Composition-Diamond Lemma

Normal S-words

Composition-Diamond Lemma

Normal S-words Let u ∈ Φ(X), S ⊂ RB(X) and s ∈ S. Then u|s := uxi→s is called an s-word or S-word.

Composition-Diamond Lemma

Normal S-words Let u ∈ Φ(X), S ⊂ RB(X) and s ∈ S. Then u|s := uxi→s is called an s-word or S-word. An S-word u|s is normal S-word if u|s = u|s.

Composition-Diamond Lemma

Normal S-words Let u ∈ Φ(X), S ⊂ RB(X) and s ∈ S. Then u|s := uxi→s is called an s-word or S-word. An S-word u|s is normal S-word if u|s = u|s.

• Example. If u = P(x1)x2P2(x1)x4P(x5), then

u|s = P(x1)x2P2(s)x4P(x5) is an s-word, also a normal S-word.

Composition-Diamond Lemma

Normal S-words Let u ∈ Φ(X), S ⊂ RB(X) and s ∈ S. Then u|s := uxi→s is called an s-word or S-word. An S-word u|s is normal S-word if u|s = u|s.

• Example. If u = P(x1)x2P2(x1)x4P(x5), then

u|s = P(x1)x2P2(s)x4P(x5) is an s-word, also a normal S-word. Lemma Let the ordering “>” be defined as before. Then “>” is monomial in the sense that for any u, v, w ∈ Φ(X), u > v = ⇒ w|u > w|v where w|u = w|x→u and w|v = w|x→v.

Composition-Diamond Lemma

Compositions There are four kinds of compositions.

Composition-Diamond Lemma

Compositions There are four kinds of compositions. Let f , g ∈ RB(X) be monic with f = u1u2 · · · un where each ui is prime.

Composition-Diamond Lemma

Compositions There are four kinds of compositions. Let f , g ∈ RB(X) be monic with f = u1u2 · · · un where each ui is prime. (i) If un ∈ P(Φ(X)), then we define composition of right multiplication as f · u where u ∈ P(Φ(X)).

Composition-Diamond Lemma

Compositions There are four kinds of compositions. Let f , g ∈ RB(X) be monic with f = u1u2 · · · un where each ui is prime. (i) If un ∈ P(Φ(X)), then we define composition of right multiplication as f · u where u ∈ P(Φ(X)). (ii) If u1 ∈ P(Φ(X)), then we define composition of left multiplication as u · f where u ∈ P(Φ(X)).

Composition-Diamond Lemma

(iii) If there exits a w = f a = bg where fa is normal f -word and bg is normal g-word, a, b ∈ Φ(X) and deg{P}∪X (w) < deg{P}∪X (f ) + deg{P}∪X (g), then we define the intersection composition of f and g with respect to w as (f , g)w = f · a − b · g.

Composition-Diamond Lemma

(iii) If there exits a w = f a = bg where fa is normal f -word and bg is normal g-word, a, b ∈ Φ(X) and deg{P}∪X (w) < deg{P}∪X (f ) + deg{P}∪X (g), then we define the intersection composition of f and g with respect to w as (f , g)w = f · a − b · g. (iv) If there exists a w = f = u|g where u ∈ Φ(X), then we define the inclusion composition of f and g with respect to w as (f , g)w = f − u|g.

Composition-Diamond Lemma

(iii) If there exits a w = f a = bg where fa is normal f -word and bg is normal g-word, a, b ∈ Φ(X) and deg{P}∪X (w) < deg{P}∪X (f ) + deg{P}∪X (g), then we define the intersection composition of f and g with respect to w as (f , g)w = f · a − b · g. (iv) If there exists a w = f = u|g where u ∈ Φ(X), then we define the inclusion composition of f and g with respect to w as (f , g)w = f − u|g. Remark. (f , g)w < w.

Composition-Diamond Lemma

Let S ⊂ RB(X) be a set of monic polynomials.

Composition-Diamond Lemma

Let S ⊂ RB(X) be a set of monic polynomials. The composition of left (right) multiplication is called trivial mod(S) if u · f =

• i

αiui|si (f · u =

• i

αiui|si) where each αi ∈ k, si ∈ S, ui|si is normal si-word and ui|si ≤ u · f (ui|si ≤ f · u). If this is the case, then we write u · f ≡ 0 mod(S) (f · u ≡ 0 mod(S)).

Composition-Diamond Lemma

Let S ⊂ RB(X) be a set of monic polynomials. The composition of left (right) multiplication is called trivial mod(S) if u · f =

• i

αiui|si (f · u =

• i

αiui|si) where each αi ∈ k, si ∈ S, ui|si is normal si-word and ui|si ≤ u · f (ui|si ≤ f · u). If this is the case, then we write u · f ≡ 0 mod(S) (f · u ≡ 0 mod(S)). Then the composition (f , g)w is called trivial modulo (S, w) if (f , g)w =

• i

αiui|si where each αi ∈ k, si ∈ S, ui|si is normal si-word and ui|si < w. If this is the case, then we write (f , g)w ≡ 0 mod(S, w).

Composition-Diamond Lemma

Definition The set S is called a Gr¨

• bner-Shirshov basis in RB(X) if each

composition in S is trivial.

Composition-Diamond Lemma

Definition The set S is called a Gr¨

• bner-Shirshov basis in RB(X) if each

composition in S is trivial. Main Theorem

Composition-Diamond Lemma

Definition The set S is called a Gr¨

• bner-Shirshov basis in RB(X) if each

composition in S is trivial. Main Theorem (Composition-Diamond lemma for Rota-Baxter algebras) Let RB(X) be a free Rota-Baxter algebra over a field of characteristic 0 and S a set of monic polynomials in RB(X), > the monomial

• rdering on Φ(X) defined as before and Id(S) the Rota-Baxter ideal
• f RB(X) generated by S. Then the following statements are

equivalent. (I) S is a Gr¨

• bner-Shirshov basis in RB(X).

(II) f ∈ Id(S) ⇒ ¯ f = u|s for some u ∈ Φ(X), s ∈ S. (III) Irr(S) = {u ∈ Φ(X)|u = v|¯

s, s ∈ S, v|s is normal s-word} is a

k-basis of RB(X|S) = RB(X)/Id(S).

Applications

• 1. Normal form theorem

Let I be a well ordered set, X = {xi|i ∈ I} and the ordering on Φ(X) defined as before. Let S = {xixj − xjxi|i > j, i, j ∈ I} ∪ {P(u)xi − xiP(u)|u ∈ Φ(X), i ∈ I}. Then S is a Gr¨

• bner-Shirshov basis in RB(X). By our CD-lemma,

Irr(S) = Y1 ∪ Y2 ∪ Y3 is a normal form of the free commutative Rota-Baxter algebra RB(X|S) generated by X with weight λ, where Y1 = {x1x2 · · · xn ∈ S(X)|n ∈ N +, xi ∈ X, x1 ≤ x2 ≤ · · · ≤ xn}, Y2 = {Pl1(u1Pl2(· · · (ut−1Plt(ut)) · · · )) ∈ Φ(X)| uj ∈ Y1, t ≥ 1, lj ≥ 1, j = 1, 2, . . . , t}, Y3 = {uv ∈ Φ(X)|u ∈ Y1, v ∈ Y2}.

Applications

• 2. Embedding theorem

Every countably generated Rota-Baxter algebra with weight 0 can be embedded into a two-generated Rota-Baxter algebra. Proof: We may assume that A has a k-basis X = {xi|i = 1, 2, . . .}. Then A can be expressed as A = RB(X|S), where S = {xixj = {xi, xj}, P(xi) = {P(xi)} | i, j ∈ N +}. For u ∈ Φ(X), we denote supp(u) the set of xi ∈ X appearing in the word u. Let H = RB(X, a, b|S1), where S1 consists of the following relations: I. xixj = {xixj}, i, j ∈ N +, II. P(xi) = {P(xi)}, i ∈ N +, III. aabiab = xi, i ∈ N +, IV. P(t) = 0, t ∈ Φ′(X, a, b), where Φ′(X, a, b) = {u ∈ Φ(X, a, b)|∃u′ ∈ Φ(X, a, b), u′ ∈ Irr({III}), supp(u′) ∩ {a, b} = ∅, s.t., u′ ≡ u mod({III}, u)}.

Applications

Then it is easy to check that S1 is a Gr¨

• bner-Shirshov basis in

RB(X, a, b). By our CD-lemma, A can be embedded into the Rota-Baxter algebra H which is generated by {a, b}.

Applications

Then it is easy to check that S1 is a Gr¨

• bner-Shirshov basis in

RB(X, a, b). By our CD-lemma, A can be embedded into the Rota-Baxter algebra H which is generated by {a, b}. The above results come from the paper

• 9. L.A. Bokut, Yuqun Chen and Xueming Deng, Gr¨
• bner-Shirshov

bases for Rota-Baxter algebras, Siberian Math J, to appear. arxiv:0908.2281

• 3. PBW theorem

A dendriform algebra is a k-module D with two binary operations ≺ and ≻ such that for any x, y, z ∈ D, (x ≺ y) ≺ z = x ≺ (y ≺ z + y ≻ z) (x ≻ y) ≺ z = x ≻ (y ≺ z) (x ≺ y + x ≻ y) ≻ z = x ≻ (y ≻ z)

Applications

Suppose that (D, ≺, ≻) is a dendriform algebra over k with a linear basis X = {xi|i ∈ I}. Let xi ≺ xj = {xi ≺ xj}, xi ≻ xj = {xi ≻ xj}, where {xi ≺ xj} and {xi ≻ xj} are linear combinations of x ∈ X. Then D has an expression by generator and defining relations D = D(X|xi ≺ xj = {xi ≺ xj}, xi ≻ xj = {xi ≻ xj}, xi, xj ∈ X). Denote by U(D) = RB(X|xiP(xj) = {xi ≺ xj}, P(xi)xj = {xi ≻ xj}, xi, xj ∈ X). Then U(D) is the universal enveloping Rota-Baxter algebra of D.

Applications

Suppose that (D, ≺, ≻) is a dendriform algebra over k with a linear basis X = {xi|i ∈ I}. Let xi ≺ xj = {xi ≺ xj}, xi ≻ xj = {xi ≻ xj}, where {xi ≺ xj} and {xi ≻ xj} are linear combinations of x ∈ X. Then D has an expression by generator and defining relations D = D(X|xi ≺ xj = {xi ≺ xj}, xi ≻ xj = {xi ≻ xj}, xi, xj ∈ X). Denote by U(D) = RB(X|xiP(xj) = {xi ≺ xj}, P(xi)xj = {xi ≻ xj}, xi, xj ∈ X). Then U(D) is the universal enveloping Rota-Baxter algebra of D. Similar to a classical problem (PBW theorem) involving associative and Lie algebras, L. Guo post the following conjecture: each dendriform algebra can be embedded into its universal enveloping Rota-Baxter algebra.

Applications

In a recent paper

• 27. Yuqun Chen and Qiuhui Mo, Embedding dendriform algebra into

its universal enveloping Rota-Baxter algebra, Proc. Amer. Math. Soc., to appear. arxiv:1005.2717 by using CD-lemma for Rota-Baxter algebras, we give a positive answer for the conjecture when the base field is of characteristic 0.

Applications

In a recent paper

• 27. Yuqun Chen and Qiuhui Mo, Embedding dendriform algebra into

its universal enveloping Rota-Baxter algebra, Proc. Amer. Math. Soc., to appear. arxiv:1005.2717 by using CD-lemma for Rota-Baxter algebras, we give a positive answer for the conjecture when the base field is of characteristic 0.

• 3. PBW theorem

Every dendriform algebra over a field of characteristic 0 can be embedded into its universal enveloping Rota-Baxter algebra.

Composition-Diamond lemma for differential algebras

We had established Gr¨

• bner-Shirshov bases theory for differential

algebras in the following papers.

Composition-Diamond lemma for differential algebras

We had established Gr¨

• bner-Shirshov bases theory for differential

algebras in the following papers.

• 1. Chen Yuqun, Yongshan Chen and Yu Li, Composition-Diamond

lemma for differential algebras, The Arabian Journal for Science and Engineering, 34(2A)(2009),135-145. arXiv:0805.2327v2

Composition-Diamond lemma for differential algebras

We had established Gr¨

• bner-Shirshov bases theory for differential

algebras in the following papers.

• 1. Chen Yuqun, Yongshan Chen and Yu Li, Composition-Diamond

lemma for differential algebras, The Arabian Journal for Science and Engineering, 34(2A)(2009),135-145. arXiv:0805.2327v2

• 2. Jianjun Qiu and Yuqun Chen, Composition-Diamond lemma for

λ-differential associative algebras with multiple operators, Journal of Algebra and its Applications, 9(2010), 223-239. arXiv:0904.0836

We had also established Gr¨

• bner-Shirshov bases theory for some

algebras.

We had also established Gr¨

• bner-Shirshov bases theory for some

algebras.

• 3. L.A. Bokut, Yuqun Chen and Yu Li, Groebner-Shirshov bases for

Vinberg-Koszul-Gerstenhaber right-symmetric algebras, Fundamental and Applied Mathematics, 14(8)(2008), 55-67. (in Russian) English version: Journal of Mathematical Sciences, 166(2010), 603-612. arXiv:0903.0706

We had also established Gr¨

• bner-Shirshov bases theory for some

algebras.

• 3. L.A. Bokut, Yuqun Chen and Yu Li, Groebner-Shirshov bases for

Vinberg-Koszul-Gerstenhaber right-symmetric algebras, Fundamental and Applied Mathematics, 14(8)(2008), 55-67. (in Russian) English version: Journal of Mathematical Sciences, 166(2010), 603-612. arXiv:0903.0706

• 4. L.A. Bokut, Yuqun Chen and Yu Li, Anti-commutative

Groebner-Shirshov bases of a free Lie algebra, Science in China, 52(2)(2009), 244-253. arXiv:0804.0914

We had also established Gr¨

• bner-Shirshov bases theory for some

algebras.

• 3. L.A. Bokut, Yuqun Chen and Yu Li, Groebner-Shirshov bases for

Vinberg-Koszul-Gerstenhaber right-symmetric algebras, Fundamental and Applied Mathematics, 14(8)(2008), 55-67. (in Russian) English version: Journal of Mathematical Sciences, 166(2010), 603-612. arXiv:0903.0706

• 4. L.A. Bokut, Yuqun Chen and Yu Li, Anti-commutative

Groebner-Shirshov bases of a free Lie algebra, Science in China, 52(2)(2009), 244-253. arXiv:0804.0914

• 5. L.A. Bokut, Yuqun Chen and Jianjun Qiu, Groebner-Shirshov bases

for associative algebras with multiple operations and free Rota-Baxter algebras, Journal of Pure and Applied Algebra, 214(2010), 89-100. arXiv:0805.0640v2

• 6. L.A. Bokut, Yuqun Chen and Yongshan Chen,

Composition-Diamond lemma for tensor product of free algebras, Journal of Algebra, 323(2010), 2520-2537. arXiv:0804.2115

• 6. L.A. Bokut, Yuqun Chen and Yongshan Chen,

Composition-Diamond lemma for tensor product of free algebras, Journal of Algebra, 323(2010), 2520-2537. arXiv:0804.2115

• 7. Chen Yuqun, Yongshan Chen and Chanyan Zhong,

Composition-Diamond lemma for modules, Czech J Math, 60(135)(2010), 59-76. arXiv:0804.0917

• 6. L.A. Bokut, Yuqun Chen and Yongshan Chen,

Composition-Diamond lemma for tensor product of free algebras, Journal of Algebra, 323(2010), 2520-2537. arXiv:0804.2115

• 7. Chen Yuqun, Yongshan Chen and Chanyan Zhong,

Composition-Diamond lemma for modules, Czech J Math, 60(135)(2010), 59-76. arXiv:0804.0917

• 8. L.A. Bokut, Yuqun Chen and Cihua Liu, Groebner-Shirshov bases

for dialgebras, International Journal of Algebra and Computation, 20(3)(2010), 391-415. arXiv:0804.0638

• 6. L.A. Bokut, Yuqun Chen and Yongshan Chen,

Composition-Diamond lemma for tensor product of free algebras, Journal of Algebra, 323(2010), 2520-2537. arXiv:0804.2115

• 7. Chen Yuqun, Yongshan Chen and Chanyan Zhong,

Composition-Diamond lemma for modules, Czech J Math, 60(135)(2010), 59-76. arXiv:0804.0917

• 8. L.A. Bokut, Yuqun Chen and Cihua Liu, Groebner-Shirshov bases

for dialgebras, International Journal of Algebra and Computation, 20(3)(2010), 391-415. arXiv:0804.0638

• 9. L.A. Bokut, Yuqun Chen and Xueming Deng, Groebner-Shirshov

bases for Rota-Baxter algebras, Siberian Math J, to appear. (in Russian) arxiv.org/abs/0908.2281

• 10. Yuqun Chen and Jiapeng Huang, Groebner-Shirshov bases for

L-algebras, submitted. arxiv:1005.0118

• 10. Yuqun Chen and Jiapeng Huang, Groebner-Shirshov bases for

L-algebras, submitted. arxiv:1005.0118

• 11. L.A. Bokut, Yuqun Chen and Yongshan Chen, Groebner-Shirshov

bases for Lie algebras over a commutative algebra, arXiv:1006.3217

• 10. Yuqun Chen and Jiapeng Huang, Groebner-Shirshov bases for

L-algebras, submitted. arxiv:1005.0118

• 11. L.A. Bokut, Yuqun Chen and Yongshan Chen, Groebner-Shirshov

bases for Lie algebras over a commutative algebra, arXiv:1006.3217

• 12. Yuqun Chen, Jing Li and Mingjun Zeng, Composition-Diamond

lemma for non-associative algebras over a commutative algebra, preprint.

• 10. Yuqun Chen and Jiapeng Huang, Groebner-Shirshov bases for

L-algebras, submitted. arxiv:1005.0118

• 11. L.A. Bokut, Yuqun Chen and Yongshan Chen, Groebner-Shirshov

bases for Lie algebras over a commutative algebra, arXiv:1006.3217

• 12. Yuqun Chen, Jing Li and Mingjun Zeng, Composition-Diamond

lemma for non-associative algebras over a commutative algebra, preprint.

• 13. L.A. Bokut, Yuqun Chen and Yu Li, Groebner-Shirshov bases for

categories, preprint.

• 10. Yuqun Chen and Jiapeng Huang, Groebner-Shirshov bases for

L-algebras, submitted. arxiv:1005.0118

• 11. L.A. Bokut, Yuqun Chen and Yongshan Chen, Groebner-Shirshov

bases for Lie algebras over a commutative algebra, arXiv:1006.3217

• 12. Yuqun Chen, Jing Li and Mingjun Zeng, Composition-Diamond

lemma for non-associative algebras over a commutative algebra, preprint.

• 13. L.A. Bokut, Yuqun Chen and Yu Li, Groebner-Shirshov bases for

categories, preprint.

• 14. L.A. Bokut, Yuqun Chen and Guangliang Zhang,

Composition-Diamond lemma for associative n-conformal algebras, arXiv:0903.0892

The following papers are some applications of Groebner-Shirshov bases theory.

The following papers are some applications of Groebner-Shirshov bases theory.

• 15. Chen Yuqun and Zhong Chanyan, Groebner-Shirshov basis for

HNN extensions of groups and for the alternative group, Comm. Algebra, 36(1)(2008), 94-103. arXiv:0804.0642

The following papers are some applications of Groebner-Shirshov bases theory.

• 15. Chen Yuqun and Zhong Chanyan, Groebner-Shirshov basis for

HNN extensions of groups and for the alternative group, Comm. Algebra, 36(1)(2008), 94-103. arXiv:0804.0642

• 16. Chen Yuqun, Groebner-Shirshov basis for Schreier extensions of

groups, Comm. Algebra, 36(5)(2008), 1609-1625. arXiv:0804.0641

The following papers are some applications of Groebner-Shirshov bases theory.

• 15. Chen Yuqun and Zhong Chanyan, Groebner-Shirshov basis for

HNN extensions of groups and for the alternative group, Comm. Algebra, 36(1)(2008), 94-103. arXiv:0804.0642

• 16. Chen Yuqun, Groebner-Shirshov basis for Schreier extensions of

groups, Comm. Algebra, 36(5)(2008), 1609-1625. arXiv:0804.0641

• 17. Chen Yuqun, Groebner-Shirshov basis for extensions of algebras,

Algebra Colloq., 16(2)(2009), 283-292. arXiv:0804.0643

The following papers are some applications of Groebner-Shirshov bases theory.

• 15. Chen Yuqun and Zhong Chanyan, Groebner-Shirshov basis for

HNN extensions of groups and for the alternative group, Comm. Algebra, 36(1)(2008), 94-103. arXiv:0804.0642

• 16. Chen Yuqun, Groebner-Shirshov basis for Schreier extensions of

groups, Comm. Algebra, 36(5)(2008), 1609-1625. arXiv:0804.0641

• 17. Chen Yuqun, Groebner-Shirshov basis for extensions of algebras,

Algebra Colloq., 16(2)(2009), 283-292. arXiv:0804.0643

• 18. Chen Yuqun, Hongshan Shao and K. P. Shum, On Rosso-Yamane

theorem on PBW basis of Uq(AN), CUBO A Mathematical Journal, 10(3)(2008), 171-194. arXiv:0804.0954

• 19. Chen Yuqun and Jianjun Qiu, Groebner-Shirshov basis for the

Chinese monoid, Journal of Algebra and its Applications,7(5)(2008), 623-628. arXiv:0804.0972

• 19. Chen Yuqun and Jianjun Qiu, Groebner-Shirshov basis for the

Chinese monoid, Journal of Algebra and its Applications,7(5)(2008), 623-628. arXiv:0804.0972

• 20. Yuqun Chen, Wenshu Chen and Runai Luo, Word problem for

Novikov’s and Boone’s group via Groebner-Shirshov bases, Southeast Asian Bull Math., 32(5)(2008), 863-877.

• 19. Chen Yuqun and Jianjun Qiu, Groebner-Shirshov basis for the

Chinese monoid, Journal of Algebra and its Applications,7(5)(2008), 623-628. arXiv:0804.0972

• 20. Yuqun Chen, Wenshu Chen and Runai Luo, Word problem for

Novikov’s and Boone’s group via Groebner-Shirshov bases, Southeast Asian Bull Math., 32(5)(2008), 863-877.

• 21. L.A. Bokut, Yuqun Chen and Xiangui Zhao, Groebner-Shirshov

beses for free inverse semigroups, International Journal of Algebra and Computation, 19(2)(2009), 129-143. arXiv:0804.0959

• 19. Chen Yuqun and Jianjun Qiu, Groebner-Shirshov basis for the

Chinese monoid, Journal of Algebra and its Applications,7(5)(2008), 623-628. arXiv:0804.0972

• 20. Yuqun Chen, Wenshu Chen and Runai Luo, Word problem for

Novikov’s and Boone’s group via Groebner-Shirshov bases, Southeast Asian Bull Math., 32(5)(2008), 863-877.

• 21. L.A. Bokut, Yuqun Chen and Xiangui Zhao, Groebner-Shirshov

beses for free inverse semigroups, International Journal of Algebra and Computation, 19(2)(2009), 129-143. arXiv:0804.0959

• 22. Yuqun Chen and Qiuhui Mo, Artin-Markov normal form for braid

group, Southeast Asian Bull Math, 33(2009), 403-419. arXiv:0806.0877

• 19. Chen Yuqun and Jianjun Qiu, Groebner-Shirshov basis for the

Chinese monoid, Journal of Algebra and its Applications,7(5)(2008), 623-628. arXiv:0804.0972

• 20. Yuqun Chen, Wenshu Chen and Runai Luo, Word problem for

Novikov’s and Boone’s group via Groebner-Shirshov bases, Southeast Asian Bull Math., 32(5)(2008), 863-877.

• 21. L.A. Bokut, Yuqun Chen and Xiangui Zhao, Groebner-Shirshov

beses for free inverse semigroups, International Journal of Algebra and Computation, 19(2)(2009), 129-143. arXiv:0804.0959

• 22. Yuqun Chen and Qiuhui Mo, Artin-Markov normal form for braid

group, Southeast Asian Bull Math, 33(2009), 403-419. arXiv:0806.0877

• 23. Chen Yuqun and Zhong Chanyan, Groebner-Shirshov basis for

some one-relator groups, Algebra Colloq., to appear. arXiv:0804.0643

• 24. Chen Yuqun and Yu Li, Some remarks for the Akivis algebras and

Pre-Lie algebras, Czech Math J, to appear. arXiv:0804.0915

• 24. Chen Yuqun and Yu Li, Some remarks for the Akivis algebras and

Pre-Lie algebras, Czech Math J, to appear. arXiv:0804.0915

• 25. L.A. Bokut, Yuqun Chen and Qiuhui Mo, Groebner-Shirshov

bases and embeddings of algebras, International Journal of Algebra and Computation, to appear. arxiv:0908.1992

• 24. Chen Yuqun and Yu Li, Some remarks for the Akivis algebras and

Pre-Lie algebras, Czech Math J, to appear. arXiv:0804.0915

• 25. L.A. Bokut, Yuqun Chen and Qiuhui Mo, Groebner-Shirshov

bases and embeddings of algebras, International Journal of Algebra and Computation, to appear. arxiv:0908.1992

• 26. Yuqun Chen and Chanyan Zhong, Groebner-Shirshov bases for

braid groups in Adyan-Thurston generators, Algebra Colloq., to

• appear. arxiv:0909.3639

• 24. Chen Yuqun and Yu Li, Some remarks for the Akivis algebras and

Pre-Lie algebras, Czech Math J, to appear. arXiv:0804.0915

• 25. L.A. Bokut, Yuqun Chen and Qiuhui Mo, Groebner-Shirshov

bases and embeddings of algebras, International Journal of Algebra and Computation, to appear. arxiv:0908.1992

• 26. Yuqun Chen and Chanyan Zhong, Groebner-Shirshov bases for

braid groups in Adyan-Thurston generators, Algebra Colloq., to

• appear. arxiv:0909.3639
• 27. Yuqun Chen and Qiuhui Mo, Embedding dendriform algebra into

its universal enveloping Rota-Baxter algebra, Proc. Amer. Math. Soc., to appear. arxiv:1005.2717

• 24. Chen Yuqun and Yu Li, Some remarks for the Akivis algebras and

Pre-Lie algebras, Czech Math J, to appear. arXiv:0804.0915

• 25. L.A. Bokut, Yuqun Chen and Qiuhui Mo, Groebner-Shirshov

bases and embeddings of algebras, International Journal of Algebra and Computation, to appear. arxiv:0908.1992

• 26. Yuqun Chen and Chanyan Zhong, Groebner-Shirshov bases for

braid groups in Adyan-Thurston generators, Algebra Colloq., to

• appear. arxiv:0909.3639
• 27. Yuqun Chen and Qiuhui Mo, Embedding dendriform algebra into

its universal enveloping Rota-Baxter algebra, Proc. Amer. Math. Soc., to appear. arxiv:1005.2717

• 28. Yuqun Chen and Bin Wang, Hilbert series of dentriform algebra,

preprint.

Thank you!