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¨ obner-Shirshov basis?

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

Introduction What is a Gr¨ obner-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¨ obner-Shirshov basis of the ideal I if (i) S is an ideal generator: Id ( S ) = I ; f ∈ I ⇒ ¯ sb for some s ∈ S and X -words a , b , where ¯ (ii) f = a ¯ f is the leading term of the polynomial f . Remark 1: Gr¨ obner-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¨ obner bases and Gr¨ obner-Shirshov bases were invented independently by

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

Introduction By using Gr¨ obner-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¨ obner-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¨ obner-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¨ obner-Shirshov bases theory is a powerful tool to solve the following classical problems.

Introduction By using Gr¨ obner-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¨ obner-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 operator P : A → A satisfy P ( x ) P ( y ) = P ( P ( x ) y ) + P ( xP ( y )) + λ P ( 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 operator P : A → A satisfy P ( x ) P ( y ) = P ( P ( x ) y ) + P ( xP ( y )) + λ P ( 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 ( ∪ r ≥ 0 ( YP ( Z )) r Y ) ∪ ( ∪ r ≥ 1 ( YP ( Z )) r ) Λ P ( Y , Z ) = ∪ ( ∪ r ≥ 0 ( P ( Z ) Y ) r P ( Z )) ∪ ( ∪ r ≥ 1 ( P ( Z ) Y ) r ) , where for a set T , T 0 means the empty set. Remark In Λ P ( Y , Z ), there are no words with a subword P ( z 1 ) P ( z 2 ) where z 1 , z 2 ∈ Z .

Free Rota-Baxter algebras Define Φ 0 = S ( X ) . . . . . . Φ n = Λ P (Φ 0 , Φ n − 1 ) . . . . . . Let Φ( X ) = ∪ n ≥ 0 Φ n

Free Rota-Baxter algebras Define Φ 0 = S ( X ) . . . . . . Φ n = Λ P (Φ 0 , Φ n − 1 ) . . . . . . Let Φ( X ) = ∪ n ≥ 0 Φ n Remark For any u ∈ Φ( X ), u has a unique form u = u 1 u 2 · · · u n where u i ∈ X ∪ P (Φ( X )) , i = 1 , 2 , . . . , n , and u i , u i +1 can not both have forms as p ( u ′ i ) and p ( u ′ i +1 ). Such u i is called prime.

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

Free Rota-Baxter algebras Let k Φ( X ) be a free k -module with k -basis Φ( X ) and λ ∈ k . Extend linearly P : k Φ( X ) → k Φ( 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 ′ )) + λ P ( u ′ · v ′ ) , if u = P ( u ′ ) , v = P ( v ′ ); uv , otherwise . Secondly, for any u = u 1 u 2 · · · u s , v = v 1 v 2 · · · v l ∈ Φ( X ) where u i , v j are prime, i = 1 , 2 , . . . , s , j = 1 , 2 , . . . , l , define u · v = u 1 u 2 · · · u s − 1 ( u s · v 1 ) v 2 · · · v l .

Free Rota-Baxter algebras Let k Φ( X ) be a free k -module with k -basis Φ( X ) and λ ∈ k . Extend linearly P : k Φ( X ) → k Φ( 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 ′ )) + λ P ( u ′ · v ′ ) , if u = P ( u ′ ) , v = P ( v ′ ); uv , otherwise . Secondly, for any u = u 1 u 2 · · · u s , v = v 1 v 2 · · · v l ∈ Φ( X ) where u i , v j are prime, i = 1 , 2 , . . . , s , j = 1 , 2 , . . . , l , define u · v = u 1 u 2 · · · u s − 1 ( u s · v 1 ) v 2 · · · v l . Then RB ( X ) := k Φ( 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 = u 1 u 2 · · · u n ∈ Φ( X ) and for a set T ⊆ X ∪ { P } , denote by deg T ( u ) the number of occurrences of t ∈ T in u . Let wt ( u ) = ( deg { P }∪ X ( u ) , deg { P } ( u ) , u 1 , · · · , u n ) .

Composition-Diamond lemma From now on, k is always a field of characteristic 0. Order Φ( X ) For any u = u 1 u 2 · · · u n ∈ Φ( X ) and for a set T ⊆ X ∪ { P } , denote by deg T ( u ) the number of occurrences of t ∈ T in u . Let wt ( u ) = ( deg { P }∪ X ( u ) , deg { P } ( u ) , u 1 , · · · , u n ) . For any u , v ∈ Φ( X ), define by induction u > v ⇔ wt ( u ) > wt ( v ) lexicographically , where P ( u ) > P ( v ) ⇔ u > v .