Some constructions and applications of Rota-Baxter algebras I

Transcription

1 Some constructions and applications of Rota-Baxter algebras I Li Guo Rutgers University at Newark joint work with Kurusch Ebrahimi-Fard and William Keigher

2 1. Basics of Rota-Baxter algebras Definition Let k be a commutative ring (with identity). Fix a λ in k. A Rota-Baxter k-algebra (RBA) (of weight λ) is a pair (A, P) where A is a k-algebra and P is a k-linear operator P : A A such that P(x)P(y) = P(P(x)y) + P(xP(y)) + λp(xy), x, y A. P(x)P(y) + θp(xy) = P(P(x)y) + P(xP(y)), x, y A where θ = λ. λ is more convenient for algebraic constructions. θ(= 1) is commonly used in physics applications. Basic properties of a RBA (A, P). * If P has weight 1, then λp has weight λ. * λi P is a Rota-Baxter operator of weight λ; * P(A) is a subalgebra. In fact, x P y := xp(y)+p(x)y+λxy is associative and (A, P, P) is a RBA, called the double or derived algebra. * If (A, P) is a Rota-Baxter algebra with θ = 1 and idempotent P (P 2 = P), then A = P(A) (id P)(A), and P(A) and (id P)(A) are subalgebras of A. In general, a subdirect decomposition holds;

3 History Started in 1960 by Glen Baxter on fluctuation theory in probability. Studied by Pierre Cartier, Gian-Carlo Rota and his school in 1960 s and early 70 s. Independently studied by physicists in 1980s under the name classical Yang-Baxter relations (after Rodney Baxter). Promoted by Rota in 1990s. Found applications in quantum field theory, number theory, combinatorics, Hopf algebras and operads in 1990 present.

4 First examples: Integration: A = Cont(R) (ring of continuous functions on R). P : A A, P[f](x) := x Then P is a weight 0 Rota-Baxter operator: Let F(x) = P[f](x) = x 0 0 f(t)dt. f(t) dt, G(x) = P[g](x) = Then the integration by parts formula states x 0 F(t)G (t)dt = F(x)G(x) x 0 x 0 F (t)g(t)dt (F(0) = G(0) = 0), that is, P[P[f]g](x) = P[f](x)P[g](x) P[fP[g]](x). Iterations of P give iterated integrals in multiple zeta values. g(t) dt.

5 Summation: On a suitable class of functions, define P(f)(x) := n 1 f(x + n). Then P is a Rota-Baxter operator of weight 1, since ( )( ) f(x + n) g(x + m) = f(x + n)g(x + m) = n 1 ( n>m 1 k 1 + m 1 m>n 1 + m=n 1 n 1,m 1 ) f(x + n)g(x + m) ( = f(x + k } + {{ m } ) ) ( g(x + m) + g(x + k } + {{ n } ) ) f(x + n) m 1 =n n 1 =m + n 1 f(x + n)g(x + n) = P(P(f)g)(x) + P(fP(g))(x) + P(fg)(x). Its iterations give multiple zeta values and multiple polylogarithms. k 1

6 Partial sum: Let A be the set of sequences {a n } with values in k. Then A is a k-algebra with termwise sum, product and scalar product. Define P : A A, P(a 1, a 2, ) = (a 1, a 1 + a 2, ). Then P is a Rota-Baxter operator of weight 1. Iterations of this gives Z-sums, and MZVs. Matrices: On the algebra of upper triangular matrices Mn u (k), define P((c kl )) ij = δ ij c ik. k i Then P is a Rota-Baxter operator of weight 1. Scalar product: Let A be a k-algebra. For a given λ k, define P λ : A A, x λx, x A. Then (A, P λ ) is a Rota-Baxter algebra of weight λ. In particular, id is a Rota-Baxter operator of weight -1.

7 QFT dimensional regularization: Let R = C[t 1, t]] be the ring of Laurent series n= k a n t n, k 0. Define P( a n t n 1 ) = a n t n. n= k n= k Then P is a Rota-Baxter operator of weight -1. Evaluation for cut-off regularization: R(F(x,Λ)) = F(µ,Λ) is multiplicative and idempotent, so is a Rota-Baxter operator. Others Classical and associative Yang-Baxter equations, divided powers, Hochschild homology ring, dendriform algebras, rooted trees, quasishuffles, Chen integral symbols,...

8 2. Free commutative Rota-Baxter algebras Let A be a k-algebra. Let + X (A) = n 0 A n. Consider the following products on + X (A). Define 1 k k to be the unit. Let a = a 1 a m A m and b = b 1 b n A n. Stuffles: Cartier (1972) on free commutative Rota-Baxter algebras, Ehrenborg (1996) on monomial quasi-symmetric functions and Bradley (2004) on q-multiple zeta values. Given a triple (k, α, β) where 1 k m + n, α : [m] [m + n] and β : [n] [m + n] are order preserving injections with im(α) im(β) = [k], define Φ k,α,β (a, b) = (c 1,, c k ) A k by a u, if α 1 (j) = u, β 1 (j) = ; c j = b v, if β 1 (j) = v, α 1 (j) = ; a u b v, if α 1 (j) = u, β 1 (j) = v. Let S c (m, n) be the set of such triples (k, α, β). Define a b = Φ k,α,β (a, b). (k,α,β) S c (m,n)

9 Delannoy paths: Fares (1999) on coalgebras, Aguiar-Hsiao (2004) on quasi-symmetric fuctions and Loday (2005) on Zinbiel operads. Let D(m, n) be the set of lattice paths from (0,0) to (m, n) consisting steps either to the right, to the above, or to the above-right. For d D(m, n) define d(a, b) to be the path d with a = (a 1,, a m ) (resp. b = (b 1, b n )) sequentially labelling the horizontal (resp. vertical) and diagonal segments of d. Define a d b = d(a, b). d D(m,n)

10 Quasi-shuffle product: Hoffman (2000) on multiple zeta values. Write a = a 1 a, b = b 1 b. Recursively define (a 1 a ) (b 1 b ) = a 1 (a (b 1 b )))+b 1 ((a 1 a ) b )+a 1 b 1 (a b ), with the convention that if a = a 1, then a multiple as the identity. Example. a 1 (b 1 b 2 ) = a 1 (a (b 1 b 2 )) + b 1 (a 1 b 2 ) + (a 1 b 1 ) (a b 2 ) = a 1 (b 1 b 2 ) + b 1 (a 1 b 2 ) + (a 1 b 1 ) b 2. = a 1 b 1 b 2 + b 1 a 1 b 2 + b 1 b 2 a 1 + b 1 a 1 b 2 + a 1 b 1 b 2.

11 Mixable shuffle product: Guo-Keigher (2000) on Rota-Baxter algebras and Goncharov (2002) on motivic shuffle relations. A shuffle of a = a 1... a m and b = b 1... b n is a tensor list of a i and b j without change the order of the a i s and b j s. A mixable shuffle is a shuffle in which some pairs a i b j are merged into a i b j. Define (a 1... a m ) (b 1... b n ) to be the sum of mixable shuffles of a 1... a m and b 1... b n. Example: a 1 (b 1 b 2 ) = a 1 b 1 b 2 + b 1 a 1 b 2 + b 1 b 2 a 1 (shuffles) + a 1 b 1 b 2 + b 1 a 1 b 2 (merged shuffles). Theorem All the above products define the same algebra on X + (A).

12 A free Rota-Baxter algebra over another algebra A is a Rota-Baxter algebra X(A) with an algebra homomorphism j A : A X(A) such that for any Rota-Baxter algebra R and algebra homomorphism f : A R, there is a unique Rota-Baxter algebra homomorphism making the diagram commute A j A f X(A) When A = k[x], we have the free Rota-Baxter algebra over X. Recall ( X + (A), ) is an associative algebra. Then the tensor product algebra (scalar extension) X(A) := A X + (A) is an A-algebra. f R Theorem X(A) with the shift operator P(a) := 1 a is the free commutative RBA over A. X + (A) can be recovered from X(A) as its derived algebra (the double): X + (A) = P( X(A)).

13 3. Rota-Baxter algebra of rooted trees Planar rooted trees and forests A planar rooted tree is a plane rooted tree with a fixed embedding into the plane. Two drawings of planar rooted trees. Let T be the set of planar rooted trees and let F be the free semigroup generated by T in which the product is denoted by. Thus each element in F is a noncommutative product of elements in T, called a forest. So a forest is of the form T 1 T n consisting of trees T 1,, T n (concatenation). T 1 T n the tree with a new root and with the trees T 1,, T n as branches (grafting). The depth (or height) d(t) of a forest is the length of longest path in the forest. Let y(t) be the number of leaves of T.

14 Universal property of rooted forests. A mapped semigroup is a semigroup U together with a map α : U U. A morphism from a mapped semigroup (U, α) to a mapped semigroup (V, β) is a semigroup homomorphism f : U V such that f α = β f. U α f β V U f V A free mapped semigroup on a set X is a mapped semigroup (U X, α X ) together with a map j X : X U X with the property that, for any mapped semigroup (V, β) together with a map f : X V, there is a unique morphism f : (U X, α X ) (V, β) of mapped semigroups such that f = f j X. X j X U X f f V

15 Theorem Let j : { } F be the natural embedding j ( ) = T. 1. The triple (F,( ), j ) is the free mapped semigroup on the generator. 2. The triple (k F,( ), j ) is the free mapped nonunitary algebra on one generator. Note: The non-planar trees in the Connes-Kreimer Hopf algebra is given by the free mapped commutative semigroup generated by.

16 Rota-Baxter algebra on rooted forests. Note that k F is also the free noncommutative polynomials over T with coefficients in k and with product. Now define another product on k F, making it into a unitary RBA. We do this by recursively defining : F F k F and then extending bilinearly. Consider forests F = T 1 T b and F = T 1 T b with T i, T j T. Define F F = T 1 T b 1 (T b T 1 ) T 2 T b, where T 1 T 2 = T 1, if T 2 =, T 2, if T 1 =, F 1 F 2 + F 1 F 2 + λ F 1 F 2, if T 1 = F 2, T 2 = F 2. Theorem Define P T (F) = F. Then (k F,, P T ) is a unitary RBA of weight λ. Also, y(f 1 F 2 ) = y(f 1 ) + y(f 2 ) 1.

17 4. Free Rota-Baxter algebra over a set Angularly decorated forests Let X be a set. Let F X be the set of forests with each angle between two adjacent leaves decorated by an element of X.... Each such an angularly decorated forest is uniquely determined by the forest F and the list x 1,, x n of decorations. Then n = y(f) 1 where y(f) is the number of leaves of F. Notation: U = (F;(x 1,, x n )). Example: ( ; x ) =, ( ; x y ) =, ( ; x y ) = x, ( ; a ) = a. There is a canonical decomposition U = U 0 x 1 U 1 U m 1 x m U m where each U i is an angularly decorated tree.

18 RB bracketed words (RBWs) X. These are bracketed words with letters from X {1} such that no two pairs of brackets are next to each other. For x, y, z X, x y z, x y z are RBWs, while x y is not. There is a one-one correspondence Φ between angularly decorated trees F with decoration set X and RB bracketed words X given by Φ( ) = 1, Φ(U 0 x 1 x m U m+1 ) = Φ(U 1 )x 1 x m Φ(U m ), Φ( U ) = Φ(U).,, x, a. x 1, xy, x y, a The decomposition of a forest into trees corresponds to decomposition of a RB word into indecomposable words. x = x 1 x 2 x b with each x i in X or in X. For example, x 1 x 2 x 3 = x 1 x 2 x 3, x 1 x 2 x 3 x 4 = x 1 x 2 x 3.

19 Product on angularly decorated forests. Let X NC (X) be the free module generated by F X. Define a product on X NC (X). In terms of forests: For decorated forests (S;(u 1,, u k )) and (T;(v 1,, v l )), define (S;(u 1,, u k )) (T;(v 1,, v l )) = (S T;(u 1,, u m, v 1,, v n )).

20 In terms of A-decorated forests: For forests U = U (0) x1 xm U (m), V = V (0) y1 yn V (n) F X, U (i), V (j) T, define U V = U (0) ( x1 xm U (m) V (0) ) y1 yn V (n), where U (m), if V (0) =, U (m) V (0) = V (0), if U (m) =, U (m) V (0) +U (m) V (0) +λu (m) V (0), if U (m), V (0) Here U (m) = U (m), V (0) = V (0).. x 1 = x + x 1 + x. Define P X (x) = x. Theorem The triple (X NC (X),, P X ) is the free Rota-Baxter algebra over X.

21 5. Rota-Baxter algebra on controlled forests. T planar rooted trees. F r forests in which the empty tree only occurs as the outmost branches of a vertex or the outmost trees: T 1 T 2 T n with no middle T i =. For example,,,,, are in F r, while, are not.

22 Product r : First define to be the identity. Next for controlled forests U = U (0) U (m), V = V (0) V (n) F r, U (i), V (j) { } F r, define if U (m) = V (1) =. U r V = U (0) U (m 1) V (2) V (n), If U (m) or V (0), define U r V = U (0) ( U (m) V (0)) V (n), where U (m), if U (m), V (0) =, U (m) r V (0) = V (0), if U (m) =, V (0), U (m) V (0) +U (m) V (0) +λu (m) V (0), if U (m), V (0). Here U (m) = U (m), V (0) = V (0). Theorem Define P r F (U) = U. The triple (k Fr, r, P A ) is a Rota-Baxter algebra. Homomorphism of tree RBAs. Define δ : F F r by deleting all leaves not on the edge. Then δ : k F k F r is a surjective Rota-Baxter algebra homomorphism.

23 6. Free Rota-Baxter algebras on an algebra Let A be the unitarization of a k-algebra Ǎ: A = k Ǎ. FA r planar trees from F with each adjacent pair of leaves decorated by Ǎ.,,, x, w z x x y x x y z w[[x]y] z More precisely, they are elements in the tensor product Ǎ F := (F; Ǎ) := {(F; a) a Ǎ (l 1) } with the same k-module structure as Ǎ (l 1). Define : (F; Ǎ) ( F ; Ǎ), (F; a) = ( F ; a).

24 Product r on decorated trees. X NC,0 (A) := F F r(f; Ǎ). First define ( ;1) to be the identity. For A-decorated forests U = U (0) x1 xm U (m), V = V (0) y1 yn V (n) (F r ; A). If U (m) = V (1) =, then define If not, then define U r V = U (0) x1 U (m 1) xm y 1 V (2) yn V (n). U V = U (0) ( x1 xm U (m) V (0) ) y1 yn V (n), where U (m), if U (m), V (0) =, U (m) V (0) = V (0), if U (m) =, V (0), U (m) V (0) +U (m) V (0) +λu (m) V (0), if U (m), V (0). Here U (m) = U (m), V (0) = V (0). Theorem Define P A (x) = x. The triple (X NC,0 (A),, P A ) is the free Rota-Baxter algebra over the algebra A.

25 7 Hopf algebras of Connes-Kreimer and Loday-Ronco Loday-Ronco Hopf algebra V k-module with basis X. n 0. Y n planar binary trees with n + 1 leaves. Y n,x planar binary trees with n+1 leaves and with vertices decorated by X. So Y 0 = Y 0,X = { }. Here are some undecorated trees.,,,, For T Y m,x, U Y n,x and x X, the grafting of T and U over x is T x U Y m+n+1,x. K[Y n,x ] the K-vector space generated by Y n,x. DD(V ) the graded vector space n 1 K[Y n,x]. Define operations and on DD(V ): Let T Y m,x, U Y n,x with m 1, n 1. Then T = T l x T r, U = U l y U r with x, y X and T l, T r, U l, U r i 0 Y i,x. Define T U : = T U : = { T l x (T r U + T r U), if T r, T l x U, if T r =. { (T U l + T U l ) y U r, if U l, T y U r, if U l =.

26 Theorem (Loday-Ronco) (DD(V ),, ) is the free dendriform dialgebra on V. In particular, H LR = k1 + DD(V ) is a Hopf algebra. Natural embedding. Define a map φ : DD(V ) X NC (V ) (of weight 0) recursively. First φ( x ) = ( ;x) = x (= x). Further, any T Y n,x, n 1 can be uniquely written as T = T l x T r with x X and T l, T r 0 i<n Y i,x. We then define φ(t l ) x φ(t r ), T l 1, T r 1, x φ(t φ(t) = r ), T l = 1, T r 1, φ(t l ) x, T l 1, T r = 1, x, T l = 1, T r = 1. ( φ x y z) = x y z = ( ;(x, y, z)). The image is the subset of binary a-decorated trees.

27 Theorem. Let = +. φ : (DD(V ), ) NC X (V ) is an injective algebra homomorphism whose image is the subalgebra of binary angularly decorated forests. Similar result holds for dendriform trialgebras.

28 Relation with Connes-Kreimer Hopf algebras Theorem (Holtkamp and Foissy) The noncommutative Connes-Kreimer Hopf algebra NCK is isomorphic to the Loday-Ronco Hopf algebra H LR = k1 DD(V ). Composing with φ : H LR X NC (V ), we have an injection ψ : NCK X NC (V ). Using the explicit map NCK H LR of Aguiar and Sottile, we have, for example, ( ψ c b a ) (= ψ(b + a (B + b (c)))) = c b a, ( ψ t 1 t 2 t 3 ) a (= ψ(b a + (t 1 t 2 t 3 ))) = t 1 t 2 t 3 a. Here we use the notation B + a of Connes and Kreimer. In general, vertical vertices are listed with descending parentheses; horizontal vertices are listed with increasing parentheses.

29 References: Aguiar and W. Moreira, Combinatorics of the free Baxter algebra, to appear in Electrical Journal of Combinatorics, arxiv: math.co/ K. Ebrahimi-Fard, L. Guo, On free Rota-Baxter algebras, arxiv: math.ra/ K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras and dendriform algebras, arxiv:math.ra/ , prepublication, IHES, M/05/11, L. Guo and W. Keigher, Baxter algebras and shuffle products, Advances in Mathematics 150 (2000), no.1, , arxiv:math.ra/ L. Guo and W. Keigher: On free Baxter algebras: completions and the internal construction, Advances in Mathematics 151 (2000), no.1, , arxiv:math.ra/ L. Guo and W. Yu Sit, Enumenation of Rota-Baxter words, to appear in Proceedings ISSAC 2006, Genoa, Italy, ACM Press, arxiv: math.ra/