Rota-Baxter Algebra I

1 Rota-Baxter Algebra I Li GUO Rutgers University at Newark

. From differentiation to integration. d Differential operator: dx (f)(x) = lim h differential equations differential geometry differential topology differential algebra differential algebraic geometry f(x + h) f(x) h d (Leibniz rule) dx (f g) = f d dx (g) + d dx (f)g. A differential operator in general is a linear operator d satisfying the Leibniz rule: d(fg) = f d(g) + d(f)g. Integration operator: x f(t)dt = limit of Riemann sums.??? x (integration by parts) F Gdt = FG x x FG dt. An integration operator in general should be a linear operator P satisfying the integration by parts formula. 2

1. Rota-Baxter algebras: definition and examples A Rota-Baxter operator or a Baxter operator on an algebra R is a linear map P : R R such that P(x)P(y) = P(xP(y)) + P(P(x)y) + λp(xy), x, y R. Examples: Integration: A = Cont(R) (ring of continuous functions x on R). P : R R, P[f](x) := f(t)dt. Then P is a weight Rota-Baxter operator: Let F(x) = P[f](x) = x f(t) dt, G(x) = P[g](x) = Then the integration by parts formula states x (F() = G() = ), that is, F(t)G (t)dt = F(x)G(x) x x F (t)g(t)dt P[P[f]g](x) = P[f](x)P[g](x) P[fP[g]](x). 3 g(t) dt.

4 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 ( )( ) P[f](x)P[g](x) = f(x + n) g(x + m) = n 1,m 1 ( = n>m 1 n 1 f(x + n)g(x + m) + m>n 1 + m=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 k 1 =n n 1 k 1 =m + f(x + n)g(x + n) n 1 = P(P(f)g)(x) + P(fP(g))(x) + P(fg)(x).

5 Partial sum: Let R be the set of sequences {a n } with values in k. Then R is a k-algebra with termwise sum, product and scalar product. Define P : R R, P(a 1, a 2, ) = (a 1, a 1 + a 2, ). Then P is a Rota-Baxter operator of weight 1. Matrices: On the algebra of upper triangular n n matrices Mn(k), u define P((c kl )) ij = δ ij c ik. k i Then P is a Rota-Baxter operator of weight 1.

6 Scalar product: Let R be a k-algebra. For a given λ k, define P λ : R R, x λx, x R. Then (R, P λ ) is a Rota-Baxter algebra of weight λ. In particular, id is a Rota-Baxter operator of weight -1. Laurent series (QFT dimensional regularization): Let R = C[t 1, t]] be the ring of Laurent series n= k a nt n, k. Define 1 P( a n t n ) = a n t n. n= k n= k Then P is a Rota-Baxter operator of weight -1.

7 Classical Yang-Baxter equation: Let g be a Lie algebra with a perfect pairing g g k. Then g 2 = g g = End(g). Let r 12 g 2 be anti-symmetric. Then r 12 is a solution (r-matrix) of the classical Yang-Baxter equation (CYB) CYB(r) := [r 12, r 13 ] + [r 12, r 23 ] + [r 13, r 23 ] = if and only if the corresponding P End(g) is a (Lie algebra) Rota-Baxter operator of weight : [P(x), P(y)] = P[P(x), y] + P[x, P(y)] Associative Yang-Baxter equations Let A be an associative algebra and let r := i u i v i R R be a solution of the AYB equation r 13 r 12 r 12 r 23 + r 23 r 13 =. Then P r (x) := i u ixv i defines a Rota-Baxter operator of weight. Others Divided powers, Hochschild homology ring, dendriform algebras, rooted trees, quasi-shuffles, Chen integral symbols,...

8 MZV Hopf Algebra pqft Rota Baxter Algebra Operad Yang Baxter Equation Combi natorics

2. Free commutative Rota-Baxter algebras 9 Let A be a commutative k-algebra. Let X + (A) = n A n (= T(A)). 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. Quasi-shuffle product: Hoffman (2) 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 multiples as the identity. It defines the shuffle product without the third term. 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.

1 Mixable shuffle product: Guo-Keigher (2) on Rota-Baxter algebras, Goncharov (22) on motivic shuffle relations and Hazewinckle on overlapping shuffle products. 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).

Stuffles: Cartier (1972) on free commutative Rota-Baxter algebras, Ehrenborg (1996) on monomial quasi-symmetric functions and Bradley (24) on q-multiple zeta values. Given a triple (k,α,β) where max{m, n} k m + n, α : [m] [k] and β : [n] [k] order preserving injections with im(α) im(β) = [k], define Φ k,α,β (a,b) = c 1 c k A k by (taking a = b = 1) c j = a α 1 (j)b β 1 (j). Let S c (m, n) be the set of such triples (k,α,β). Define a b = Φ k,α,β (a,b) = a α 1 (1) b β 1 (1) a α 1 (k) b β 1 (k) (k,α,β) S c(m,n) (k,α,β) S c(m,n) Example: a = a 1, m = 1, b = b 1 b 2, n = 2. k α β Φ k,α,β 3 α(1) = 1 β(1) = 2,β(2) = 3 a 1 b 1 b 2 3 α(1) = 2 β(1) = 1,β(2) = 3 b 1 a 1 b 2 3 α(1) = 3 β(1) = 1,β(2) = 2 b 1 b 2 a 1 2 α(1) = 1 β(1) = 1,β(2) = 2 a 1 b 1 b 2 2 α(1) = 2 β(1) = 1,β(2) = 2 b 1 a 1 b 2 11

Delannoy paths: Fares (1999) on coalgebras, Aguiar-Hsiao (24) on quasi-symmetric functions and Loday (25) on Zinbiel operads. Let D(m, n) be the set of lattice paths from (, ) 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 labeling the horizontal (resp. vertical) and diagonal segments of d. Define a d b = d(a,b). d D(m,n) b2 b2 b2 b2 b2 b1 b1 b1 b1 b1 a1 a1 a1 a1 a1 a 1 b 1 b 2 b 1 a 1 b 2 b 1 b 2 a 1 a 1 b 1 b 2 b 1 a 1 b 2 Theorem All the above products define the same algebra on X + (A) (of weight λ = 1). 12

13 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 j A A X(A) f f R 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. Theorem (Guo-Keigher) X(A) with the shift operator P(a) := 1 a is the free commutative RBA over A.

14 Hopf algebras and free Rota-Baxter algebras Theorem. Then the sextuple (R : R λ := X λ (k),µ,η,,ε, S) is a Hopf algebra. Here (i) R = ke n, is the free k-module on the set {e n } n, n= (ii) µ = µ λ : R R R, e m e n (iii) η = η λ : k R, 1 e, (iv) = λ : R R R, e n (v) ε = ε λ : R k, e n m ( )( λ k m+n k m ) e m+n k k= m n n k ( λ) k e i e n k i, k= i= 1, n =, λ1, n = 1,, n 2 n ( (vi) S = S λ : R R, e n ( 1) n v= n 3 v 3 ) λ n v e v. k

15 Here for any positive or negative integer x, ( ) x k is defined by the ( generating function (1 + z) x x ) = z k. k= When λ =, we have the divided power algebra. More generally, if A is a Hopf algebra, then X(A) is a Hopf algebra. In particular, X(X) = X(k[X]) is a Hopf algebra. k

3. Spitzer s Identity 16 Theorem (Spitzer, 1956) Let (A, P) be a unital commutative Rota Baxter Q-algebra of weight λ. Then for a A, we have ( exp P ( )) log(1 + λat) = λ in the ring of power series A[[t]]. P ( ap(ap( (ap(a)))) ) }{{} n-times n= Geometric series. Let P = id on R. It is a Rota-Baxter operator of weight -1. Applying Spitzer, we have 1/(1 at) = n (at) n. t n

Integration In LHS of Spitzer, 1 λ log(1 + λat) = 1 ( λat + (λat) 2 /2 +... ) λ at. λ So we have weight Spitzer s identity exp (P(at))) = t n P ( ap(ap( (ap(a)))) ) }{{} n= n-times Note that RHS is the solution to y = 1 + tp(ay)(= 1 + tp(a) + t 2 P(aP(ay)) = ). Now take A = Cont(R) and P(f)(x) = x f(u)du. Then Spitzer says that the IVP y = tay, y() = 1 has unique solution ( x ) exp t y(u)du. Taking t = 1, this known in calculus. 17

18 Expanding ( x ) x exp t y(u)du = 1 + t y(u)dx + t2 ( x 2 y(u)dy) + 2 and comparing with the right hand side of weight Spitzer: x x ( u1 1 + t y(u)du + t 2 y(u 1 ) y(u 2 )du 2 )du 1 +... we get ( x ( x ) 3= x y(u)dy 3! ) 2 x y(u)dy = 2 u1 y(u 1 ) u1 y(u 1 ) y(u 2 ) u2 y(u 2 )du 2 du 1. y(u 3 )du 3 du 2 du 1. ( x ) n y(u)dy = n! y(u n ) y(u 1 ) du n du 1. }{{} u n u 1 x

Fluctuation theory Let A be the Banach algebra { ϕ(t) := e itx df(x) } df(x) <, F( ) < of characteristic functions of random variables X. Let P(ϕ)(t) = eitx df(x) + F() F( ), giving the characteristic function of random variables max{, X}. Then (A, P) is a Rota-Baxter algebra of weight 1. Let {X k } be a sequence of independent random variables with identical distribution function F(x) and characteristic function ψ(s) = e isx df(x). Let S n = X 1 +... + X n and let M n = max(, S 1, S 2,..., S n ). Let F n (x) = Prob(M n < x) (Prob for probability) be the distribution function of M n. Applying Spitzer to a = ψ(s), we obtain the original identity of Spitzer: ( ( e isx df n (x) = exp e isx df(x) + F()) ). n= k=1 19

Waring s formula Let X = {x n }, x n variables. Let A = Q[X] and A = A N = {(a n ) a n A, n 1}. Let P((a n )) = (a 1, a 1 + a 2,, k n a k, ). Then P is a Rota-Baxter operator of weight -1. Applying Spitzer to a = (x 1,, x n, ), we have the well-known Waring s formula ( ) exp ( 1) k t k p k (x 1,..., x m )/k = e n (x 1,..., x m )t n, m 1. k=1 n= Here p k (x 1,...,x m ) = x1 k + x 2 k +... + x m k are the power sum symmetric function. e n (x 1,, x m ) = x i1 x in 1 i 1 i n m are the elementary symmetric function. 2

Multiple zeta values Multiple Hurwitz values are defined by 1 ζ(s 1,, s k ; x) = (n 1 + x) s 1 (nk + x) s, s i 1, s 1 2 k n 1 >n 2 > >n k 1 Taking x = gives multiple zeta values. For k 2, let a = f k := 1/x k. With the summation operator P(f)(x) = n 1 f(x + n), we have P(log(1 + at)) = P ( ( 1) i 1 ( ) t i ) ( 1) i 1 = i x k ζ(ki; x + 1)t i. i i=1 Similarly, the right hand side of Spitzer becomes 1 + ζ(k,, k; x + 1)t i. }{{} i=1 i So evaluating at x =, we get the well-known identity ( ) exp ( 1) i 1 ζ(ik) ti = 1 + ζ(k,, k)t i. i }{{} i=1 i=1 i 21 i=1

General form of Spitzer s identity leads to Algebraic Birkhoff Decomposition of Connes and Kreimer; Factorization theorem of Barron, Huang and Lepowsky for Lie algebras and vertex operator algebras; Even-odd decomposition of Aguiar, Bergeron and Sottile in combinatorial Hopf algebras; Lie algebra polar decomposition of Munthe-Kaas et. al. in numerical differential equations; Decomposition in integrable systems by Reyman and Semenov-Tian-Shansky; Magnus expansional formula for noncommutative differential equations; Renormalization of multiple zeta values. 22