Free Commutative Integro-differential Algebras

Transcription

1 1 Free Commutative Integro-differential Algebras Li GUO Rutgers University (Joint work with G. Regensburger and M. Rosenkranz, and X. Gao and S. Zheng)

2 Free differential algebras Differential algebra d(xy) = d(x)y + xd(y)+λd(x)d(y). d(uv) φ d(u)v + ud(v)+λd(u)d(v), u, v R. This leads to normal forms w with no products in d. Thus free commutative differential algebra (of weight λ) on a set X is of the form k{x} := k[ X], X := {x (n) x X, n 0} with concatenation product. Define d X : k{x} k{x} as follows. Let w = u 1 u k, u i (X), 1 i k, be a commutative word from the alphabet set (X). If k = 1, so that w = x (n) (X), define d X (w) = x (n+1). If k > 1, recursively define d X (w) = d X (u 1 )u 2 u k + u 1 d X (u 2 u k )+λd X (u 1 )d X (u 2 u k ). Further define d X (1) = 0. Then (k{x}, d X ) is the free commutative differential algebra of weight λ on the set X. 2

3 Integral algebra 3 What about integral algebra? What is an integral algebra? It is a special case of Rota-Baxter algebra. Let k be a commutative ring. Let λ k be fixed. A Rota-Baxter operator of weight λ on a k-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. References: 1. L. Guo, WHAT IS a Rota-Baxter Algebra, Notice of Amer. Math. Soc. 56 (2009), L. Guo, An Introduction to Rota-Baxter Algebra, International Press, 2012.

4 The integration operator I For continuous functions f(x) and g(x), define F(x) := I[f](x) := x Then F (x) = f(x), G (x) = g(x). The integration by parts formula x can be rewritten as 0 x 0 0 f(s)ds, G(x) := I[g](x) := x F (t)g(t)dt = F(t)G(t) x x 0 F(t)G (t) dt f(t)g(t) dt = F(x)G(x) 0 x 0 0 F(t)g(t) dt. Using Eq. (1), get I[f I[g]](x) = I[f]I[g](x) I[I[f] g](x). g(s)ds. (1) I[f I[g]] = I[f]I[g] I[I[f] g], I[f]I[g] = I[f I[g]]+I[I[f] g]. An integral algebra is an algebra R together with a linear operator I : R R that satisfies I[f]I[g] = I[f I[g]]+I[g I[f]], f, g R. 4

5 Free commutative Rota-Baxter algebras 5 Rota-Baxter algebra P(x)P(y) = P(P(x)y) + P(xP(y)) + λp(xy). P(x)P(y) P P(P(x)y)+P(xP(y)) +λp(xy). This leads to normal forms w with no subwords of the form P(x)P(y). In a commutative Rota-Baxter algebra, this means The product is given by a = a 0 P(a 1 P(a 2 P( P(a n ) ))), a i A. a = a 0 a 1 a n A (n+1). ab = (a 0 b 0 ) ( (a 1 a n ) (b 1 b m ) ). is a shuffle like product, called mixable shuffle product.

6 Mixable shuffle product 6 Let A be a commutative k-algebra. Let X + (A) = n 0 A n (= T(A)). Consider the following products on X + (A). Mixable shuffle product: Guo-Keigher (2000) on Rota-Baxter algebras, Goncharov (2002) 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).

7 7 Quasi-shuffle product: Hoffman (2000) on multiple zeta values. Let X + (A) = n 0 A n (= T(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. 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.

8 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 8 By Stuffles: Cartier (1972) on free commutative Rota-Baxter algebras, Ehrenborg (1996) on monomial quasi-symmetric functions and Bradley (2004) on q-multiple zeta values. By Delannoy paths: Fares (1999) on coalgebras, Aguiar-Hsiao (2004) on quasi-symmetric functions 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 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

9 Free commutative Rota-Baxter algebras Theorem All the above products define the same algebra on X + (A). 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. 9

10 where X is the shuffle product. 10 Examples The free commutative Rota-Baxter algebra on k (i.e., on the empty set) is X( ) = k 1 ka k, a m a n = min(m,n) r=0 ( m+n r m )( m r ) λ r 1 (m+n r). When λ = 0, we obtain the divided powers. The free commutative integral algebra (Rota-Baxter algebra of weight 0) on k[x] (i.e., on one generator x): Let I := k 1 Nk 0. For I = (i 0,, i k ) I, denote x I := x i 0 x i k. Then X(k[x]) = I I kx I. For x I = x i 0 x I and x J = x j 0 x J, we have ( x I x J = x i 0+j 0 x I Xx J),

11 Differential Rota-Baxter algebra Then (X(A), d A, P A ) is the free commutative differential Rota-Baxter algebra on A. Let X be a set. The differential Rota-Baxter algebra (X(k{X}), d k{x}, P k{x} ) is the free differential Rota-Baxter algebra on X. It is the algebra of differential Rota-Baxter polynomials in X. 11 A differential Rota-Baxter algebra (DRB) is a triple (R, d, P) where (R, d) is a differential algebra (of weight λ), (R, P) is a Rota-Baxter algebra (of weight λ) such that d P = id R. These give three rewriting rules that imply that a normal form for the DRB algebra is of the form x := x 0 x 1 x n, x i X. More generally, let (A, d) be a differential algebra of weight λ. On the free commutative Rota-Baxter algebra (X(A), P A ), define d A : X(A) X(A), d A (x 0 x 1 x n ) = d 0 (x 0 ) x 1... x n +x 0 x 1 x 2... x n +λd 0 (x 0 )x 1 x 2... x n

12 Integro-differential algebras 12 Note that the integral by parts formula in Rota-Baxter algebra is a purified version (so that only I occurs) of the original formula I(f)I(g) = I(fI(g))+I(I(f)g) FG = I(F G)+I(FG ) where the interaction of differentiation and integration is taken out of the picture. This needs to be put back in order to understand fully the algebraic structure in differential equations, in particular, to compute Green s operators of boundary problems for linear ordinary/partial differential equations.

13 Definition of Integro-differential Algebras 13 An integro-differential k-algebra of weight λ (also called a λ-integro-differential k-algebra) is a differential k-algebra (R, D) of weight λ with a linear operator Π: R R such that D Π = id R and the initialization satisfies J: = Π D J(x)J(y) = J(x)y + xj(y) J(xy) for all x, y R.

14 Equivalent conditions 14 Let (R, D) be a differential algebra of weight λ with a linear operator Π on R such that D Π = id R. Denote J = Π D, called the initialization, and E = id R J, called the evaluation. Then the following statements are equivalent: 1. (R, D, Π) is an integro-differential algebra; 2. E(xy) = E(x)E(y) for all x, y R; 3. ker E = imj is an ideal; 4. J(xJ(y)) = xj(y) and J(J(x)y) = J(x)y for all x, y R; 5. J(xΠ(y)) = xπ(y) and J(Π(x)y) = Π(x)y for all x, y R; 6. xπ(y) = Π(D(x)Π(y))+Π(xy)+λΠ(D(x)y) and Π(x)y = Π(Π(x)D(y))+Π(xy)+λΠ(xD(y)) for all x, y R; 7. (R, D, Π) is a differential Rota-Baxter algebra and Π(E(x)y) = E(x)Π(y) and Π(xE(y)) = Π(x)E(y) for all x, y R; 8. (R, D, Π) is a differential Rota-Baxter algebra and J(E(x)J(y)) = E(x)J(y) and J(J(x)E(y)) = J(x)E(y) for all x, y R.

15 Integral by parts revisited Working in the free differential Rota-Baxter algebra X(A) where (A, d) is a differential algebra, this means that d(x) should not appear in Π. More precisely, in a = a 0 a 1 a n, a 1,, a n 1 A should be in complement of d(a), i.e., in A T such that A = imd A T. Such an A is called regular. 15 (R, D,Π) is an integro-differential algebra if and only if (R, D) is a differential algebra, D Π = id R and Π(D(x)Π(y)) xπ(y)+π(xy)+λπ(d(x)y) = 0, x, y R. Theorem Let (A, D) be a differential algebra. Let I ID be the differential Rota-Baxter ideal of X(A) generated by elements in the above equations. Then the quotient differential Rota-Baxter algebra X(A)/I ID is the free integro-differential algebra on (A, D). The last equation suggests the rewriting rule Π(D(x)Π(y)) ID xπ(y) Π(xy) λπ(d(x)y).

16 Example of regular differential algebras 16 Let A = k{u} = k[u i, i 0] be the ring of differential polynomials in one variable. Then a monomial of A is of the form u i := u i 0 0 u i k k with i := (i0,, i k ), i j 0, 0 j k, i k 0, k 0. Such a monomial is said to have order k. Also say u () := 1 to have order 1. A monomial u i is called functional if either k 0 or i k > 1. Then the linear span A T is a linear complement of ker d. Let X be a well ordered set. Let u C( X) be in the form u = u j 0 0 u j k k, where u 0,, u k X, u 0 > > u k and j 0,, j k Z 1 Call u functional if either u {1}, or u k X, or j k > 1. Let A = k[ X] and A T be the linear span of the functional monomials. Then k[ X] = A T imd. Thus k{x} is a regular differential algebra.

17 Construction of free integro-differential algebras Let (A, d) be a regular differential algebra. Thus A = imd A T and A T is a nonunitary subalgebra of A when λ 0. Let n X T (A) = n 0 A A n T = A (A A T ) be the k-submodule of X(A). Since A T is assumed to be a nonunitary subalgebra if λ 0, X T (A) is a subalgebra of X(A). Let A ε := {ε(a) a A} be another copy of the algebra A, but with the zero derivation. Then both A and A ε are K -algebras for K := ker d. Define ID(A) := A ε K X T (A) = A ε K (A X + (A T )) to be the tensor product algebra. Define d A on ID(A) by the product rule on the tensor product. For a A, define P A (a) = φ(a) ε(φ(a))+1 ψ(a). For a := a 0 a n A (A ψ ) n, write a = a 0 a,a A n ψ. Define P A (a) = φ(a 0 ) a P A (φ(a 0 )a)+1 ψ(a 0 ) a. Theorem (Guo-Regensburger-Rosenkranz) The triple (ID(A), d u, P u ) is the free commutative integro-differential algebra on (A, d). 17

18 Normal forms of integro-differential algebras 18 Back to the rewriting rule Π(D(x)Π(y)) ID xπ(y) Π(xy) λπ(d(x)y). Then in the free differential Rota-Baxter algebra X(k{X}). Elements of the form d(x) should not appear in Π( pπ(v)). So for a = a 0 a 1 a n to be normal, should have a i A T, 1 i n 1. This is quite hard to verify directly.

19 of the inclusion and the quotient map is a linear bijection. Thus X(A) f gives an explicit construction of the free integro-differential algebra X(A)/I ID. 19 Free integro-differential algebras by normal forms By the method of Gröbner-Shirshov basis, we obtain. Theorem(Gao-Guo-Zheng) Let X be a nonempty well-ordered set and A := k{x}. Let X(k{X}) = X(k[ X]), with the derivation d and Rota-Baxter operator P, be the free commutative differential Rota-Baxter algebra of weight λ on X. Let I ID be the differential Rota-Baxter ideal of X(k{X}) generated by S := {P(d(u)P(v)) up(v)+p(uv)+λp(d(u)v) u, v X(k{X})}. Let A f be the submodule of A = k{x} spanned by functional monomials. Then the composition X(A) f := A k 0 A A k f A X(A) X(A)/I ID

20 Comparison 20 Compare the two constructions: ID(A) := A ε K X T (A) = A ε K (A X + (A T )) = (A ε K A) (A ε A A T ) (A ε A A T A T ). X(A) f := A k 0 A A k f A. = A (A A) (A A T A).

21 21 References: * X. Gao, L. Guo and S. Zheng, Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases, preprint. * L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. Math 150 (2000), * L. Guo and W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra 212 (2008), * L. Guo, G. Regensburger and M. Rosenkranz, On integro-differential algebras, to appear in J. Pure Appl. Algebra. * M. Rosenkranz and G. Regensburger, Solving and factoring boundary problems for linear ordinary differential equations in differential algebra, J. Symbolic Comput. 43 (2008), Thank You!