Séminaire Lotharingien de Combinatoire 57 (2007), Article B57b

AN IDENTITY IN ROTA–BAXTER ALGEBRAS

RAFAEL DÍAZ AND MARCELO PÁEZ

Abstract. We give explicit formulae and study the combinatorics of an identity holding in all Rota–Baxter algebras. We describe the specialization of this identity for a couple of examples of Rota–Baxter algebras.

1. Introduction

The study of Rota–Baxter algebras was initiated by Baxter in his works [2, 3]. The theory was later taken over by Rota [11], who gave an explicit construction of free Rota– Baxter algebras and uncovered the relationship with symmetric functions. Soon after, Cartier studied free Rota–Baxter algebras in [4]. In the last few years the theory of Rota–Baxter algebras has received a great impulse, mainly because of its applications to renormalization, as formalized by Connes and Kreimer in [5, 6, 7]. New techniques and applications of Rota–Baxter algebras have been found by an active group of researches in a number of important works, among which we cite just a few [1, 8, 9, 10].

In this work we consider the seemingly naive problem of writing an element of the form P^{a}(x)P^{b}(y) in a Rota–Baxter algebra as a linear combination of terms of the form P^{j}(xP^{i}(y)) and P^{j}(P^{i}(x)y), with i and j varying. The existence of such a linear combination is an immediate consequence of the Rota–Baxter identity satisfied by the operator P. The actual problem is to determine the coefficients involved in such an expression as explicitly as possible. We provide a solution to this problem, and take a look at its meaning in a couple of Rota–Baxter algebras. We approach our problem from a rather pedestrian point of view using a graphical notation to illustrate our ideas.

2. Basic ideas

Let us fix a field k of characteristic zero. A Rota–Baxter algebra is a triple (A, λ, P) where A is an associative k-algebra, λ is a constant in k, and P : A ⟶ A is a k-linear operator satisfying the identity

P (x)P (y) = P (xP (y)) + P (P (x)y) + λP (xy)

for x, y ∈ A. We find it convenient to use a graphical notation to express our results.

2 RAFAEL DÍAZ AND MARCELO PÁEZ

We represent the product on A by and the Rota–Baxter operator by The Rota–Baxter identity satisfied by P is represented graphically by For example using the graphical form of the Rota–Baxter identity one can see that A further application of the graphical Rota–Baxter identity yields Thus we have shown that the following identity holds in any Rota–Baxter algebra:

P^{2}(x)P (y) = P^{2}(xP (y)) + P^{2}(P (x)y) + λP^{2}(xy) + P (P^{2}(x)y) + λP (P (x)y). The symbol T (a, b, c) has two different meanings in this work:

• On the one hand it stands for the operator P^{c}(m ∘ (P^{a} ⊗ P^{b})) : A ⊗ A ⟶ A where m denotes the product on A.

• On the other hand it represents the tree with a dots on the left leg, b dots on the

right leg, and c dots on the neck. The tree T (a, b, c) is drawn as follows For example the tree T (1, 2, 3) is represented graphically as follows:

It is clear from the graphical Rota–Baxter identity that each tree

can be written as a linear combination with coefficients in k[λ] of trees of the form

Indeed each application of the graphical Rota–Baxter relation replaces the tree

by the sum of trees

From an algorithmic point of view the graphical Rota–Baxter identity can be described as the application of three weighted moves:

(1) A weight 1 move where a dot from the left leg moves up.

(2) A weight 1 move where a dot from the right leg moves up.

(3) A weight λ move where a couple of dots, one from the left leg and another one from the right leg, merge and move up as one dot.

We are ready to formulate our main results.

3. Restricted case λ = 0

The case of Rota–Baxter algebras with λ = 0 simplifies considerably. We report on this special case because of its applications and elegant proof.

Theorem 1. Let a, b > 1 and c ≥ 0 be integers. The following identity holds in any Rota–Baxter algebra:

T (a, b, c) = ∑_{i=1}^{b} \binom{a − 1 + b − i}{a − 1}T (0, i, a+b+c−i)+∑_{i=1}^{a} \binom{b − 1 + a − i}{b − 1}T (i, 0, a+b+c−i).

4 RAFAEL DÍAZ AND MARCELO PÁEZ

Proof. We justify only the left summand, the right summand is justified in an analogous way. For λ = 0 only move 1 and move 2 are allowed. With each move a dot from one of the legs moves up. Suppose that after applying several times the Rota–Baxter identity to T (a, b, c) we arrive at a tree of the form T (0, i, j). Then a total of a + b − i dots from the legs have moved up so j = a + b + c − i. Necessarily the last dot moving up comes from the left leg. The other dots moved up in an arbitrary order, so this explain the factor

\binom{a − 1 + b − i}{a − 1}.

□

Consider the Rota–Baxter algebra (C(ℝ), 0, P), where C(ℝ) denotes the algebra of continuous functions on ℝ and P is the Riemann integral operator given by

P (f)(y) = ∫_{0}^{y} f(x)dx.

For real numbers 0 ≤ x ≤ y and a ∈ ℕ_{+}, let Δ_{x,a}^{y} be the convex polytope

Δ_{x,a}^{y} = {(x_{1}, ..., x_{a}) ∈ (ℝ_{≥0})^{n} | x ≤ x_{1} ≤ ... ≤ x_{a} ≤ y}.

For a ≥ 1 we let v_{a}(x, y) be the volume of Δ_{x,a}^{y}. By convention we set v_{0}(x, y) = 1. It is easy to check that

P^{a+1}(f)(y) = ∫_{0}^{y} f(x)v_{a}(x, y)dx.

Theorem 1 implies the following result.

Theorem 2.

∫_{0}^{z} ∫_{0}^{z}f(x)g(y)v_{a}(x, z)v_{b}(y, z)dxdy

= ∑_{i=1}^{b+1} \binom{a + b + 1 − i}{a} ∫_{0≤x≤y≤z} g(x)f(y)v_{i−1}(x, y)v_{a+b+1−i}(y, z)dxdy

+ ∑_{i=1}^{a+1} \binom{b + a + 1 − i}{b} ∫_{0≤x≤y≤z} f(x)g(y)v_{i−1}(x, y)v_{b+a+1−i}(y, z)dxdy.

4. Generic case

Now we consider the generic situation, i.e., a Rota–Baxter algebra with λ ≠ 0.

Theorem 3. Let a, b ≥ 1 and c ≥ 0 be integers. The following identity holds in any Rota–Baxter algebra:

T (a, b, c) = ∑_{(i,j)∈D_{1}} c_{1}(a, b; i, j)T (0, i, c + j)

+ ∑_{(i,j)∈D_{2}} c_{2}(a, b; i, j)T (0, i, c + j)

+ ∑_{(i,j)∈D_{3}} c_{2}(a, b; i, j)T (i, 0, c + j)

+ ∑_{(i,j)∈D_{4}} c_{4}(a, b; i, j)T (i, 0, c + j)

+ ∑_{j∈D_{5}} c_{5}(a, b; j)T (0, 0, c + j)

where

D_{1} = {(i, j) ∈ ℕ_{+} × ℕ_{+} | 1 ≤ i ≤ b, a ≤ j, b − i + 1 ≤ j, j ≤ a + b − i},
D_{2} = {(i, j) ∈ ℕ_{+} × ℕ_{+} | 1 ≤ i ≤ b − 1, a ≤ j, b ≤ j, j ≤ a + b − 1},
D_{3} = {(i, j) ∈ ℕ_{+} × ℕ_{+} | 1 ≤ i ≤ a, a − i + 1 ≤ j, b ≤ j, j ≤ a + b − i},
D_{4} = {(i, j) ∈ ℕ_{+} × ℕ_{+} | 1 ≤ i ≤ a − 1, a − i ≤ j, b ≤ j, j ≤ a + b − i − 1},
D_{5} = {j ∈ ℕ | a ≤ j, b ≤ j, j ≤ a + b − 1},

and

c_{1}(a, b; i, j) = \binom{j − 1}{i + j − b − 1, j − a, a + b − i − j}λ^{a+b−i−j},
c_{2}(a, b; i, j) = \binom{j − 1}{j − b, j − a, a + b − j − 1}λ^{a+b−j},
c_{3}(a, b; i, j) = \binom{j − 1}{j − b, i + j − a − 1, a + b − i − j}λ^{a+b−i−j},
c_{4}(a, b; i, j) = \binom{j − 1}{j − b, i + j − a, a + b − i − j − 1}λ^{a+b−i−j},
c_{5}(a, b; j) = \binom{j − 1}{j − b, j − a, a + b − j − 1}λ^{a+b−j}.

Proof. Notice that with each move a dot is added to the neck. If starting from the tree T (a, b, c) we arrive using the allowed moves to the graph T (0, i, c + j), then necessarily we must have applied j moves and the last move must have been either move 1 or move 3. Let us consider the case were the last move is of type 1. The other j − 1 moves are distributed into k_{1} moves of type 1, k_{2} moves of type 2, and k_{3} moves of type 3, giving

6 RAFAEL DÍAZ AND MARCELO PÁEZ

rise to the combinatorial number

\binom{j − 1}{k_{1}, k_{2}, k_{3}}.

The numbers k_{1}, k_{2} and k_{3} are subject to the constraints

k_{1} + k_{2} + k_{3} = j − 1, k_{1} + k_{3} = a − 1 and k_{2} + k_{3} = b − i − 1.

Solving this linear system of equations we find that

k_{1} = i + j − b − 1, k_{2} = j − a and k_{3} = a + b − i − j.

This justifies the expression for c_{1}(a, b, c; i, j) from the statement of the Theorem. We proceed to justify the expression for c_{2}(a, b, c; i, j) which arises when the last move taken in the path towards T (0, i, j) is of type 3. The remaining new j − 1 dots in the neck move up as consequence of the application of any of the moves, giving rise to the factor \binom{j−1}{k_{1},k_{2},k_{3}} where k_{1}, k_{2} and k_{3} are subject to the constraints

k_{1} + k_{2} + k_{3} = j − 1, k_{1} + k_{3} = a − 1, k_{2} + k_{3} = b − i − 1.

Solving this equations we find that

k_{1} = i + j − b, k_{2} = j − a, k_{3} = a + b − i − j − 1.

Thus we have justified the factor

\binom{j − 1}{j − b, j − a, a + b − i − j − 1}λ^{a+b−i−j−1}

appearing in the formula for c_{2}(a, b, c; i, j). The formulas for c_{3}(a, b, c; i, j) and c_{4}(a, b, c; i, j) are derived in a fairly similar way. Let us consider the formula for c_{5}(a, b, c; j). In this case the last move is necessarily of type 3 and gives rise to the factor \binom{j−1}{k_{1},k_{2},k_{3}}, where k_{1}, k_{2} and k_{3} satisfy the constraints

k_{1} + k_{2} + k_{3} = j − 1, k_{1} + k_{3} = a − 1, k_{2} + k_{3} = b − 1.

We find that

k_{1} = j − b, k_{2} = j − a, k_{3} = a + b − j − 1,

which justifies the factor

\binom{j − 1}{j − b, j − a, a + b − j − 1}λ^{a+b−j}

appearing in the formula for c_{5}(a, b, c; j).

□

Consider the Rota–Baxter algebra ({f : ℕ ⟶ ℂ}, −1, P), where the operator

P : {f : ℕ ⟶ ℂ} ⟶ {f : ℕ ⟶ ℂ}

is given by the Riemann sum

P (f)(m) = ∑_{n=1}^{m} f(n).

For a ≥ 1 one can check that

P^{a}f(m) = ∑_{1≤n_{1}≤...≤n_{a}≤m} f(n_{1}).

In particular,

P^{a}1(m) = |Ω_{a}^{m}|,

where

Ω_{a}^{m} = {(n_{1}, ..., n_{a}) ∈ ℕ_{+} | 1 ≤ n_{1} ≤ ... ≤ n_{a} ≤ m}.

It is not hard to show using the Chu–Vandermonde identity that

|Ω_{a}^{m}| = ∑_{s=1}^{m} \binom{m}{s}\binom{a}{s} = \binom{a + m}{m} − 1.

As a consequence of Theorem 3 we get that the numbers |Ω_{a}^{m}| satisfy the following identity.

Theorem 4. For integers a, b ≥ 1, we have

|Ω_{a}^{m}||Ω_{b}^{m}| = ∑_{(i,j)∈D_{1}}[c_{1}(a, b; i, j)(−1)]|Ω_{i+j}^{m}|

+ ∑_{(i,j)∈D_{2}}[c_{2}(a, b; i, j)(−1)]|Ω_{i+j}^{m}|

+ ∑_{(i,j)∈D_{3}}[c_{2}(a, b; i, j)(−1)]|Ω_{i+j}^{m}|

+ ∑_{(i,j)∈D_{4}}[c_{4}(a, b; i, j)(−1)]|Ω_{i+j}^{m}|

+ ∑_{j∈D_{5}}[c_{5}(a, b; j)(−1)]|Ω_{j}^{m}|.

5. Acknowledgment

We thank Dominique Manchon for helpful comments and suggestions. Thanks also to Takasi Kimura, José Mijares, Eddy Pariguan and Sylvie Paycha.

References

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[4] P. Cartier, On the structure of Baxter algebras, Adv. in Math. 9 (1972) 253-265.

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[8] K. Ebrahimi-Fard, L. Guo, On free Rota–Baxter algebras, arχiv:math.NT/0510266.

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[10] D. Manchon, S. Paycha, Chen sums of symbols and renormalized multiple zeta functions,

   arχiv:math.NT/0702.135.

[11] G.C. Rota, Gian-Carlo Rota on Combinatorics, J. Kung (ed.), Birkhäuser, Boston and Basel, 1995.

Escuela de Matemáticas, Universidad Central de Venezuela, Caracas 1020, Venezuela.

ragadiaz@gmail.com, mathmarcelo@gmail.com.