Planar Shuffle Product, Co-Addition and the non-associative Exponential

Commutative modified Rota-Baxter algebras, shuffle products and Hopf algebras

In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.

Planar Prop of Differential Operators

We propose a definition of differential operators of an associative algebra $A$ in the spirit of Hochschild cohomology. Specifically we define $D(A)$ as the zero cohomology of a certain bicomplex formed by Hom-spaces $\mathrm{Hom}(A^{\otimes q}, A^{\otimes p})$. We show that it has a structure of a planar prop, i.e. each differential operator has multiple inputs and outputs and they can be composed along planar graphs. Furthermore, for a formally smooth algebra we have the surjective symbol map from $D(A)$ to the space of poly-derivations. We also consider another planar prop $E(A)$ generated …

兼容的结合积与树

Compatible associative products and trees

研究自由代数中两个兼容结合积的分级组件维度，并通过平面根树解释这些代数结构。

Mersenne Binomials and the Coefficients of the non-associative Exponential

The non-associative exponential series $exp(x)$ is a power series with monomials from the magma $M$ of finite, planar rooted trees. The coefficient $a(t)$ of $exp(x)$ relative to a tree $t$ of degree $n$ is a rational number and it is shown that $$\hat{a}(t) := \frac{a(t)}{2^{n-1}\cdot \prod^{n-1}_{i=1}(2^i - 1)}$$ is an integer which is a product of Mersenne binomials. One obtains summation formulas $$\sum \hat{a}(t) = ω(n)$$ where the sum is extended over all trees $t$ in $M$ of degree $n$ and $$ω(n) = \frac{2^{n-1}}{n!} \prod^{n-1}_{i=1} (2^i - 1).$$ The prime factorization of $ω(n)$ is des…

The planar Chain Rule and the Differential Equation for the planar Logarithms

A planar monomial is by definition an isomorphism class of a finite, planar, reduced rooted tree. If $x$ denotes the tree with a single vertex, any planar monomial is a non-associative product in $x$ relative to $m-$array grafting. A planar power series $f(x)$ over a field $K$ in $x$ is an infinite sum of $K-$multiples of planar monomials including the unit monomial represented by the empty tree. For every planar power series $f(x)$ there is a universal differential $d f(x)$ which is a planar power series in $x$ and a planar polynomial in a variable $y$ which is the differential $d x$ of $x$. …

Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals

This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with exponentially bounded nonnegative integer coefficients. Then we summarize some related facts on algebras with polynomial identity. Further we discuss the problem how to find series $a(t)$ which are rational/algebraic/transcendental over ${\mathbb Q}(t)$. Applying a classical result of Fatou we conclude that if a finitely generated graded algebra has a finite G…