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arXiv · 2109.14070

Demi-shuffle duals of Magnus polynomials in a free associative algebra

Hiroaki Nakamura
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Abstract

We study two linear bases of the free associative algebra $\mathbb{Z}\langle X,Y\rangle$: one is formed by the Magnus polynomials of type $(\mathrm{ad}_X^{k_1}Y)\cdots(\mathrm{ad}_X^{k_d}Y) X^k$ and the other is its dual basis (formed by what we call the `demi-shuffle' polynomials) with respect to the standard pairing on the monomials of $\mathbb{Z}\langle X,Y\rangle$. As an application, we show a formula of Le-Murakami, Furusho type that expresses arbitrary coefficients of a group-like series $J\in \mathbb{C}\langle\langle X,Y\rangle\rangle$ by the `regular' coefficients of $J$.

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Hiroaki Nakamura. 2021-09-28. Demi-shuffle duals of Magnus polynomials in a free associative algebra. https://arxiv.org/abs/2109.14070

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