On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866910022353551360 |
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| author | Furusho, Hidekazu Komiyama, Nao Raphael, Elise Schneps, Leila |
| author_facet | Furusho, Hidekazu Komiyama, Nao Raphael, Elise Schneps, Leila |
| contents | The goal of this article is to define a linearized or depth-graded version $\mathfrak{lkv}$, and a closely related elliptic version $\mathfrak{krv}_{ell}$, of the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara-Vergne problem. We show how the elliptic Lie algebra $\mathfrak{krv}_{ell}$ is related to earlier constructions of elliptic versions $\mathfrak{grt}_{ell}$ and $\mathfrak{ds}_{ell}$ of the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$ and the double shuffle Lie algebra $\mathfrak{ds}$. In particular we show that there is an injective Lie morphism $\mathfrak{ds}_{ell}\hookrightarrow \mathfrak{krv}_{ell}$, and an injective Lie algebra morphism $\mathfrak{krv}\rightarrow \mathfrak{krv}_{ell}$ extending the known morphisms $\mathfrak{grt}\hookrightarrow\mathfrak{grt}_{ell}$ (Enriquez section) and $\mathfrak{ds}\rightarrow\mathfrak{ds}_{ell}$ (Écalle map). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1706_08299 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra Furusho, Hidekazu Komiyama, Nao Raphael, Elise Schneps, Leila Quantum Algebra The goal of this article is to define a linearized or depth-graded version $\mathfrak{lkv}$, and a closely related elliptic version $\mathfrak{krv}_{ell}$, of the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara-Vergne problem. We show how the elliptic Lie algebra $\mathfrak{krv}_{ell}$ is related to earlier constructions of elliptic versions $\mathfrak{grt}_{ell}$ and $\mathfrak{ds}_{ell}$ of the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$ and the double shuffle Lie algebra $\mathfrak{ds}$. In particular we show that there is an injective Lie morphism $\mathfrak{ds}_{ell}\hookrightarrow \mathfrak{krv}_{ell}$, and an injective Lie algebra morphism $\mathfrak{krv}\rightarrow \mathfrak{krv}_{ell}$ extending the known morphisms $\mathfrak{grt}\hookrightarrow\mathfrak{grt}_{ell}$ (Enriquez section) and $\mathfrak{ds}\rightarrow\mathfrak{ds}_{ell}$ (Écalle map). |
| title | On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/1706.08299 |