Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.00554 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929230895382528 |
|---|---|
| author | Wolff, Vincent |
| author_facet | Wolff, Vincent |
| contents | We study the deformation complex of a canonical morphism $i$ from the properad of (degree shifted) Lie bialgebras $\mathbf{Lieb}_{c,d}$ to its polydifferential version $\mathcal{D}(\mathbf{Lieb}_{c,d})$ and show that it is quasi-isomorphic to the oriented graph complex $\mathbf{GC}^{\text{or}}_{c+d+1}$, up to one rescaling class. As the latter complex is quasi-isomorphic to the original graph complex $\mathbf{GC}_{c+d}$, we conclude that the space of homotopy non-trivial infinitesimal deformations of the canonical map $i$ can be identified with the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$; moreover, every such an infinitesimal deformation extends to a genuine deformation of the canonical morphism $i$ from $\mathbf{Lieb}_{c,d}$ to $\mathcal{D}(\mathbf{Lieb}_{c,d})$. The full deformation complex is described with the help of a new graph complex of so called entangled graphs, whose suitable quotient complex is shown to contain the tensor product $H(\mathbf{GC}_c) \otimes H(\mathbf{GC}_d)$ of cohomologies of Kontsevich graph complexes $\mathbf{GC}_c \otimes \mathbf{GC}_d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00554 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polydifferential Lie bialgebras and graph complexes Wolff, Vincent Quantum Algebra We study the deformation complex of a canonical morphism $i$ from the properad of (degree shifted) Lie bialgebras $\mathbf{Lieb}_{c,d}$ to its polydifferential version $\mathcal{D}(\mathbf{Lieb}_{c,d})$ and show that it is quasi-isomorphic to the oriented graph complex $\mathbf{GC}^{\text{or}}_{c+d+1}$, up to one rescaling class. As the latter complex is quasi-isomorphic to the original graph complex $\mathbf{GC}_{c+d}$, we conclude that the space of homotopy non-trivial infinitesimal deformations of the canonical map $i$ can be identified with the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$; moreover, every such an infinitesimal deformation extends to a genuine deformation of the canonical morphism $i$ from $\mathbf{Lieb}_{c,d}$ to $\mathcal{D}(\mathbf{Lieb}_{c,d})$. The full deformation complex is described with the help of a new graph complex of so called entangled graphs, whose suitable quotient complex is shown to contain the tensor product $H(\mathbf{GC}_c) \otimes H(\mathbf{GC}_d)$ of cohomologies of Kontsevich graph complexes $\mathbf{GC}_c \otimes \mathbf{GC}_d$. |
| title | Polydifferential Lie bialgebras and graph complexes |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2402.00554 |