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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2401.17771 |
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| _version_ | 1866917791296126976 |
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| author | Umble, Ronald |
| author_facet | Umble, Ronald |
| contents | A *Gerstenhaber-Schack (G-S) bialgebra* consists of a graded Hopf algebra $H$ together with multilinear operations $\{ω^1_3,ω^2_2,ω^3_1\}\subset \{Hom^{-1}(H^{\otimes m},H^{\otimes n}): m+n=4\},$ whose sum is the degree $-1$ component of a $2$-cocycle in the G-S complex of $H$. A *G-S extension* of a graded Hopf algebra $H$ is a G-S bialgebra containing $H$. G-S extensions of $H$ are classified up to isomorphism by the degree $-1$ component of the G-S cohomology group $H_{GS}^{2}(H;H)$. We exhibit a space $X$ and a non-trivial topologically induced G-S bialgebra structure on $H^{\ast}\left( ΩX;\mathbb{Z}_{2}\right) .$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gerstenhaber-Schack Bialgebras Umble, Ronald Algebraic Topology Rings and Algebras 16S80, 16T10, 32G99, 55P35, 55P48, 52B05, 52B11 A *Gerstenhaber-Schack (G-S) bialgebra* consists of a graded Hopf algebra $H$ together with multilinear operations $\{ω^1_3,ω^2_2,ω^3_1\}\subset \{Hom^{-1}(H^{\otimes m},H^{\otimes n}): m+n=4\},$ whose sum is the degree $-1$ component of a $2$-cocycle in the G-S complex of $H$. A *G-S extension* of a graded Hopf algebra $H$ is a G-S bialgebra containing $H$. G-S extensions of $H$ are classified up to isomorphism by the degree $-1$ component of the G-S cohomology group $H_{GS}^{2}(H;H)$. We exhibit a space $X$ and a non-trivial topologically induced G-S bialgebra structure on $H^{\ast}\left( ΩX;\mathbb{Z}_{2}\right) .$ |
| title | Gerstenhaber-Schack Bialgebras |
| topic | Algebraic Topology Rings and Algebras 16S80, 16T10, 32G99, 55P35, 55P48, 52B05, 52B11 |
| url | https://arxiv.org/abs/2401.17771 |