Cohomology of perfect Lie algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917845203419136 |
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| author | Burde, Dietrich Wagemann, Friedrich |
| author_facet | Burde, Dietrich Wagemann, Friedrich |
| contents | We study the adjoint cohomology of perfect Lie algebras over the complex numbers. For the family of perfect Lie algebras $\mathfrak{g}=\mathfrak{sl}_2(\Bbb C)\ltimes V_m$ we obtain some explicit results for $H^k(\mathfrak{g},\mathfrak{g})$ with $k\ge 0$. Here $V_m$ is the irreducible representation of $\mathfrak{sl}_2(\Bbb C)$ of dimension $m+1$. For the computation of the cohomology we use the Hochschild-Serre formula, a long exact sequence in the cohomology and explicit formulas for the multiplicities of $V_k$ in the exterior product $Λ^j(V_m)$ for $j\le 4$. In general we cannot determine the total adjoint cohomology for $\mathfrak{sl}_2(\Bbb C)\ltimes V_m$, but for some small $m$ this is possible. We also give a classification of complex perfect Lie algebras $\mathfrak{g}$ of dimension $n\le 9$ and explicitly compute the cohomology spaces $H^k(\mathfrak{g},\mathfrak{g})$ with $k=0,1,2$ for all Lie algebras from the classification list. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14952 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomology of perfect Lie algebras Burde, Dietrich Wagemann, Friedrich Representation Theory 17B10, 17B56 We study the adjoint cohomology of perfect Lie algebras over the complex numbers. For the family of perfect Lie algebras $\mathfrak{g}=\mathfrak{sl}_2(\Bbb C)\ltimes V_m$ we obtain some explicit results for $H^k(\mathfrak{g},\mathfrak{g})$ with $k\ge 0$. Here $V_m$ is the irreducible representation of $\mathfrak{sl}_2(\Bbb C)$ of dimension $m+1$. For the computation of the cohomology we use the Hochschild-Serre formula, a long exact sequence in the cohomology and explicit formulas for the multiplicities of $V_k$ in the exterior product $Λ^j(V_m)$ for $j\le 4$. In general we cannot determine the total adjoint cohomology for $\mathfrak{sl}_2(\Bbb C)\ltimes V_m$, but for some small $m$ this is possible. We also give a classification of complex perfect Lie algebras $\mathfrak{g}$ of dimension $n\le 9$ and explicitly compute the cohomology spaces $H^k(\mathfrak{g},\mathfrak{g})$ with $k=0,1,2$ for all Lie algebras from the classification list. |
| title | Cohomology of perfect Lie algebras |
| topic | Representation Theory 17B10, 17B56 |
| url | https://arxiv.org/abs/2411.14952 |