The Weil algebra of a double Lie algebroid
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866916496829054976 |
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| author | Meinrenken, Eckhard Pike, Jeffrey |
| author_facet | Meinrenken, Eckhard Pike, Jeffrey |
| contents | Given a double vector bundle $D\to M$, we define a bigraded `Weil algebra' $\mathcal{W}(D)$, which `realizes' the algebra of smooth functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebras of $D$ and those of the double vector bundles $D',\ D"$ obtained by duality operations. In particular, we show that double-linear Poisson structures on $D$ can be described alternatively as Gerstenhaber brackets on $\mathcal{W}(D)$, vertical differentials on $\mathcal{W}(D')$, or horizontal differentials on $\mathcal{W}(D")$. We also give a new proof of Voronov's result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that $\mathcal{W}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad-Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multivector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_00230 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The Weil algebra of a double Lie algebroid Meinrenken, Eckhard Pike, Jeffrey Differential Geometry Given a double vector bundle $D\to M$, we define a bigraded `Weil algebra' $\mathcal{W}(D)$, which `realizes' the algebra of smooth functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebras of $D$ and those of the double vector bundles $D',\ D"$ obtained by duality operations. In particular, we show that double-linear Poisson structures on $D$ can be described alternatively as Gerstenhaber brackets on $\mathcal{W}(D)$, vertical differentials on $\mathcal{W}(D')$, or horizontal differentials on $\mathcal{W}(D")$. We also give a new proof of Voronov's result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that $\mathcal{W}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad-Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multivector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras. |
| title | The Weil algebra of a double Lie algebroid |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1901.00230 |