On super Plücker embedding and cluster algebras
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Ekaterina Shemyakova [1] ; Theodore Voronov [2]
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[1] University of Toledo
University of Toledo
City of Toledo, Estados Unidos
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[2] University of Manchester, Reino Unido; Tomsk State University, Rusia
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 2, 2022
- Idioma: inglés
- DOI: 10.1007/s00029-021-00756-w
- Enlaces
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Resumen
We define a super analog of the classical Plücker embedding of the Grassmannian into a projective space. One of the difficulties of the problem is rooted in the fact that super exterior powers Λr|s(V) are not a simple generalization from the completely even case (this works only for r|0 when it is possible to use Λr(V)). To construct the embedding we need to non-trivially combine a super vector space V and its parity-reversion ΠV. Our “super Plücker map” takes the Grassmann supermanifold Gr|s(V) to a “weighted projective space” P1,−1(Λr|s(V)⊕Λs|r(ΠV)) with weights +1,−1. A simpler map Gr|0(V)→P(Λr(V)) works for the case s=0. We construct a super analog of Plücker coordinates, prove that our map is an embedding, and obtain “super Plücker relations”. We analyze another type of relations (due to Khudaverdian) and show their equivalence with the super Plücker relations for r|s=2|0. We discuss application to much sought-after super cluster algebras and construct a super cluster structure for G2(R4|1) and G2(R5|1).
