2026/07/19 by Alexandru Chirvasitu
Mathematics · #math.AG #math.GR #math.RT #math.SG
Consider a reductive complex algebraic group \mathbbG equipped with an action by a linearly reductive affine group scheme \mathbbK. The extension of \mathfrakp^* by \mathfrakp induced by an (\mathbbK,\mathfrakg)-invariant symmetric non-degenerate bilinear form on \mathfrakg:=Lie(\mathbbG), for a \mathbbK-invariant parabolic ℙ≤ \mathbbG, is \mathbbK-equivariantly isomorphic to the extension obtained via the standard bialgebra structure attached to a \mathbbK-invariant Cartan/Borel pair ℍ≤ \mathbbB≤ ℙ≤ \mathbbG and the same bilinear form. Associating bundle extensions on an elliptic curve E to said \mathfrakp-module extensions, this identifies Poisson structures on the smooth locus of the principal-ℙ-bundle moduli space over E respectively defined by Balduzzi (using the former extension) and Feigin-Odesskii (via the standard bialgebra structure). This in particular verifies Feigin-Odesskii's identification of the bialgebra-induced symplectic leaves with the loci of bundles mutually isomorphic after forgetting structure along ℙ≤ \mathbbG.