Global splittings and super Harish-Chandra pairs for affine supergroups (Q2790614)

Let \(k\) be a commutative unital ring. A \(k\)-supermodule \( \mathfrak g =\mathfrak g_0\oplus \mathfrak g_1\) is a Lie superalgebra with bilinear multiplication \([x,y]\) such that standard skew-symmetry and Jacobi identities hold. Moreover \([w,w]= [z,[z,z]]=0\) for all even \(a\) and for all add \(z\). Also there exists a unary quadratic operation \(z^{(2)}\) mapping \(\mathfrak g_1\to \mathfrak g_0 \) such that NEWLINE\[NEWLINE (z_1+z_2)^{(2)} = z_1^{(2)} + z_2^{(2)} + [z_1,z_2],\quad [z^{(2)}x]=[z,[z,x]]. NEWLINE\]NEWLINE For a affine supergroup \(G\) one can consider representing commutative Hopf \(k\)-superalgebra \(H=H_0\oplus H_1\). It is shown that in most cases the following property of being strongly split holds, namely, \(W^H=H_1/H_0^+H_1\) is \(k\)-free and there exists an isomorphism NEWLINE\[NEWLINE H\to (H/HH_1H)\otimes_k \left(\wedge W^H\right)NEWLINE\]NEWLINE of super counital \(\left (H/HH_1H\right)\)-comodule algebras. A Harish-Chandra pair over \(k\) consists of an affine \(k\)-group scheme \(G_+\) and a Lie superalgebra \(\mathfrak g\) where \(\mathfrak G_1\) is a free \(k\)-module of a finite rank. Moreover the Lie algebra of \( G_+\) is equal to \(\mathfrak g_0\) and there an action of \(G_+\) on \(\mathfrak g_0\) is extended to an action on \(\mathfrak g\). In a standard way with super Hopf algebra \(H\) one can associate super Lie algebra \(\text{Lie}(G)\). There exists a functor \(\Phi\) from the category of supergroups in which \(\mathfrak g_1\) for its Lie algebra is a free module of a finite rank to the category of Harish-Chandra pairs, namely \(\Phi(G)= (G_0, \text{Lie} G)\). One of the main results show that \(\Phi\) has an inverse functor and there is given an explicit for of this inverse.