Hopf 2-cocycles for certain affine algebraic reductive groups

2026/07/21 by Shlomo Gelaki
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Abstract

Motivated by the open problem of classifying Hopf 2-cocycles for affine algebraic reductive groups G over ℂ, in particular by \cite[Question 7.2]EG1 which concerns minimal Hopf 2-cocycles, we classify (minimal) Hopf 2-cocycles for affine algebraic reductive groups G whose connected component of the identity is a torus T. To achieve this we utilize \cite[Proposition 5.4]ENO in our situation to classify finite indecomposable semisimple module categories over the infinite semisimple equivariantization tensor category \Rep(T)^K≃ \Rep(G), where K:=G/T. We then use it to classify the module categories over \Rep(G) of rank 1, and show that the corresponding fiber functors on \Rep(G) are classical (that is, preserve dimensions), which implies that they are in bijection with Hopf 2-cocycles for G.

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