Hopf $2$-cocycles for certain affine algebraic reductive groups

Abstract

Let G be a reductive affine algebraic group over ℂ with toral identity component T and finite component group K. We describe finite indecomposable semisimple module categories over Rep(G) using equivariant data that incorporate the extension class. We prove that every fiber functor is classical and determine the image of its Hopf $2$-cocycle support in K and its intersection with T. This yields a minimality criterion involving a finite stabilizer cocycle and an alternating bicharacter on a sublattice of T̂ of index at most |K|. When dim T=1, every Hopf $2$-cocycle for G has finite support of order at most |K|². We compute all gauge classes for the split and nonsplit extensions with component group ℤ/2ℤ and inversion action, and compare the support formulas with the Künneth coordinates for commutative direct products. Finally, for an arbitrary reductive group G, a minimal Hopf $2$-cocycle in a regular deformation of the trivial Hopf $2$-cocycle forces G to be a torus. This uses locality and rigidity of algebraic matrix coefficients on the semisimple quotient of G.

Zeta functions of quasi-split reductive algebraic groups over local fields
Marcus du Sautoy
arXiv2 Nov 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
AV-modules on affine algebraic groups
Yuly Billig, Colin Ingalls
arXiv1 Nov 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Algebraic and Symmetrized Affine star-Ricci Solitons for the Bott Connection on Three-Dimensional Lorentzian Lie Groups
Yuxin Ding
arXiv14 Oct 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
Twisting of affine algebraic groups, III
Shlomo Gelaki
arXiv21 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Intrinsic Non-Reductive Geometric Invariant Theory and Moduli Spaces of Principal Bundles with Non-Reductive Structure Groups
George Cooper, Ludvig Modin
arXiv28 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{gelaki2026hopf,
  title = {{Hopf $2$-cocycles for certain affine algebraic reductive groups}},
  author = {Shlomo Gelaki},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.18599},
  eprint = {2607.18599},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.18599},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Hopf $2$-cocycles for certain affine algebraic reductive groups
AU  - Shlomo Gelaki
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2607.18599
UR  - https://arxiv.org/abs/2607.18599
AB  - Let G be a reductive affine algebraic group over ℂ with toral identity component T and finite component group K. We describe finite indecomposable semisimple module categories over Rep(G) using equivariant data that incorporate the extension class. We prove that every fiber functor is classical and determine the image of its Hopf $2$-cocycle support in K and its intersection with T. This yields a minimality criterion involving a finite stabilizer cocycle and an alternating bicharacter on a sublattice of T̂ of index at most |K|. When dim T=1, every Hopf $2$-cocycle for G has finite support of order at most |K|^2. We compute all gauge classes for the split and nonsplit extensions with component group ℤ/2ℤ and inversion action, and compare the support formulas with the Künneth coordinates for commutative direct products. Finally, for an arbitrary reductive group G, a minimal Hopf $2$-cocycle in a regular deformation of the trivial Hopf $2$-cocycle forces G to be a torus. This uses locality and rigidity of algebraic matrix coefficients on the semisimple quotient of G.
ER  -

Abstract

Related

Cite