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Generation conjecture for finite-dimensional pointed Hopf algebras

Let LL be a cosemisimple Hopf algebra and let AA be a pointed Hopf algebra with diagram R\mathcal R and infinitesimal braiding V=R1\mathcal V=\mathcal R^1 in LLYD{}_{L}^{L}{\mathcal{YD}}. In the case L=kGL=\Bbbk G for a finite group, finite-dimensional Nichols algebras are expected to be fundamentally finite, meaning that every finite-dimensional pre- or post-Nichols algebra of VV is isomorphic to B(V){\mathcal B}(V). Generation conjecture. Every finite-dimensional pointed Hopf algebra is generated by group-like and skew-primitive elements. This is equivalent, in the stated group-algebra setting, to the fundamental finiteness of every VV with dim⁡B(V)<∞\dim {\mathcal B}(V)<\infty; it would imply that the relevant finite-dimensional graded connected Hopf algebras are Nichols algebras.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The generation conjecture for finite-dimensional pointed Hopf algebras

    Let k\Bbbk be an algebraically closed field of characteristic zero, and let HH be a finite-dimensional pointed Hopf algebra over k\Bbbk. An element g∈Hg\in H is grouplike if Δ(g)=g⊗g\Delta(g)=g\otimes g, and an element x∈Hx\in H is skew primitive if there are grouplike elements g,g′∈Hg,g'\in H such that Δ(x)=g⊗x+x⊗g′\Delta(x)=g\otimes x+x\otimes g'. The generation conjecture. The Hopf algebra HH is generated by its grouplike and skew primitive elements. The converse of the implication that generation by grouplike and skew primitive elements implies pointedness is expected in the finite-dimensional case; the conjecture concerns the reverse implication over an algebraically closed field of characteristic zero.

    source: Pavel Etingof and Chelsea Walton, “Pointed Hopf actions on fields, I”, arXiv:1403.4673 (2015).

Sources & referencesView supporting material

Primary source

Nicolás Andruskiewitsch and Iván Angiono, “On Finite dimensional Nichols algebras of diagonal type”, arXiv:1707.08387 (2017).

hopf-algebrasnichols-algebraspointed-hopf-algebras

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