Fomin–Kirillov algebras: equality with Nichols algebras and finite-dimensionality for m ≥ 6

Ascertain, for m ≥ 6, whether the Fomin–Kirillov algebra FK_m coincides with the Nichols algebra B(V) of the braided vector space associated to the conjugacy class of transpositions in S_m with the standard cocycle q, and determine whether FK_m is finite-dimensional.

Background

The Fomin–Kirillov algebra is defined as a quadratic quotient of the tensor algebra T(V) for the braided vector space V built from the transposition rack of S_m with cocycle q. For m=3,4,5, FK_m equals the Nichols algebra and is finite-dimensional with known dimensions; however, the situation for m ≥ 6 is unresolved and has attracted sustained interest.