• Deduce that hence in any congurations associated to a nite-
dimensional orbifold, only black A1 or A2 (at most one in- dependent node) may be coupled with one of the Dynkin dia- grams of the list above in white (or by Doi twist vice-versa in white/black)!
• Clarify, whether in most cases if one black and n ≥ 4 white
elements are not mutually commuting forming an n-branch in
the graph, then the generated abelian subrack (over G!) con-
tains an n-branch and hence had innite dimension!
• Clarify, whether in most cases, if one black and three white
elements are not mutually commuting, forming aD4 with white center, then the reection on this center generates a 3-cycle in
Gand hence cannot occur over 2-groups by Heckenberger's list! • Deduce a general statement (probably including several excep-
tions from above), linking the graphs to simply-laced ane Dynkin diagrams, with the black nodes precisely the addi- tonal nodes from anization.
• Try to construct examples or contradict such indecomposable
nite-dimensional Nichols algebras associated to ane Dynkin diagrams and with nontrivial actionG0 prescribed as above pre-
cisely on the additional anization nodes.
3. Nichols Algebras Over Nilpotent Groups Of Class ≥3
Finally we need to target nilpotent groups of class ≥ 3. While the
case > 3 seems to be discarded rather easily by extending a result
of Schneider and Heckenberger regarding the rank 2 case in general groups [HS10], in class 3 there is a class of more resilient groups cor-
responding to the case G4 in cit. loc.. If there were an orbifold over this group, it had to be from a certain non-minimally indecomposable Nichols algebra over D4 found in the preceding section. The author has no opinion, what the Dynkin diagram might be, or whether this yields new cases or can be negated by a more skillful approach!
The author's study of the case of general nilpotent group G started
with the following observations about Nichols algebras B(Oχ s ⊕ O
ρ t)of
rank 2 of nite dimension, that are strong consequences of [HS08] in the nilpotent case and have implications on the structure of G, if an
160 OUTLOOK: 3 CONJECTURAL STEPS TO ALL NILPOTENT GROUPS indecomposable nite-dimensional B(L
iO χi
gi)exists:
• The general stst=tsts implies forg := [s, t] that stg =g−1
• However, a nilpotent group is direct product of Sylow-subgroups G(p). Considers,¯ ¯tin the largest quotient of odd orderG/G(2);
since there squaring is invertible, we get from the condition on the other hand s¯t¯= ¯t¯s.
• Suppose now there exists an indecomposable L
iO χi
gi where
any two discommuting s = gi and t = gj commute as de-
manded above, then h{[ ¯si]}i generate also G/G(2), but there
they all commute. Hence G/G(2) is abelian.
The behaviour of the rank 2 Nichols algebrasB(Oχ s⊕O
ρ
t)in question of
course depends greatly on the order ofg = [s, t]. Suppose the nilpotency
class ofGto be just 2 (=commutators are central), then our rst obser-
vation shows that always g2 = 1 meaning G0 is 2-elementary-abelian. In this work, we construct large examples of Nichols over such groups and clarify conversely existence of minimally indecomposable nite- dimensional Nichols algebras (even connected) at least for G0 ∼=Z2.
The author had also put considerable eort into clarifying the other cases g2n = 1 with (st)g = g−1 6= g. Before deciding the existence of
an orbifold (in one class lower), one has to exclude possible examples with nontrivial g-actions, that are no Doi twist of trivial action; al-
though the author was able to derive certain conditions, the particular lack of cohomology for some groups has prevented to derive the aimed conclusion.
Example 8.3. Particularly resisent was the following case of the qua- sidihedral group D8˜ , as the author announced in a mini-talk given
at the Oberwolfach Conference 2010.
˜
D8 :=ha, b|b2 =a8 = 1, ba=a3bi
˜
M :=Oa⊕ Oa2b Σ3g2 = [a, a2b]2 =a4
In contrast to Q16, only one Oa powers to g2, which proves g2 to act
trivial only there; this are few relations (compare section 5.3. On the other hand, in contrast to D8, there is not enough cohomology to generate the remaining case (g2.x
3. NICHOLS ALGEBRAS OVER NILPOTENT GROUPS OF CLASS ≥3 161
Towards the end of this dissertation, Schneider suggested in this context the recent paper [HS10]. Heckenberger and himself had proven strong implications for the structure of such groups in the general caseG0 =Zn
for B(Oχ s ⊕ O
ρ
t)nite-dimensional. More specic, they proved that the
support hOs,Oti for Os,Ot discommuting (exceptional pair) has to
be a quotient of one of the following three innite groups:
G2 :=hs, t, g|sg =g tg =gi
G3 :=hs, t, g|sg =g−1 tg =g g3 = 1i
G4 :=hs, t, g|sg =g−1 tg =g g4 = 1i Note that no nonabelian quotient of G3 may be nilpotent!
Problem 8.4. Decide Class 3: (such as the above groups of order 16). If such a Nichols algebra were an orbifold by Σ = hg2i of an ex- ample with ¯g2 = 1 in Γ = G/Σ, the smaller Nichols algebra (by some
calculations) of a non-minimally indecomposable Nichols algebra with both trivial and non-trivial Σ∗-action over D4, hence cannot be gen-
erally deorbifoldized further to the abelian case. The necessary edges between discommuting nodes severely restricts the possible diagrams. A good guess might be A3 ∪ A3 → C2 between conjugacy classes of orders 2, 4 (for q = −1 of dimension 212), which could even lead to a familyDn∪Dn→Cn−1 over central products with extraspecial groups. A second possibility would be A(1)3 over conjugacy classes of both order
4. We would require the discussions of the preceding section to decide
it's existence.
Problem 8.5. Discard Class >3: By the result quoted above, there
may not be any rank 2 Nichols algebra with support already class >3;
namely G2,G4 have class2,3. Hence the only possibility were a situation
Oa⊕ Ob ⊕ Oc with ha, bi,ha, ci,hb, ci of class ≤ 3, such that e.g. g ∈ [a, b]2 (which is central in ha, bi) is nontrivially conjugated upon by c. We now sketch an argument, that requires the knowledge of the Dynkin diagram in the class 3 case above: By possibly using a Weyl reection
on Oa we may suppose b2 = [a, b], but then hb, ci could not have been