The Gelfand-Kirillov dimension of rank 2 Nichols algebras of diagonal type
Most interests in the theory of Nichols algebras emerged from the the theory of pointed Hopf algebras. For their classification it is an essential step to classify finite (Gelfand-Kirillov-) dimensional Nichols-algebras under some finiteness conditions. Nichols algebras have been discussed iby vari...
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|Most interests in the theory of Nichols algebras emerged from the the theory of pointed Hopf algebras. For their classification it is an essential step to classify finite (Gelfand-Kirillov-) dimensional Nichols-algebras under some finiteness conditions.
Nichols algebras have been discussed iby various authors. Especially, those of diagonal type which yielded interesting applications, for example as the positive part of quantized enveloping algebra of a simple finite-dimensional Lie algebras g. Finite-dimensional Nichols algebras of diagonal type have been classified in a series of papers. One important step for this has been the introduction of the root-system and the associated Weyl groupoid.In this context the following implications were observed:
(1) If a Nichols algebra is of finite dimension, then the corresponding Weyl grouppoid is finite.
(2) If the Weyl grouppoid of a Nichols algebra is finite, the Gelfand-Kirillov dimension of a Nichols algebra is finite.
For (1) the converse is true under some circumstances. The converse of (2) has been conjectured to be true. Recently, the topic of finite Gelfand-Kirillov dimensional Nichols algebras has received increased attention. In particular rank 2 Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension over a field of characteristic zero have been classified and were used to also classify finite Gelfand-Kirillov-dimensional Nichols algebras over abelian groups.
The goal of this work is to extend this result to any characteristic. Note that there are more braidings yielding a finite root system in positive characteristic, especially there are examples with simple roots a yielding X(a,a) = 1 where X denotes the corresponding bicharacter. Roots of this kind imply infinite Gelfand-Kirillov dimension in characteristic zero. Hence new tools have to be developed generalizing the results for characteristic zero in addition.