On finite GK-dimensional Nichols algebras of diagonal type
Nicol\'as Andruskiewitsch, Iv\'an Angiono, Istv\'an Heckenberger
TL;DR
This paper proves that rank 2 Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension have finite root systems, and shows that affine Cartan type Nichols algebras have infinite Gelfand-Kirillov dimension.
Contribution
It confirms a conjecture for rank 2 Nichols algebras and distinguishes affine Cartan types by their infinite Gelfand-Kirillov dimension.
Findings
Rank 2 Nichols algebras with finite GK dimension have finite root systems.
Affine Cartan type Nichols algebras have infinite GK dimension.
Abstract
It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show that a Nichols algebra of affine Cartan type has infinite Gelfand-Kirillov dimension.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Nonlinear Waves and Solitons