Classification of arithmetic root systems

Abstract

Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the construction of quantized enveloping algebras, in the noncommutative differential geometry of quantum groups, and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In the present paper arithmetic root systems are classified in full generality. As a byproduct many new finite dimensional pointed Hopf algebras are obtained.

Factorization of Graded Traces on Nichols Algebras

“…The root system ∆ + of a Nichols algebra B(M) in the sense of [1,2] directly presents a factorization of B(M) as a graded Yetter-Drinfel'd module: this completely explains the factorization of the graded trace of an endomorphism Q that respects the root system grading.…”

“…In the abelian case, Heckenberger (e.g., [2]) introduced q-decorated diagrams, with each node corresponding to a simple Yetter-Drinfel'd module decorated by q ii , and each edge decorated by τ 2 = q ij q ji and edges are drawn if the decoration is = 1; it turns out that this data is all that is needed to determine the respective Nichols algebra. Theorem 1.…”

“…However, several additional exotic examples of finite-dimensional Nichols algebras exist that possess unfamiliar Dynkin diagrams, such as a multiply-laced triangle, and where Weyl reflections may connect different diagrams (yielding a Weyl groupoid). Heckenberger completely classified all Nichols algebras over abelian G in [2].…”

“…The root system theory and PBW-basis of Nichols algebras developed by [1,2] precisely explains a factorization of B(M) as graded vector space (even Yetter-Drinfel'd module) into Nichols subalgebras of rank 1 associated with each root. This explains the complete factorization for Nichols algebras over abelian groups.…”

Lifting via cocycle deformation

“…of [H,Table 1,row 9]. Consider a matrix (q ij ) 1≤i,j≤2 corresponding to (5.9), that is q 11 = −ζ, q 22 = ζ 3 and q 12 q 21 = ζ 7 .…”