Classification of arithmetic root systems

By István Heckenberger. 2006

250 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9d387fbb678449bbc04696f92861f588ee0c5a14

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Citations in the corpus

Nichols algebras of group type with many quadratic relations (2010, methodology): “The proof of this remarkable result makes heavily use of the classification of Nichols algebras of diagonal type [16, 17].”
On finite-dimensional Hopf algebras (2014, background): “[44] The classification of all braided vector spaces of diagonal type with finite-dimensional Nichols algebra is known.”
A generalization of Coxeter groups, root systems, and Matsumoto’s theorem (2006, background): “However Kac–Moody algebras and a very large class of their super and quantized analogues are still covered, together with so far less understood examples appearing in the classification of Nichols algebras of diagonal type [11].”
Root systems and Weyl groupoids for Nichols algebras (2008, background): “In this setting, finite-dimensional Nichols algebras have been classified in [13].”
On finite dimensional Nichols algebras of diagonal type (2017, methodology): “From the list of [46], there are still 12 examples whose GRS could not be identified in Lie theory; we call them UFO’s.”
Hopf Algebras and Root Systems (2020, methodology): “The classiﬁcation in [ Hec09 ] of all ﬁnite-dimensional braided vector spaces of diagonal type and with ﬁnite Cartan graph is beyond the scope of this book.”