I. Heckenberger

The Weyl groupoid of a Nichols algebra of diagonal type

The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding.

The Locally Finite Part of the Dual Coalgebra of Quantized Irreducible Flag Manifolds

For quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra

Examples of finite-dimensional rank 2 Nichols algebras of diagonal type

\overline{U}

Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

of

Finite Weyl groupoids of rank three

U_q ( \mathfrak{g} )

Spin geometry on quantum groups via covariant differential calculi

. On the way the coradical filtration of

Differential Calculus on Quantum Homogeneous Spaces

\overline{U}

Symmetrizer and Antisymmetrizer of the Birman–Wenzl–Murakami Algebras

is determined. A graded version of the duality between

Differential forms via the Bernstein–Gelfand–Gelfand resolution for quantized irreducible flag manifolds

\overline{U}

Nichols algebras with many cubic relations

and the quantized coordinate ring is established. This leads to a natural construction of several examples of quantized vector spaces. As an application, covariant first order differential calculi on quantized irreducible flag manifolds are classified.