# On Fomin–Kirillov algebras for complex reflection groups

@article{Laugwitz2016OnFA, title={On Fomin–Kirillov algebras for complex reflection groups}, author={Robert Laugwitz}, journal={Communications in Algebra}, year={2016}, volume={45}, pages={3653 - 3666} }

ABSTRACT In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857…

## 3 Citations

On the dimension of the Fomin-Kirillov algebra and related algebras

- Mathematics
- 2020

Let Em be the Fomin-Kirillov algebra, and let BSm be the Nichols-Woronowicz algebra model for Schubert calculus on the symmetric group Sm which is a quotient of Em, i.e. the Nichols algebra…

On the dimension of the Fomin-Kirillov algebra and related algebras

- Mathematics
- 2020

Let $\mathcal{E}_m$ be the Fomin-Kirillov algebra, and let $\mathcal{B}_{\mathbb{S}_m}$ be the Nichols-Woronowicz algebra model for Schubert calculus on the symmetric group $\mathbb{S}_m$ which is a…

On the quadratic dual of the Fomin–Kirillov algebras

- MathematicsTransactions of the American Mathematical Society
- 2019

<p>We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"…

## References

SHOWING 1-10 OF 39 REFERENCES

Nichols–Woronowicz algebra model for Schubert calculus on Coxeter groups

- Mathematics
- 2004

Abstract We realise the cohomology ring of a flag manifold, more generally the coinvariant algebra of an arbitrary finite Coxeter group W, as a commutative subalgebra of a certain Nichols–Woronowicz…

The Weyl–Brandt 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 semisimple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols…

A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups

- Mathematics
- 2017

Over fields of arbitrary characteristic we classify all braid-indecomposable tuples of at least two absolutely simple Yetter-Drinfeld modules over non-abelian groups such that the group is generated…

The Weyl groupoid of a Nichols algebra of diagonal type

- Mathematics
- 2006

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…

The classification of Nichols algebras over groups with finite root system of rank two

- Mathematics
- 2013

We classify all groups G and all pairs (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the support of the direct sum of V and W generates G, the square of the braiding between V…

Connected braided Hopf algebras

- Mathematics
- 2007

Abstract We prove a braided version of Kostant–Cartier–Milnor–Moore theorem: The category of connected τ-cocommutative ( τ 2 = id ) braided Hopf algebras over a field of zero characteristic is…

Braided doubles and rational Cherednik algebras

- Mathematics
- 2007

We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal…

Nichols algebras with many cubic relations

- Mathematics
- 2015

Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All…

On the classification of finite-dimensional pointed Hopf algebras

- Mathematics
- 2005

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elementsG.A/ is abelian such…

Noncommutative Differentials and Yang-Mills on Permutation Groups Sn

- MathematicsHopf Algebras in Noncommutative Geometry and Physics
- 2019

We study noncommutative differential structures on the group of permutations $S_N$, defined by conjugacy classes. The 2-cycles class defines an exterior algebra $\Lambda_N$ which is a super analogue…