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… 
On the dimension of the Fomin-Kirillov algebra and related algebras
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
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
<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
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
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
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
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
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
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
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
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
  • S. Majid
  • Mathematics
    Hopf 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
...
1
2
3
4
...