Related papers: Finite Weyl groupoids

For the root system of type $B_l$ and $C_l$, we generalize the result of \cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the…

We introduce a new class of graded rings extending the class of generalized Weyl algebras. These rings are orders in crossed products of the most general type, and we introduce their basic structure theory. We provide an extensive list of…

The Weyl group of the Cuntz algebra O_n, with n finite, is investigated. This is (isomorphic to) the group of polynomial automorphisms of O_n, namely those induced by unitaries that can be written as finite sums of words in the canonical…

The Weyl groups of the fine gradings with infinite universal grading group on $\mathfrak{e}_6$ are given.

It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show…

In this talk we describe very briefly how Gritsenko and Nikulin classified in \cite{3} the generalised Cartan matrices of rank 3, of elliptic type (so they have the generalised lattice Weyl vector), which are twisted to symmetric…

We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and…

This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter group in real vector spaces. We settle the question for finite Coxeter groups, irreducible affine Weyl…

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…

We use the author's combinatorial theory of full heaps (defined in math.QA/0605768) to categorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of…

We derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated to the affine Weyl group E$_8^{(1)}$. Five such systems are obtained, three of which turn out to be…

For any finite-dimensional complex semisimple Lie algebra two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations and the Weyl group acts on the sets of all their Diophantine…

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.

The orbits of Weyl groups W(A(n)) of simple A(n) type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of A(n). Matrices transforming points of the orbits of W(An) into points of subalgebra…

For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1) (m>=2), in terms of numerical and group-theoretical invariants. Our main tool is automorphism…

We study a class of algebraic surfaces of degree 3n in the complex projective space with only ordinary double points. They are obtained by using bivariate polynomials with complex coefficients related to the generalized cosine associated to…

Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. For any given rank $n$, we prove that there are only finitely many isomorphism classes of representations…

There are several researches on Lie algebras and Lie superalgebras graded by finite root systems. In this paper, we study Leibniz algebras graded by finite root systems and obtain some results in simply-laced cases.

There are many different ways that the exponents of Weyl groups of irreducible root systems have been defined and put into practice. One of the most classical and algebraic definitions of the exponents is related to the eigenvalues of…

In the present paper we introduce and study a canonical ${\cal E}$-lattice structure on the set of element orders of some finite groups. We show that a finite abelian group is uniquely determined by this canonical ${\cal E}$-lattice.