Related papers: On the classification of hyperbolic root systems o…

Here we prove classification results announced in Part I (alg-geom/9711032). We classify maximal hyperbolic root systems of the rank 3 having restricted arithmetic type and a generalized lattice Weyl vector $\rho$ with $\rho^2\ge 0$ (i.e.…

See Parts I and II in alg-geom/9711032 and alg-geom/9712033. Here we classify maximal hyperbolic root systems of the rank three having restricted arithmetic type and a generalized lattice Weyl vector $\rho$ with $\rho^2<0$ (i. e. of the…

Sometimes a hyperbolic Kac-Moody algebra admits an automorphic correction, meaning a generalized Kac-Moody algebra with the same real simple roots and whose denominator function has good automorphic properties; these for example allow one…

A Kac-Moody algebra is called hyperbolic if it corresponds to a generalized Cartan matrix of hyperbolic type. We study root subsystems of root systems of hyperbolic algebras. In this paper, we classify maximal rank regular hyperbolic…

The paper purposes to contribute to the classification of pointed Hopf algebras by the method of Andruskiewitsch and Schneider. The structure of arithmetic root systems is enlightened such that their relation to ordinary root systems…

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding…

Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices…

We study the Weyl groups of hyperbolic Kac-Moody algebras of `over-extended' type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed division algebras K=R,C,H,O, respectively. A crucial role is played by integral…

We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Such a classification of an irreducible root system is closely related to the classification of the regular…

The classification of Nichols algebras is an essential step in the classification theory of pointed Hopf algebras by lifting method of N. Andruskiewitsch and H.-J. Schneider. Arithmetic root systems are invariants of Nichols algebras of…

We present a classification of the hyperbolic Kac-Moody (HKM) superalgebras. The HKM superalgebras of rank larger or equal than 3 are finite in number (213) and limited in rank (6). The Dynkin-Kac diagrams and the corresponding simple root…

Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains…

Let $\Delta$ be a rank 2 hyperbolic root system. Then $\Delta$ has generalized Cartan matrix $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ indexed by $a,b\in\mathbb{Z}$ with $ab\geq 5$. If $a\neq b$, then…

For an even, integral hyperbolic lattice $L$, the symmetry group of $L$ is the quotient of the group of isometries of $L$ by the Weyl subgroup of $(-2)$-reflections. Following Nikulin, the exceptional lattice of $L$ is defined as the…

We show that, given a rank 3 affine root system $\Phi$ with Weyl group $W$, there is a unique oriented matroid structure on $\Phi$ which is $W$-equivariant and restricts to the usual matroid structure on rank 2 subsystems. Such oriented…

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…

This is a continuation of our "Lecture on Kac--Moody Lie algebras of the arithmetic type" \cite{25}. We consider hyperbolic (i.e. signature $(n,1)$) integral symmetric bilinear form $S:M\times M \to {\Bbb Z}$ (i.e. hyperbolic lattice),…

We completely determine the structure constants between real root vectors in a rank 2 Kac--Moody algebra $\mathfrak{g}$. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the…

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing…

We discuss a general theory of Lorentzian Kac--Moody algebras which should be a hyperbolic analogy of the classical theories of finite-dimensional semi-simple and affine Kac-Moody algebras. First examples of Lorentzian Kac-Moody algebras…