Related papers: Classification of marked elliptic root systems wit…
K. Saito (Publ. RIMS 21 (1), 1985, 75-179) has introduced a class of root systems called elliptic root systems which lies in the real vector space $F$ with a metric $I$ whose signature is of type $(l,2,0)$. He also classified the pair…
We consider a natural generalization of both locally finite irreducible root systems and extended affine root systems defined by Saito. We classify the systems.
Extended affine root systems appear as the root systems of extended affine Lie algebras. A subclass of extended affine root systems, whose elements are called ``minimal" turns out to be of special interest mostly because of the geometric…
This is an exposition in order to give an explicit way to understand (1) a non-topological proof for an existence of a base of an affine root system, (2) a Serre-type definition of an elliptic Lie algebra with rank =>2, and (3) the…
Any maximal root subsystem of a finite crystallographic reduced root system is either a closed root subsystem or its dual is a closed root subsystem in the dual root system. In this article, we classify the maximal root subsystems of an…
We consider the problem of finding $1$-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in $\mathbb{Z}$. We classify all such families when the degree of the coefficients (in the…
We obtain several determinant evaluations, related to affine root systems, which provide elliptic extensions of Weyl denominator formulas. Some of these are new, also in the polynomial special case, while others yield new proofs of the…
In order to construct the inverse mapping of the period mapping for the primitive form for the semi-universal deformation of a simple elliptic singularity, K.Saito introduced in [5] the ``flat structure'' for the extended affine root…
A root systems in Carroll spaces with degenerate metric are defined. It is shown that their Cartan matrices and reflection groups are affine. With the help of the geometric consideration the root system structure of affine algebras is…
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…
By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott, we know…
A root system in which all roots have same norm is known as a simply laced root system. We present a simple method of classifying all simply laced root systems.
We associate a generalized root system in the sense of Kyoji Saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra. We generalize results by Saito--Takebayshi…
The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root…
Following the definition of a root basis of an affine root system, we define a base of the root system of an affine Lie superalgebra to be a linearly independent subset $B$ of its root system such that each root can be written as a linear…
We study a combinatorial object, which we call a GRRS (generalized reflection root system); the classical root systems and GRSs introduced by V. Serganova are examples of finite GRRSs. A GRRS is finite if it contains a finite number of…
For each $t\in\mathbb{Q}\setminus\{-1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*} E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ as a function of $t$ and assuming some standard…
A root lattice is a finite rank $\mathbb{Z}$-lattice generated by elements $x$ satisfying $x\cdot x=2$. It is well-known that the root lattices have an $ADE$ classification and they play a prominent role in the study of even unimodular…
Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…
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…