Jun Morita’s research while affiliated with University of Tsukuba and other places

Minimal locally affine Lie algebras, minimal LALAs for short, are classified. In particular, we will have a condition for a minimal LALA to be isomorphic to a standard minimal LALA containing the so-called degree derivation. In fact, we will see that there are many kinds of non-standard minimal LALAs.

In infinite dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.

We provide explicit generators and relations for the affine Kac-Moody groups, as well as a realization of them as (twisted) loop groups by means of Galois descent considerations.

The paper is devoted to model-theoretic properties of Kac-Moody groups with the focus on elementary equivalence of Kac-Moody groups. We show that elementary equivalence of (untwisted) affine Kac-Moody groups implies coincidence of their generalized Cartan matrices and the elementary equivalence of their ground fields. We also show that elementary equivalence of arbitrary Kac-Moody groups over finite fields implies coincidence of these fields and an isomorphism of their twin root data. The similar result is established for Kac-Moody groups defined over infinite subfields of the algebraic closures of finite fields.

We obtain new examples of simple Kac-Moody groups with trivial Schur multipliers.

The vertices of the four dimensional 120-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group $H_{4}$. There are two special coordinate representations of this root system in which they and their corresponding Coxeter groups involve only rational numbers and the golden ratio $\tau$. The two are related by the conjugation $\tau \mapsto\tau' = -1/\tau$. This paper investigates what happens when the two root systems are combined and the group generated by both versions of $H_{4}$ is allowed to operate on them. The result is a new, but infinite, `root system' $\Sigma$ which itself turns out to have a natural structure of the unitary group $SU(2,\mathcal R)$ over the ring $\mathcal R = \mathbb Z[\frac{1}{2},\tau]$ (called here golden numbers). Acting upon it is the naturally associated infinite reflection group $H^{\infty}$, which we prove is of index 2 in the orthogonal group $O(4,\mathcal R)$. The paper makes extensive use of the quaternions over $\mathcal R$ and leads to highly structured discretized filtration of SU(2). We use this to offer a simple and effective way to approximate any element of SU(2) to any degree of accuracy required using the repeated actions of just five fixed reflections, a process that may find application in computational methods in quantum mechanics.

The vertices of the four dimensional 120-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group $H_{4}$. There are two special coordinate representations of this root system in which they and their corresponding Coxeter groups involve only rational numbers and the golden ratio $\tau$. The two are related by the conjugation $\tau \mapsto\tau' = -1/\tau$. This paper investigates what happens when the two root systems are combined and the group generated by both versions of $H_{4}$ is allowed to operate on them. The result is a new, but infinite, `root system' $\Sigma$ which itself turns out to have a natural structure of the unitary group $SU(2,\mathcal R)$ over the ring $\mathcal R = \mathbb Z[\frac{1}{2},\tau]$ (called here golden numbers). Acting upon it is the naturally associated infinite reflection group $H^{\infty}$, which we prove is of index 2 in the orthogonal group $O(4,\mathcal R)$. The paper makes extensive use of the quaternions over $\mathcal R$ and leads to highly structured discretized filtration of SU(2). We use this to offer a simple and effective way to approximate any element of SU(2) to any degree of accuracy required using the repeated actions of just five fixed reflections, a process that may find application in computational methods in quantum mechanics.

The paper is a short survey of recent developments in the area of word maps evaluated on groups and algebras. It is aimed to pose questions relevant to Kac--Moody theory.

We give a new relationship between several simple automata and formal power series as word invariants. Such an invariant is derived from certain combinatorics and algebraic structures. We review them and, especially, we deal with a connection to Lie theory through tilings.