Jun Morita

Octagonal quasicrystals and a formula for shelling

Octagonal quasicrystals will be realized in a cyclotomic field, and a formula for shelling will be given using number theory.

Gauss decompositions of kac-moody groups

We present an axiomatic approach to both a Gauss decomposition of a Kac-Moody group and a Gauss decomposition of the associated Steinberg group. We study also a prescribed version in case of rank 2.

Some Factorizations in Universal Enveloping Algebras of Three Dimensional Lie Algebras and Generalizations

We introduce the notion of Lie algebras with plus-minus pairs as well as regular plus-minus pairs. These notions deal with certain factorizations in universal enveloping algebras. We show that many important Lie algebras have such pairs and we classify, and give a full treatment of, the three dimensional Lie algebras with plus-minus pairs.

Local properties, bialgebras and representations for one-dimensional tilings

For any one-dimensional tiling, we discuss finite-dimensional standard modules for the associated tiling bialgebra. We will note that such modules are completely reducible, and we will parametrize finite-dimensional irreducible ones, using the set of all patches. Furthermore, we will discuss the associated completed groups and Iwasawa-type decompositions. We also characterize for one-dimensional tilings to be locally nondistinguishable by the associated tiling bialgebra structures.

K2SL2 of Euclidian domains, generalized Dennis-Stein symbols and a certain three-unit formula

We introduce a generalized Dennis-Stein symbol, and give a sufficient condition for a Euclidean domain to be weakly quasi-universal for SL2. Using this, we obtain how to get an explicit presentation of SL(2, Z[1p1,…,1pn]). Also we will establish a certain formula using three units, and determine the group structure of K2(2,Z[13]) explicitly.

Discretization of SU(2) and the orthogonal group using icosahedral symmetries and the golden numbers

ABSTRACT The vertices of the four-dimensional 600-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group H4. 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 τ. The two are related by the conjugation τ↦τ′ = −1∕τ. This paper investigates what happens when the two root systems are combined and the group generated by both versions of H4 is allowed to operate on them. The result is a new, but infinite, ‘root system’ Σ which itself turns out to have a natural structure of the unitary group SU(2,ℛ) over the ring (called here golden numbers). Acting upon it is the naturally associated infinite reflection group H∞, which we prove is of index 2 in the orthogonal group O(4,ℛ). The paper makes extensive use of the quaternions over ℛ and leads to a 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.

Words, Automata and Lie Theory for Tilings

We will 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 via through tilings.

A certain algebraic construction of quasicrystals and their isomorphism classes

Certain quasicrystals will be realized in cyclotomic fields, and their isomorphism structures will be given in the case of seven-fold or 30-fold symmetry.

Sheel structure of dodecagonal quasicrystals associated with root system F4 and cyclotomic field Q(ζ12)

Using a Coxeter transformation of type F4, we will construct quasicrystals of dimension 2 with twelvefold symmetry. Then, such quasicrystals will be realized as subsets of a certain cyclotomic field. And we will study a typical shell structure algebraically. In particular, we can give its mathematical meaning in terms of elementary number theory.

The group SL(2, Z s) is either perfect or prosolvable

We will show the prosolvability of SL2 over Z and some other rings. Also we will discuss about braid groups.